THE MECHANICAL 

ENGINEER'S 
REFERENCE BOOK 



A HAND-BOOK OF 

TABLES, FORMULAS, AND METHODS 

FOR ENGINEERS, STUDENTS, AND DRAFTSMEN 



BY 

HENRY HARRISON SUPLEE, B.Sc, M.E. 

MEMBER AMERICAN SOCIETY OF MECHANICAL ENGINEERS 

MEMBRE DE LA SOCIETE DES INGENIEURS CIVILS DE FRANCE 

MITGLIED DES -/ZllEIN^S I DSUTSC^ER INCFN T EURE 

MEMBER OF TLL FRANKLIN INSTIT-TF 




PHILADELPHIA & LONDON 
J. B. L1PPINCOTT COMPANY 

1904 



^ 



y -. 1 A 






Copyright, 1903, 

BY 

J B. Lippincott Company 



Electrotyped and Printed by 
J. B Lippincott Company, Philadelphia, U. S. A. 



PREFACE 



IN preparing a hand-book for engineering reference it is 
necessary to select from among a great mass of detailed 
information the matter which shall be most generally 
available. Naturally, the differentiation which has taken 
place in the science of engineering makes it desirable that 
some one department of work shall predominate, and, as 
indicated in the title, this book is devoted principally to the 
presentation of tables, formulas, and reference data for me- 
chanical engineers. It is, therefore, purposely full in the 
portions relating to machine design and to such information 
as will render it useful in the drawing room and in the de- 
signing department, the intention being to render it available 
broadly in furnishing a record of general principles, as well as 
of detailed methods. 

The many and varying rules and formulas existing in this 
connection have been carefully examined, and only those 
which in the judgment of the author are most generally 
applicable have been given, since the presentation of a mass 
of data, much of it contradictory, throws the burden of 
selection upon the user. In this portion of the work the 
author has sought to relieve the user of the necessity of 
selecting from among a mass of contradictory information the 
matter of the most general value, leaving special work to be 
conducted — as it should be — under the control of special 
investigation. 

In view of the fact that the metric system has been under 
active discussion of late, a number of the tables have been 
presented in both British and metric units, so that those 
engineers who are desirous of using the latter system may do 
so. Among these tables may be mentioned the metric steam 
tables, which render it convenient for steam computations to 
be made in the metric system. 



Pbefack. 



This work is intended to be a successor to the well-known 
pocket-book written many years ago by the late John W. 
Nystrom, and published by Messrs. J. B. Lippincott Com- 
pany. The plates and stock of that valuable work having 
been destroyed by fire in 1899, certain of the information 
therein contained has been utilized, with such modifications 
as are necessary to meet engineering problems and needs of 
the present. 

Among the valuable works to which acknowledgments are 
due in the preparation of this hand-book may be mentioned 
Reuleaux's "Constructor," Unwin's "Machine Design," 
Weisbach's "Ingenieur," "Des Ingenieurs Taschenbuch 
Hiitte," the Smithsonian Physical Tables, and the hand- 
books of the Pencoyd Iron Works and the Passaic Steel 
Company, as well as the various authorities mentioned in 
the text. 

HENRY HARRISON SIJPLEE. 



CONTENTS 



PAGE 

Mathematics 5 

Factor Table 7 

Powers and Koots 24 

Interest 56 

Weights and Measures 58 

Monetary Systems 61 

Metric System 63 

Metric Conversion Tables 64 

Algebra 76 

Binomial Theorem 76 

Arithmetical Progression 77 

Geometrical Progression 77 

Equations 78 

Logarithms 79 

Table of Logarithms 82 

Geometry 105 

Table of Chords 107 

Table of Polygons 108 

The Circle 109 

Tables of Circles 110 

Arcs and Segments of Circles 117 

The Ellipse 122 

The Parabola 124 

The Hyperbola 126 

Areas of Plane Figures 127 

Surfaces of Solids 129 



vi Contents. 

Mathematics. — Continued. PAGE 

Volumes of Solids 130 

Trigonometry 132 

Trigonometric Tables 133 

Natural Trigonometric Functions 134 

Logarithmic Trigonometric Functions. ... 179 

Differential and Integral Calculus 230 

Mechanics 234 

Statics 234 

Funicular Polygons 237 

Centre of Gravity 242 

Statics of Framed Structures 247 

Wind Stresses 254 

Framed Beams 255 

Bridge Trusses . 257 

Motion 259 

Falling Bodies 259 

Accelerated Motion 259 

Retarded Motion 260 

Tables of Falling Bodies 262 

Leverage 267 

Dynamics 268 

Revolving Bodies 272 

Radius of Gyration 273 

Central Forces 274 

Pendulum 276 

Impact 278 

Friction 280 

Journal Friction 281 

Materials of Engineering 282 

Specific Gravity 284 

Weight of Iron 286 



Contents. vii 

Materials of Engineering, — Continued. page 

Weight of Sheet-metal 294 

Weight of Spheres 297 

Weight of Cast-iron Pipe 299 

Weight of Rivets 301 

Weight of Bolts 302 

Screw Threads 304 

Pipe Dimensions . . 308 

Boiler Tubes 311 

Wire Gauges 312 

Weight of Wire 313 

Chain 315 

Glass 316 

Slate 317 

Tin Plate 318 

Lead Pipe 319 

Corrugated Iron 320 

Timber 321 

Board Measure 322 

Spikes and Nails 323 

Screws 324 

Extra Strong Pipe 325 

Riveted Hydraulic Pipe 327 

Standard Flanges 329 

Cast-iron Pipe Fittings 331 

Angle, Globe, and Check Valves 335 

Gate Valves 336 

Pipe Unions 337 

Eye Bars 340 

Wire Rope 342 

Steel Wire 345 

Strength of Materials 346 

Tension ; 348 



viii Contents. 

Strength of Materials.— Continued. page 

Compression 350 

Shearing 350 

Bending 351 

Elements of Kolled Sections 353 

Structural Sections 356 

Kolled Beams 360 

Kolled Channels 362 

Bending Moments of Beams 364 

Struts 368 

Z-Bars 376 

Tee Bars 379 

Angle Bars 380 

Cast-iron Columns 391 

Torsion 393 

Internal Pressure 397 

Springs 399 

Specifications for Structural Steel 403 

Timber 405 

Wooden Beams 408 

Wooden Pillars 410 

Tables of Strength of Materials 411 

Machine Design 416 

Kiveting 416 

Bolts 420 

Keyed Fastenings 423 

Journals 424 

Pivots 427 

Shafting 428 

Bearings 434 

Couplings 436 

Levers 439 

Cranks 441 



Contents. ix 

Machine Design.— Continued. page 

Connecting Kods 442 

Eccentrics 445 

Cross-heads 447 

Gearing 448 

Belts and Pulleys 467 

Kope Transmission 478 

Heat 481 

Thermometers 483 

Coefficients of Expansion 486 

Fusing-points ' 489 

Expansion of Gases 489 

Heat Units 490 

Specific Heat 495 

Latent Heat 496 

Kadiation 497 

Heat Emission 499 

Air 500 

Compression arid Expansion of Air 501 

Air Transmission 504 

Compressed Air 505 

Flow of Air 508 

Movement of Air 509 

Air Friction 510 

Atmospheric Pressure 514 

Barometric Tables -. 515 

Water 519 

Tables of Properties of Water 520 

Water-heads and Pressures 525 

Water-heads and Velocities 528 

Flow of Water through Pipes 529 



x Contents. 

Water. — Continued. PAGE 

Flow of Water in Open Channels 535 

Weir Measurement of Water 539 

Miner's Inch 541 

Water Power 542 

Contents of Pipes 543 

Tank Measurement 545 

Water-wheels 545 

Turbines 551 

Pumps 554 

Duty Trials of Pumping Engines 560 

Hydraulic Ram '. 564 

Hydrometers 564 

Hydrostatics 565 

Hydraulic Transmission of Power 569 

Fuel 572 

Calorific Values of Fuels 573 

Heating Values of Coals 574 

Liquid Fuels 579 

Gas Fuels 581 

Steam 581 

Steam Tables 583 

Flow of Steam 588 

Moisture in Steam 591 

Steam Boilers 592 

Factors of Evaporation 594 

Boiler Trials 595 

Chimneys 605 

Chimney Flues 608 

Draft Pressures 610 

Steam-boiler Details 611 



Contents. xi 

Steam Boilers. — Continued. page 

Boiler Pressures 622 

Boiler Specifications 628 

Safety Valves 632 

Incrustation in Boilers 635 

Steam Engines 638 

Hyperbolic Logarithms 640 

Expansion of Steam 644 

Economical Point of Cut-off , 646 

Multiple-expansion Engines 649 

Indicator Diagrams 650 

Engine Performance 653 

Steam-engine Testing 655 

Steam-engine Performances 675 

Yalve Gears , 678 

Link Motions 680 

Slide Valves 681 

Condensers 684 

Internal=combustion Motors 687 

Gas Engines 688 

Gas-engine Testing 690 

Electric Power 698 

Electric Cables 701 

Wire Tables 702 

National Electric Code 704 

Wiring Formulas 733 

Standardization 736 

Electric Driving c 749 

Power Required for Tools 750 

Electric Cranes 753 

Choice of Motors and Systems 755 



xii Contents. 

Electric Power.— Continued. page 

Speed Variations 756 

Motor Tests 760 

The Cost of Power 771 

Water-power Plant Costs 771 

Water-power Costs 772 

Summary of Boiler Tests 772 

Summary of Engine Tests 773 

Steam-plant Costs 774 

Steam-power Costs 775 

Gas-power Systems 775 

Gas-power Costs 777 

Electric-power Costs 778 

Works Management 780 

Cost Keeping 781 

General Expense 785 

Depreciation 786 

Appendix 789 

Index 803 



The 



Mechanical Engineer's 
Reference Book 



MATHEMATICS. 



The engineer should use mechanical appliances for mathematical com- 
putations whenever possible, including the slide-rule in some of its various 
modifications, but the following tables will also be found useful : 

MULTIPLICATION TABLE. 

By the use of the following table products of numbers from 1 to 10 by 
numbers from 1 to 100 may be obtained directly, and of larger numbers by 
successive operations, as follows : 

f 268001 
67 X 489 = 67 X 400 + 67 X 80 + 67 X 9 = < 5360 }■ = 32763. 

(. 603 J 
If both factors consist of more than three figures, one of the factors 
may be modified and the operation performed as follows : 
854 X 279 = 850 X 279 + 4 X 279. 
Here we subtract 4 from 854 and then get the product of 850 by 279 from 
the table, and add to this the product of 4 by 279, also readily taken from 
the table ; thus : 

(170000) f 800) 

850 X 279 + 4 X 279 = 1 59500 l + l 280 V- 

{ 7650 ) (, 36j 

= 237150 + 1116 = 238266. 



1 


2 


3 


4 


5 


6 


7 


8 


9 





























1 


2 


3 


4 


5 


6 


7 


8 


9 


2 


4 


6 


8 


10 


12 


14 


16 


18 


3 


6 


9 


12 


15 


18 


21 


24 


27 • 


4 


8 


12 


16 


20 


24 


28 


32 


36 


5 


10 


15 


20 


25 


30 


35 


40 


45 


6 


12 


18 


24 


30 


36 


42 


48 


54 


7 


14 


21 


28 


35 


42 


49 


56 


63 


8 


16 


24 


32 


40 


48 


56 


64 


72 


9 


18 


27 


36 


45 


54 


63 


72 


81 



Multiplication Table. 



1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


20 


30 


40 


50 


60 


70 


80 


90 


11 


22 


33 


44 


55 


66 


77 


88 


99 


12 


24 


36 


48 


60 


72 


84 


96 


108 


13 


26 


39 


52 


65 


78 


91 


104 


117 


14 


28 


42 


56 


70 


84 


98 


112 


126 


15 


30 


45 


60 


75 


90 


105 


120 


135 


16 


32 


48 


64 


80 


96 


112 


128 


144 


17 


34 


5L 


68 


85 


102 


119 


136 


153 


18 


36 


54 


72 


90 


108 


126 


144 


162 


19 


38 


57 


76 


95 


114 


133 


152 


171 


20 


40 


60 


80 


100 


120 


140 


160 


180 


21 


42 


63 


84 


105 


126 


147 


168 


189 


22 


44 


66 


88 


110 


132 


154 


176 


198 


23 


46 


69 


92 


115 


138 


161 


184 


207 


24 


48 


72 


96 


120 


144 


168 


192 


216 


25 


50 


75 


100 


125 


150 


175 


200 


225 


26 


52 


78 


104 


130 


156 


182 


208 


234 


27 


54 


81 


108 


135 


162 


189 


216 


243 


28 


56 


84 


112 


140 


168 


196 


224 


252 


29 


58 


87 


116 


145 


174 


203 


232 


261 


30 


60 


90 


120 


150 


180 


210 


240 


270 


31 


62 


93 


124 


155 


186 


217 


248 


279 


32 


64 


96 


128 


160 


192 


224 


256 


288 


33 


66 


99 


132 


165 


198 


231 


264 


297 


34 


68 


102 


136 


170 


204 


238 


272 


306 


35 


70 


105 


140 


175 


210 


245 


280 


315 


36 


72 


108 


144 


180 


216 


252 


288 


324 


37 


74 


111 


148 


185 


222 


259 


296 


333 


38 


76 


114 


152 


190 


228 


266 


304 


342 


39 


78 


117 


156 


195 


234 


273 


312 


351 


40 


80 


120 


160 


200 


240 


280 


320 


360 


41 


82 


123 


164 


205 


246 


287 


328 


369 


42 


84 


126 


168 


210 


252 


294 


336 


378 


43 


86 


129 ' 


172 


215 


258 


301 


344 


387 


44 


88 


132 


176 


220 


264 


308 


352 


396 


45 


90 


135 


180 


225 


270 


315 


360 


405 


46 


92 


138 


184 


230 


276 


322 


368 


414 


47 


94 


141 


188 


235 


282 


329 


376 


423 


48 


96 


144 


192 


240 


288 


336 


384 


432 


49 


98 


147 


196 


245 


294 


343 


392 


441 


50 


100 


150 


200 


250 


300 


350 


400 


450 


51 


102 


153 


204 


255 


306 


357 


408 


459 


52 


104 


156 


208 


260 


312 


364 


416 


468 


53 


106 


159 


212 


265 


318 


371 


424 


477 


54 


108 


162 


216 


270 


324 


378 


432 


486 


55 


110 


165 


220 


275 


330 


385 


440 


495 


56 


112 


168 


224 


280 


336 


392 


448 


504 


57 


114 


171 


228 


285 


342 


399 


456 


513 


58 


116 


174 


232 


290 


348 


406 


464 


522 


59 


118 


177 


236 


295 


354 


413 


472 


531 


60 


120 


180 


240 


300 


360 


420 


480 


540 


61 


122 


183 


244 


305 


366 


427 


488 


549 


62 


124 


186 


248 


310 


372 


434 


496 


558 


63 


126 


189 


252 


315 


378 


441 


504 


567 


64 


128 


192 


256 


320 


384 


448 


512 


576 







Multiplication Table. 




7 


1 


2 


3 


4 


5 


6 


7 


8 


9 


65 


130 


195 


260 


325 


390 


455 


520 


585 


66 


132 


198 


264 


330 


396 


462 


528 


594 


67 


134 


201 


268 


335 


402 


469 


536 


603 


68 


136 


204 


272 


340 


408 


476 


544 


612 


69 


138 


207 


276 


345 


414 


483 


552 


621 


70 


140 


210 


280 


350 


420 


490 


560 


630 


71 


142 


213 


284 


355 


426 


497 


568 


639 


72 


144 


216 


288 


360 


432 


504 


576 


648 


73 


146 


219 


292 


365 


438 


511 


584 


657 


74 


148 


222 


296 


370 


444 


518 


592 


666 


75 


150 


225 


300 


375 


450 


525 


600 


675 


76 


152 


228 


304 


380 


456 


532 


608 


684 


77 


154 


231 


308 


385 


462 


539 


616 


693 


78 


156 


234 


312 


390 


468 


546 


624 


702 


79 


158 


237 


316 


395 


474 


553 


632 


71.1 


80 


160 


240 


320 


400 


480 


560 


640 


720 


81 


162 


243 


324 


405 


486 


567 


648 


729 


82 


164 


246 


328 


410 


492 


574 


656 


738 


83 


166 


249 


332 


415 


498 


581 


664 


747 


84 


168 


252 


336 


420 


504 


588 


672 


756 


85 


170 


255 


340 


425 


510 


595 


680 


765 


86 


172 


258 


344 


430 


516 


602 


688 


774 


87 


174 


261 


348 


435 


522 


609 


696 


783 


88 


176 


264 


352 


440 


528 


616 


704 


792 


89 


178 - 


267 


356 


445 


534 


623 


712 


801 


90 


180 


270 


360 


450 


540 


630 


720 


810 


91 


182 


273 


364 


455 


546 


637 


728 


819 


92 


184 


276 


368 


460 


552 


644 


736 


828 


93 


186 


279 


372 


465 


558 


651 


744 


837 


94 


188 


282 


376 


470 


564 


658 


752 


846 


95 


190 


285 


380 


475 


570 


665 


760 


855 


96 


192 


288 


384 


480 


576 


672 


768 


864 


97 


194 


291 


388 


485 


582 


679 


776 


873 


98 


196 


294 


392 


490 


588 


686 


784 


882 


99 


198 


297 


396 


495 


594 


693 


792 


891 



FACTOR TABLE. 

It is often desirable to know whether a number is a prime number or a 
product of two or more factors. The following table gives the factors of 
all numbers not divisible by 2, 3, or 5 up to 9599, and shows all prime 
numbers up to 9595. 

If the last figure of a number is divisible by 2, the whole number is 
divisible by 2. Thus 26154 is divisible by 2. 

If the sum of the digits of which a number is composed is divisible by 
3, the number is divisible by 3. Thus the sum of the digits of 26154 is equal 
to 18, which is divisible by 3 ; hence the whole number is divisible by 3. 

Any number ending with or 5 is divisible by 5. 

It is therefore possible to discover by inspection whether a number is 
divisible by 2, 3, or 5, and such a division will bring most large numbers — 
not prime numbers— within the compass of the table. 

To use the table, look along the top lines of the successive sections for 
the hundreds, and in the vertical columns at the left for the units and 
tens. The factors will be found at the intersection. If no factors are 
given, the number is a prime. 

Thus given the number 5203, which is not divisible by 2, 3, or 5, accord- 
ing to the above rules, we find under 5200, and opposite 3. the factors 11 X 
11 X 43 = 5203. In like manner we see that 5233 is a prime number, and 
so on for any other number. 



8 




Factor Table. 








N ( 


) 


300 


600 


900 


1 




7 .43 




17 


53 


7 














11 








13 


47 






13 












11 


83 


17 












7 


131 


19 




11 


29 










23 




17 


19 


7 


89 


13 


71 


29 




7 


47 


17 


37 






31 












7 


• . 19 


37 








7 . 7 . 13 






41 




11 


31 










43 




7 . 


. 7 






23 


41 


47 
















49 7 


7 






11 


59 


13 


73 


53 
















59 












7 


137 


61 




19 


19 






31 


31 


67 








23 


29 






71 




7 


53 


11 


61 






73 












7 


139 


77 7 


11 


13 


29 










79 








7 


97 


11 


89 


83 
















89 








13 


53 


23 


43 


91 7 


13 


17 


23 










97 








17 


41 






N 1( 


)0 


400 


700 


10 


00 


j 










7 . 1 


L . 13 


3 




13 


31 


19 


37 


17 


59 


7 




11 


37 


7 


101 


19 


53 


9 
















13 




7 


59 


23 


31 






19 7 


17 














21 11 


11 






7 


103 






27 




7 


61 






13 


79 


31 








17 


43 






33 7 


19 














37 




19 


23 


11 


67 


17 


61 


39 
















43 11 


13 










7 


149 


49 








7 


107 






51 




11 


41 

















Factor Table. 






9 


N 


100 


400 


700 


1000 


57 






. 






7 


151 


61 


7 


23 












63 








7 


109 






67 








13 


59 


11 


97 


69 


13 


13 


7 . 67 










73 






11 . 43 






29 


37 


79 








19 


41 


13 


83 


81 






13 . 37 


11 


71 


23 


47 


87 


11 


17 












91 








7 


113 






93 






17 . 29 


13 


61 






97 






7 . 71 










99 








17 


47 


7 


157 


N 


200 


500 


800 


11 


00 


3 


7 


29 




11 


73 






9 


11 


19 












11 






7 . 73 






11 


101 


17 


7 


31 


11 . 47 


19 


43 






21 


13 


17 








19 


59 


23 
















27 






17 . 31 






7 . ' 


7 . 23 


29 






23 . 23 










33 






13 . 41 


7 . ' 


' . 17 


11 


103 


39 






7 . 7 . 11 






17 


67 


41 








29 


29 


7 


163 


47 


13 


19 




7 . 1 


L . 11 


31 


37 


51 






19 . 29 


23 


37 






53 


11 


23 


7 . 79 










57 












13 


89 


59 


7 


37 


13 . 43 






19 


61 


63 
















69 








11 


79 


7 


167 


71 








13 


67 






77 












11 


107 


81 






7 . 83 










83 






11 . 53 






7 . 1 


5 . 13 


87 


7 


41 












89 


17 


17 


19 . 31 


7 


127 


29 


41 


93 








19 


47 






99 


13 


23 


• 


29 


31 


11 


109 



10 






Factor Table. 








N 


1200 


1500 


1800 


2100 


1 






19 . 79 






11 


191 


7 


17 


71 


11 .137 


13 


139 


7 . 7 . 43 


11 


7 


173 












13 






17 . 89 


7 . ' 


1 . 37 






17 






37 . 41 


23 


79 


29 


73 


19 


23 


53 


7 . 7 . 31 


17 


107 


13 


153 


23 












11 


193 


29 






11 . 139 


31 


59 






31 
















37 






29 . 53 


11 


167 






41 


17 


73 


23 . 67 


7 


263 






43 


11 


113 




19 


97 






47 


29 


43 


7 . 13 . 17 






19 


113 


49 








43 


43 


7 


307 


53 


7 


179 




17 


109 






59 








11 . 1 


3 . 13 


17 


127 


61 


13 


97 


7 . 223 










67 


7 


181 








11 


197 


71 


31 


41 








13 


167 


73 


19 


67 


11 . 11 . 13 






41 


53 


77 






19 . 83 






7 


311 


79 
















83 








7 


269 


37 


59 


89 






7 . 227 






11 


199 


91 






37 . 43 


31 


61 


7 


313 


97 








7 


271 


13 . 1 


3 . 13 


N 


1300 


1600 


1900 


2200 


1 












31 


71 


3 






7 . 229 


11 


173 






7 
















9 


7 . 1 


L . 17 




23 


83 


47 


47 


13 


13 


101 












19 








19 


101 


7 


317 


21 








17 


113 






27 








41 


47 


17 


131 


31 


11 . 1 


1 . 11 


7 . 233 






23 


97 


33 


31 


43 


23 . 71 






7 . 1 


L . 29 


37 


7 


191 




13 


149 






39 


13 


103 


11 . 149 


7 


277 






43 


17 


79 


31 . 53 


29 


67 






49 


19 


71 


17 . 97 






13 


173 


51 


7 


193 


13 . 127 

















Factor Table. 






11 


N 


1300 


1600 


1900 


2200 




57 


23 


59 




19 . 103 


37 


61 


61 






11 . 151 


37 . 53 


7 . 17 . 


19 


63 


29 


47 




13 . 151 


31 


73 


67 








7 . 281 








69 


37 


37 




11 .179 








73 






7 . 239 










79 


7 


197 


23 . 73 




43 




53 


81 






41 . 41 


7 . 283 








87 


19 


73 


7 . 241 










91 


13 


107 


19 . 89 


11 . 181 


29 




79 


93 


7 


199 












97 


11 


127 












99 










11 . 11 


19 


N 


1400 


1700 


2000 


2300 


3 


23 


61 


13 . 131 




7.7. 


47 


9 








7 . 7 . 41 






11 


17 


83 


29 . 59 










17 


13 


109 


17 . 101 




7 




331 


21 


7 . 


7 . 29 




43 . 47 


11 




211 


23 








7 . 17 . 17 


23 




101 


27 






11 . 157 




13 




179 


29 






7 . 13 . 19 




17 




137 


33 








19 . 107 








39 






37 . 47 










41 


11 


131 




13 . 157 








47 








23 . 89 








51 






17 . 103 


7 . 293 








53 










13 




181 


57 


31 


47 


7 . 251 


11 . 11 . 17 








59 








29 . 71 


7 




337 


63 


7 . 1 


L . 19 


41 . 43 




17 




139 


69 


13 


113 


29 . 61 




23 




103 


71 






7 . 11 . 23 


19 . 109 








77 


7 


211 




31 . 67 








81 






13 . 137 










83 
















87 










7 . 11 . 


31 


89 














93 






11 . 163 


7 . 13 . 23 






99 






7 . 257 









12 




Factor Table. 








N 


2400 


2700 


3000 


3300 


1 


7.7.7.7 


37 . 73 








7 


29 . 83 




31 


97 






11 










7 . 11 . 43 


13 


19 . 127 




23 


. 131 






17 




11 . 13 . 19 


7 


431 


31 


107 


19 


41 . 59 












23 




7 . 389 










29 


7 . 347 




13 


233 






31 


11 . 13 . 17 




7 


433 






37 




7 . 17 . 23 






47 


71 


41 










13 


257 


43 


7 . 349 


13 


211 


17 


. 179 






47 




41 


67 


11 


277 






49 


31 . 79 










17 


197 


53 


11 . 223 






43 


71 


7 


479 


59 




31 


89 


7 . 1 


9 . 23 






61 


23 . 107 


11 


251 










67 












7 . 13 . 37 


71 


7 . 353 


17 


163 


37 


. 83 






73 




47 


59 


7 


. 439 






77 








17 


181 


11 


307 


79 


37 . 67 


7 


397 






31 


109 


83 


13 . 191 


11 . 11 . 23 






17 


199 


89 


19 . 131 












91 


47 . 53 




11 


281 






97 


11 . 227 




19 


163 


43 


79 


N 


2500 


2800 


3100 


3400 


, 


41 . 61 




7 . 443 


19 


179 


3 






29 


107 


41 


83 


7 


23 . 109 


7 . 401 


13 


239 






9 


13 . 193 


53 . 53 






7 


487 


13 


13 . 359 


29 . 97 


11 


283 






19 


11 . 229 








13 


263 


21 




7 . 13 . 31 






11 


311 


27 


7 . 19 . 19 


11 . 257 


53 


59 


23 


149 


31 




19 . 149 


31 


101 


47 


73 


33 


17 . 149 




13 


241 






37 


43 . 59 








7 


491 


39 




17 . 167 


43 


73 


19 


181 


43 






7 


449 


11 


313 


49 




7 . 11 . 37 


47 


67 






51 








23 


137 


7 . r 


" . 29 







Factor Table. 




13 


N 


2500 


2800 


3100 


3400 


57 






7 . 11 . 41 






61 


13 . 197 




29 . 109 






63 


11 . 233 


7 . 409 








67 


17 . 151 


47 . 61 








69 


7 . 367 


19 . 151 








73 


31 . 83 


13 . 13 . 17 


19 . 167 


23 


151 


79 






11 . 17 . 17 


7 . ' 


■ . 71 


81 


29 . 89 


43 . 67 




59 


59 


87 


13 . 199 






11 


317 


91 




7 . 7 . 59 








93 




11 . 263 


31 . 103 


7 


499 


97 


7 . 7 . 53 




23 . 139 


13 


269 


99 


23 . 113 


13 . 223 


7 . 457 






N 


2600 


2900 


3200 


3500 


3 


19 . 137 






31 


113 


9 








11 . 11 . 29 


11 


7 . 373 


41 . 71 


13 . 13 . 19 






17 












21 




23 . 127 




7 


503 


23 


43 . 61 


37 .79 


11 . 293 


13 


271 


27 


37 . 71 




7 . 461 






29 


11 . 239 


29 . 101 








33 




7 . 419 


53 . 61 






39 


7 . 13 . 29 




41 . 79 






41 


19 . 139 


17 . 173 


7 . 463 






47 




7 . 421 


17 . 191 






51 


11 . 241 


13 . 227 




53 


67 


53 


7 . 379 






11 . 1 


7 . 19 


57 












59 




11 . 269 








63 






13 . 251 


7 


509 


69 


17 . 157 




7 . 467 


43 


83 


71 












77 




13 . 229 


29 . 113 


7 . 7 


. 73 


81 


7 . 383 


11 . 271 


17 . 193 






S3 




19 . 157 


7 . 7 . 67 






87 




29 . 103 


19 . 173 


17 


211 


89 




7 . 7 . 61 


11 . 13 . 23 


37 


97 


93 




41 . 73 


37 ' . 89 






99 








59 


61 











Factor Table. 






15 


N 


3700 


4000 


4300 


4600 


57 


13 


. 17 . 17 














61 






31 


. 131 


7 . 7 . 89 


59 




79 


63 


53 




71 


17 


. 239 










67 








7 


. 7 . 83 


11 . 397 


13 




359 


69 








13 


. 313 


17 . 257 


7 


23 


29 


73 


7 . 


7 


. 7 . 11 














79 












29 . 151 








81 


19 




199 


7 


11 . 53 


13 . 337 


31 




151 


87 


7 




541 


61 


67 


41 . 107 


43 




109 


91 


17 




223 














93 












23 . 191 


13 


. 19 


. 19 


97 








17 


. 241 




7 


. 11 


. 61 


99 


29 




131 






53 . 83 


37 




127 


N 


3800 


4100 


4400 


4700 


3 






11 


. 373 


7 . 17 . 37 








9 


13 




293 


7 


. 587 




17 




277 


11 


37 




103 






11 . 401 


7 




673 


17 


11 




347 


23 


. 179 


7 . 631 


53 




89 


21 








13 


. 317 










23 








7 . 


19 . 31 










27 


43 




89 






19 . 233 


29 




163 


29 


7 




547 






43 . 103 








33 












11 . 13 . 31 








39 


11 




349 






23 . 193 


7 




677 


41 


23 




167 


41 


. 101 




11 




431 


47 








11 


. 13 . 29 




47 




101 


51 








7 


. 593 










53 












61 . 73 


7 


. 7 . 


97 


57 


7 . 


19 . 29 








67 




71 


59 


17 


. 227 






7 . 7 . 7 . 13 








63 






23 


. 181 




11 




433 


69 


53 


73 


11 


. 379 


41 . 109 


19 




251 


71 


7 


7 . 79 


43 


97 


17 . 263 


13 




367 


77 










11 . 11 . 37 


17 




281 


81 






37 


. 113 




7 




683 


S3 


11 


. 353 


47 


89 










87 


13 


13 . 23 


53 


79 


7 . 641 








89 






59 


71 


67 . 67 








93 


17 


. 229 


7 


. 599 










99 


7 


. 557 


13 


17 . 19 


11 . 409 









16 



Factor Table. 



N 


4800 


5100 


5400 


5700 


1 








11 


. 491 






7 


11 


. 19 . 23 








13 


439 


11 


17 


. 283 


19 . 269 


7 


. 773 






13 












29 


197 


17 






7 . 17 . 43 










19 


61 


79 








7 . 19 . 43 


23 


7 


13 . 53 


47 . 109 


11 


. 17 . 29 


59 


97 


29 


11 


. 439 


23 . 223 


61 


. 89 


17 


337 


31 






7 • . 733 




. 


11 


521 


37 


7 


. 691 


11 . 467 








• 


41 


47 


. 103 


53 . 97 










43 


29 


. 167 


37 . 139 










47 


37 


. 131 




13 


. 419 


7 


821 


49 


13 


. 373 


19 . 271 










53 


23 


. 211 




7 


19 . 41 


11 


523 


59 


43 


. 113 


7 . 11 . 67 


53 


. 103 


13 


443 


61 






13 . 397 


43 


. 127 


7 


823 


67 


31 


. 157 




7 


11 . 71 


73 


79 


71 












29 


199 


73 


11 


. 443 


7 . 739 


13 


. 421 


23 


251 


77 






31 . 167 






53 


109 


79 


7 . 


17 . 41 






. 






83 


19 


. 257 


71 . 73 










89 








11 


. 499 


7 


827 


91 


67 


73 


29 . 179 


17 


. 17 . 19 






97 


59 


. 83 




23 


. 239 


11 . 17 . 31 


N 


4900 


5200 




5500 


5800 


1 


13 


. 13 . 29 


7 . 743 










3 






11 . 11 . 43 






7 


829 


7 


7 


. 701 


41 . 127 










9 








7 


. 787 


37 


157 


13 


17 


. 17 . 17 


13 . 401 


37 


. 149 






19 






17 . 307 






11 . 2 


3 . 23 


21 


7 


19 . 37 


23 . 227 










27 


13 


. 379 












31 












7 . 7 


7 . 17 


33 








11 


. 503 


19 


307 


31 








7 


. 7 . 113 


13 


449 


39 


11 


. 449 


13 . 13 . 31 


29 


. 191 






43 






7 . 7 . 107 


23 


. 241 






49 


7 


. 7 . 101 


29 . 181 


31 


. 179 






51 






59 . 89 


7 . 


13 . 61 





























Fac 


tor Table. 






17 


N 


4900 


5200 


5500 


5800 


57 






7 


751 










61 


11 


. 11 . 41 






67 


83 






63 


7 


. 709 


19 


277 






11 . 13 . 41 


67 






23 


. 229 


19 


. 293 






69 






11 


. 479 










73 














7 


839 


79 


13 


. 383 






7 


. 797 






81 


17 


. 293 














87 






17 


. 311 


37 


. 151 


7 . 29 . 29 


91 


7 


. 23 . 31 


11 . 13 . 37 






43 


137 


93 






67 


79 


7 


17 . 47 


71 


83 


97 


19 


. 263 






29 


. 193 






99 






7 


757 


11 


. 509 


17 


. 347 


N 


5000 


5300 


5600 


5900 


3 










13 


. 431 






9 










71 


79 


19 


311 


11 






47 


113 


31 


. 181 


23 


257 


17 


29 


. 173 


13 


409 


41 


. 137 


61 


97 


21 






17 . 


313 


7 


11 . 73 


31 


191 


23 


11 


. 457 


7 






. 331 






27 


761 


17 




29 


47 


. 107 


73 


73 


13 


. 433 


7.7. 


11 . 11 


33 


7 


. 719 






43 


. 131 


17 


349 


39 






19 


281 










41 


71 


. 71 


7 . 7 


. 109 






13 


457 


47 


7 


7 . 103 










19 


313 


51 














11 


541 


53 


31 


. 163 


53 


101 










57 


13 


. 389 


11 


487 






7 . 2 


3 . 37 


59 






23 


233 






59 


101 


63 


61 


. 83 


31 


173 


7 


. 809 


67 


89 


69 


37 


. 137 


7 . 1 


3 . 59 






47 


127 


71 


11 


. 461 


41 


131 


53 


. 107 


7 


853 


77 






19 


283 


7 


. 811 


43 


139 


81 










13 


19 . 23 






83 


13 


. 17 . 23 


7 


769 






31 


193 


87 










11 


11 . 47 






89 


7 


. 727 


17 


317 






53 


113 


93 


11 


. 463 










13 


461 


99 










41 


. 139 


7 ' . 


857 



18 






Factor Table. 










N 


6000 


6300 


6600 


6900 


1 


17 


. 353 




7 


23 . 41 


67 . 103 


7 






7 . 17 . 53 










11 








11 




601 






13 


7 


. 859 


59 


107 


17 




389 


31 


223 


17 


11 


. 547 






13 




509 






19 


13 


. 463 


71 


89 








11 . 17 . 37 


23 


19 


. 317 






37 




179 


7 . 23 . 43 


29 










7 




947 


13 . 13 . 41 


31 


37 


. 163 


13 


487 


19 




349 


29 . 239 


37 
















7 


991 


41 


7 


. 863 


17 


373 


29 




229 


11 


631 


43 










7 


13 . 73 


53 


. 131 


47 






11 


577 


17 


17 . 23 






49 


23 


. 263 


7 


907 


61 




109 






53 
















17 


409 


59 


73 


83 
















61 


11 


. 19 . 29 
















67 










59 




113 






71 


13 


. 467 


23 


277 


7 




953 






73 
















19 


367 


77 


59 


. 103 


7 


911 


11 




607 






79 
















7 


997 


83 


7 


11 . 79 


13 


491 


41 




163 






89 
















29 


241 


91 






7 . 11 . 83 












97 


7 


13 . 67 




37 




181 






N 


6100 


6400 


6700 


7000 


1 






37 . 173 








3 


17 


. 359 


19 . 337 






47 


149 


7 


31 


. 197 


43 . 149 


19 


. 353 


7 . 7 


11.13 


9 


41 


. 149 


13 . 17 . 29 






43 


163 


13 






11 . 11 . 53 


7 . 


7 . 137 






19 


29 


. 211 


7 . 7 . 131 










21 








11 


13 . 47 


7 . 1 


" . 59 


27 


11 


. 557 






7 


31 . 31 






31 






59 


109 


53 


. 127 


79 


89 


33 






7 


919 








13 


541 


37 


17 


. 19 . 19 


41 


157 








31 


227 


39 


7 


. 877 


47 


137 


23 




293 






43 






17 


379 


11 




613 






49 


11 


. 13 . 43 






17 




397 


7 . 19 . 53 


51 










43 




157 


11 


641 









Factor Table. 






19 


N 


6100 


6400 


6700 


7000 


57 


47 


131 


11 


587 


29 


. 233 






61 


61 


101 


7 . 13 . 71 






23 


307 


63 






23 


281 






7 


1009 


67 


7 


881 


29 


223 


67 


. 101 


37 


191 


69 


31 


199 






.7 


. 967 






73 










13 


. 521 


11 


643 


79 


37 


167 


11 . 19 . 31 










81 


7 


883 










73 


97 


87 


23 


269 


13 


499 


11 


. 617 


19 


373 


91 


41 


151 










7 


1013 


93 


11 


563 


43 


151 






41 


173 


97 






73 


89 


7 


. 971 


47 


151 


99 






67 


97 


13 


. 523 


31 


229 


N 


6200 


6500 




6800 


7100 


3 






7 


929 










9 


7 - 


887 


23 


283 


11 


. 619 






11 






17 


383 


7 


7 . 139 


13 


547 


17 






7.7. 


7 .19 


17 


. 401 


11 


647 


21 










19 


. 359 






23 


7 . 7 


. 127 


11 


593 






17 


419 


27 


13 


479 


61 


107 






* 




29 


















33 


23 


271 


47 


139 






7 


1019 


39 


17 


367 


13 


503 


7 


. 977 


11 . 1 


1 . 59 


41 


79 


79 


31 


211 






37 


193 


47 










41 


. 167 


7 


1021 


51 


7 . 1 


9 . 47 






13 


. 17 . 31 






53 


13 . 1 


3 . 37 






7 


. 11 . 89 


23 


311 


57 






79 


83 






17 


421 


59 


11 


569 


7 


937 


19 


. 19 . 19 






63 














13 . 1 


9 . 29 


69 












• 


67 


107 


71 














71 


101 


77 










13 


. 23 . 23 






81 


11 


571 






7 


. 983 


43 


167 


83 


61 


103 


29 


227 






11 


653 


87 






7 


941 


71 


. 97 






89 


19 


331 


11 


599 


83 


83 


7 . 1 


3 . 79 


93 


7 . 2 


9 . 31 


19 


347 


61 


. 113 






99 














23 


313 



20 






Factor Table. 










N 


7200 


7500 


7800 


8100 


1 


19 


379 


13 . 577 


29 . 269 








7 










37 


211 


11 


. 11 


. 67 


11 






7 . 2 


3 . 37 


73 


107 








13 






11 


683 


13 


601 


7 


. 19 


. 61 


17 


7 


1031 
















19 






73 


103 


7. 


1117 


23 




353 


23 


31 


233 
















29 














11 




739 


31 


7 


1033 


17 


443 


41 


191 


47 




173 


37 










17 


461 


79 




103 


41 


13 


557 










7 




1163 


43 






19 


397 


11 . 23 . 31 


17 




479 


47 










7 . 19 . 59 








49 


11 


659 






47 


167 


29 




281 


53 






7 . 13 . 83 






31 




263 


59 


7 . 1 


7 . 61 




29 


271 


41 




199 


61 


53 


137 




7 


1123 








67 


13 . 1 


3 . 43 


7 . 23 . 47 












71 


11 


661 


67 . 113 


17 


463 








73 


7 


1039 








11 




743 


77 


19 


383 








13 


. 17 


. 37 


79 


29 


251 


11 . 13 . 53 












83 












7 


. 7 . 


167 


89 


37 


197 




7 . 7 


7.23 


19 




431 


91 


23 


317 




13 


607 








97 






71 . 107 


53 


149 


7 




1171 


N 


7300 


7600 


7900 


8200 


1 


7 . ' 


' . 149 


11 . 691 




59 




139 


3 


67 


109 






7 . 1129 


13 




631 


7 












29 




283 


9 






7 


1087 


11 . 719 








13 


71 


103 


23 


331 


41 . 193 


43 




191 


19 


13 


563 


19 


401 










21 










89 . 89 








27 


17 


431 


29 


263 




19 




433 


31 






13 


587 


7 . 11 . 103 








33 






17 


449 










37 


11 . 2 


3 . 29 


7 


1091 










39 


41 


179 






17 . 467 


7 . 


11 . 


107 


43 


7 


1049 






13 . 13 . 47 








49 












73 




113 


51 






7 


1093 




37 




223 









Factor Table. 






21 


N 


7300 


7600 


7900 


8200 


57 


7 


. 1051 


13 


. 19 . 31 


73 


. 109 


23 


359 


61 


17 


. 433 


47 


. 163 


19 


. 419 


11 


751 


63 


37 


. 199 


79 


97 










67 


53 


. 139 


11 


. 17 . 41 


31 


. 257 


7 


. 1181 


69 










13 


613 






73 


73 


. 101 






7 . 1 


7 . 67 






79 


47 


. 157 


7 


. 1097 


79 


101 


17 


. 487 


81 


11 . 11 . 61 






23 


347 


7 . 7 


. 13 . 13 


87 


83 


89 






7 . ' 


' . 163 






91 


19 


. 389 






61 


131 






93 






7 


7 . 157 










97 


13 


. 569 


43 


. 179 


11 


727 






99 


7 . 


7 . 151 






19 


421 


43 


193 


N 


7400 


7700 


8000 


8300 


3 


11 


. 673 






53 


151 


19 . 19 . 23 


9 


31 


. 239 


13 


. 593 






7 


1187 


11 






11 


. 701 










17 


















21 


41 


. 181 


7 


. 1103 


13 


617 


53 


157 


23 


13 


. 571 






71 


113 


7 . 2 


3 . 41 


27 


7 


. 1061 






23 


349 


11 


757 


29 


17 . 1 


9 . 23 


59 


. 131 


7 . 3: 


L . 37 






33 






11 


19 . 37 


29 


277 


13 


641 


39 


43 


. 173 


71 


. 109 






31 


269 


41 


7 


. 1063 






11 . 1 


7 . 43 


19 


439 


47 


11 


677 


61 


. 127 


13 


619 


17 


491 


51 






23 


. 837 


83 


97 


7 


1193 


53 


29 


257 














57 










7 


1151 


61 


137 


59 














13 


643 


63 


17 


439 


7 


. 1109 


11 


733 






69 


7 . 1 


1 . 97 


17 


. 457 










71 


31 


241 


19 


. 409 


7 


1153 


11 


761 


77 






7 . 


11 . 101 


41 


197 






81 






31 


. 251 






17 . 1 


7 . 29 


83 


7 


1069 


43 


. 181 


59 


137 


S3 


101 


87 






13 


. 599 










89 


















93 


59 


127 










7 . 11 


. 109 


99 






11 


. 709 


7 . IS 


. 89 


37 


227 



22 






Factor Table. 






N 


8400 


8700 


9000 


9300 


t 


31 


. 271 


7 


11 . 113 






71 . 131 


7 


7 ' 


. 1201 










41 . 227 


11 


13 


. 647 


31 


. 281 








13 


47 


. 179 










67 . 139 


17 


19 


. 443 


23 


. 379 


71 


. 127 


7 . 11 . 11 . 11 


19 










29 


. 311 




23 






11 


. 13 . 61 


7 


. 1289 




29 






7 


. 29 . 43 






19 . 491 


31 










11 


. 821 


7 . 31 . 43 


37 


11 


. 13 . 59 






7 


. 1291 




41 


23 


. 367 












43 






7 


. 1249 








47 










83 


. 109 


13 . 719 


49 


7 . 


17 . 71 


13 


. 673 








53 


79 


. 107 






11 


. 823 


47 . 199 


59 


11 


. 769 


19 


. 461 






7 . 7 . 191 


61 










13 


. 17 . 41 


11 . 23 . 37 


67 






11 


. 797 






17 . 19 . 29 


71 


43 


. 197 


7 


. 7 . 179 


47 


. 193 




73 


3? 


. 229 


31 


. 283 


43 


. 211 


7 . 13 . 103 


77 


7 


. 7 . 173 


67 


. 131 


29 


. 313 




79 


61 


. 139 






7 


. 1297 


83 . 113 


83 


17 


. 499 






31 


. 293 


11 . 853 


89 


13 


. 653 


11 


. 17 . 47 


61 


. 149 


41 . 229 


91 


7 


. 1213 


59 


. 149 








97 


29 


. 293 


19 


. 463 


11 


. 827 




N 


8500 


8800 


9100 


9400 


t 






13 


. 677 


19 


. 479 


7 . 17 . 79 


3 


11 


. 773 












7 


47 


. 181 






7 


. 1301 


23 . 409 


9 


67 


. 127 


23 


. 383 






97 . 97 


13 






7 


. 1259 


13 


. 701 




19 


7 


. 1217 






11 


. 829 




21 










7 


. 1303 




27 






7 


13 . 97 






11 . 857 


31 


19 


. 449 






23 


. 397 




33 


7 


23 . 53 


11 


. 11 . 73 








37 
















39 










13 


. 19 . 37 




43 






37 


. 239 


41 


. 223 


7 . 19 . 71 


49 


83 


. 103 






7 


. 1307 


11 . 859 


51 


17 


. 503 


53 


. 167 






13 . 727 









Factor Table. 






23 


N 


8500 


8800 


9100 


9400 


57 


43 


199 


17 


. 521 




7 


. 7 . 


193 


61 


7 


1223 














63 










7 . 7 . 11 . 17 








61 


13 


659 






89 . 103 








69 


11 . 19 . 41 


7 


7 . 181 


53 . 173 


17 




557 


13 






19 


. 467 










79 


23 


373 


13 


. 683 


67 . 137 








81 






83 


. 107 




19 




499 


87 


31 


277 








53 




179 


91 


11 . 11 . 71 


17 


. 523 


7 . 13 . 101 








93 


13 


661 






29 . 317 


11 




863 


97 






7 


31 . 41 


17 . 541 








99 






11 


. 809 




7 


23 


59 


N 


8600 


8900 


9200 


9500 


3 


7 


1229 


29 


. 307 




13 


. 17 


. 43 


9 






59 


. 151 




37 




257 


11 


79 


109 


7 


19 . 67 


61 . 151 








17 


7 


1231 


37 


. 241 


13 . 709 


31 




307 


21 


37 


233 


11 


. 811 










23 










23 . 401 


89 




107 


27 






79 


. 113 




7 




1361 


29 










11 . 839 


13 




733 


33 


89 


97 






7 . 1319 








39 


53 


163 


7 


. 1277 










41 












7 


29 


47 


47 






23 


. 389 


7 . 1321 








51 


41 


211 






11 . 29 . 29 








53 


17 


509 


7 


. 1279 


19 . 487 


41 




233 


57 


11 


787 


13 


. 13 . 53 




19 




503 


59 


7 


1237 


17 


. 17 . 31 


47 . 197 


11 


. 11 


. 79 


63 










59 . 157 


73 




131 


69 










13 . 23 . 31 


7 




1367 


71 


13 . 2 


3 . 29 






73 . 127 


17 




563 


77 






47 


. 191 




61 




157 


81 






7 


. 1283 




11 


. 13 


. 67 


.83 


19 


457 


13 


. 691 




7 


. 37 


. 37 


87 


7 . 1 


7 . 73 


11 


. 19 . 43 


37 . 251 








89 






89 


. 101 


7 . 1327 


43 




223 


93 






17 


. 23 . 23 




53 




181 


99 










17 . 547 


29 




331 



24 



Powers and Roots. 



FRACTIONS. 

There are two methods of indicating subdivisions in general use, — one 
by continual bisection, as on the common foot-rule, in which the inch is 
divided into halves, quarters, eighths, sixteenths, etc., the other by di- 
vision into tenths, hundredths, thousandths, etc. Since the latter is based 
on the same principle as our system of numeration, it is desirable for 
general use, and the following conversion table will enable the common 
fractions to be converted into their equivalent decimals. 

Fractions Reduced to Equivalent Decimals. 



1 


.015625 
.03125 
.046875 
.0625 


i 17 
64 

3% 

19 
64 

A 


.265625 
.28125 
.296875 
.3125 


II 
§i 

If 

A 


.515625 
.53125 
.546875 
.5625 


1! 
if 
§2 
H 


co cc a> i-i 


Vs 


.078125 
.09375 
.109375 
.125 


n 
hi 

fi 

y 8 


.328125 
.34375 
.359375 
.375 


a! 

if 

Vs 


.578125 
.59375 
.609375 
.625 


If 
U 

II 

Vs 


.828125 

.84375 

.859375 

.875 




.140625 
.15625 

.171875 
.1875 


M 

II 

7 
16" 


.390625 
.40625 
.421875 
.4375 


If 


.640625 
.65625 
.671875 
.6875 


57 
64 

e 

69 
B4 

H 


.890625 
.90625 
.921875 
.9375 


15 
Wi 


.203125 
.21875 
.234375 
.25 


§1 


.453125 
.46875 
.484375 
.5 


H 

| If 

! II 


.703125 
.71875 
.734375 
.75 


U 

u 
u 

1 


.953125 
.96875 
.984375 
1. 



Any common fraction may be converted into its equivalent decimal by 
dividing the numerator by the denominator, a fraction really being merely 
a form of indicating division, and the decimal being the result of the 
performance of the division thus indicated. 



POWERS AND ROOTS. 

Any number multiplied by itself is said to be raised to its second power, 
or squared ; any number multiplied by itself twice is said to be raised to 
its third power, or cubed, etc. It is clear from this that every squared 
number, or second power, is composed of two equal factors, and either one 
of these equal factors is called the square root of the number. In like 
manner every cubed number is composed of the product of three equal 
factors, and any one of these equal factors is called the cube root of the 
number. 

Since squares, cubes, square roots, and cube roots are much used, the 
following table is given for all numbers up to 1600. If much work is to be 
done in this line, reference may be made to Barlow's Tables (Spon), which 
give the squares, cubes, square roots, and cube roots of all numbers up to 
10,000. 

In the right-hand column of the following table the reciprocals of the 
numbers in the first column are given, these being the quotients resulting 
from the division of unity by the given numbers. 







Powers and Roots. 


25 




Squares. 


Cubes. 








Number. 


VRoots. 


f Roots. 


Reciprocals. 


1 


1 


1 


1.000 0000 


1.000 0000 


1.000 000 000 


2 


4 


8 


1.414 2136 


1.259 9210 


.500 000 000 


3 


• 9 


27 


1.732 0508 


1.442 2496 


.333 333 333 


4 


16 


64 


2.000 0000 


1.587 4011 


.250 000 000 


5 


25 


125 


2.236 0680 


1.709 9759 


.200 000 000 


6 


36 


216 


2.449 4897 


1.817 1206 


.166 666 667 


7 


49 


343 


2.645 7513 


1.912 9312 


.142 857 143 


8 


64 


512 


2.828 4271 


2.000 0000 


.125 000 000 


9 


81 


729 


3.000 0000 


2.080 0837 


.111 111 111 


10 


100 


1000 


3.162 2777 


2.154 4347 


.100 000 000 


11 


121 


1331 


3.316 6248 


2.223 9801 


.090 909 091 


12 


144 


1728 


3.464 1016 


2.289 4286 


.083 333 333 


13 


169 


2197 


3.605 5513 


2.351 3347 


.076 923 077 


14 


196 


2 744 


3.741 6574 


2.410 1422 


.071 428 571 


15 


225 


3 375 


3.872 9833 


2.466 2121 


.066 666 667 


16 


256 


4 096 


4.000 0000 


2.519 8421 


.062 500 000 


17 


289 


4 913 


4.123 1056 


2.571 2816 


.058 823 529 


18 


324 


5 832 


4.242 6407 


2.620 7414 


.055 555 556 


19 


361 


6 859 


4.358 8989 


2.668 4016 


.052 631 579 


20 


400 


8 000 


4.472 1360 


2.714 4177 


.050 000 000 


21 


441 


9 261 


4.582 5757 


2.758 9243 


.047 619 048 


22 


484 


10 648 


4.690 4158 


2.802 0393 


.045 454 545 


23 


529 


12167 


4.795 8315 


2.843 8670 


.043 478 261 


24 


576 


13 824 


4.898 9795 


2.884 4991 


.041 666 667 


25 


625 


15 625 


5.000 0000 


2.924 0177 


.040 000 000 


26 


676 


17 576 


5.099 0195 


2.962 4960 


.038 461 538 


27 


729 


19 683 


5.196 1524 


3.000 0000 


.037 037 037 


28 


784 


21952 


5.291 5026 


3.036 5889 


.035 714 286 


29 


841 


24 389 


5.385 1648 


3.072 3168 


.034 482 759 


30 


900 


27 000 


5.477 2256 


3.107 2325 


.033 333 333 


31 


961 


29 791 


5.567 7644 


3.141 3806 


.032 258 065 


32 


1024 


32 768 


5.656 8542 


3.174 8021 


.031 250 000 


33 


1089 


35 937 


5.744 5626 


3.207 5343 


.030 303 030 


34 


1156 


39 304 


5.830 9519 


3.239 6118 


.029 411 765 


35 


1225 


42 875 


5.916 0798 


3.271 0663 


.028 571 429 


36 


1296 


46 656 


6.000 0000 


3.301 9272 


.027 777 778 


37 


1369 


50 653 


6.082 7625 


3,332 2218 


.027 027 027 


38 


1444 


54 872 


6.164 4140 


3.361 9754 


.026 315 789 


39 


1521 


59 319 


6.244 9980 


3.391 2114 


.025 641 026 


40 


1600 


64 000 


6.324 5553 


3.419 9519 


.025 000 000 


41 


1681 


68 921 


6.403 1242 


3.448 2172 


.024 390 244 


42 


1764 


74 088 


6.480 7407 


3.476 0266 


.023 809 524 


43 


1849 


79 507 


6.557 4385 


3.503 3981 


.023 255 814 


44 


1936 


85 184 


6.633 2496 


3.530 3483 


.022 727 273 


45 


2 025 


91125 


6.708 2039 


3.556 8933 


.022 222 222 


46 


2116 


97 336 


6.782 3300 


3.583 0479 


.021 739 130 


47 


2 209 


103 823 


6.855 6546 


3.608 8261 


.021 276 600 


48 


2 304 


110 592 


6.928 2032 


3.634 2411 


.020 833 333 


49 


2 401 


117 649 


7.000 0000 


3.659 3057 


.020 408 163 


50 


2 500 


125 000 


7.071 0678 


3.684 0314 


.020 000 000 


51 


2 601 


132 651 


7.141 4284 


3.708 4298 


.019 607 843 


52 


2 704 


140 608 


7.211 1026 


3.732 5111 


.019 230 769 



26 




Powers 


and Roots 








Squares. 


Cubes. 








Number. 


^Roots. 


f Roots. 


Reciprocals. 


53 


2 809 


148 877 


7.280 1099 


3.756 2858 


.018 867 925 


54 


2 916 


157 464 


7.348 4692 


3.779 7631 


.018 518 519 


55 


3 025 


166 375 


7.416 1985 


3.802 9525 


.018 181 818 


56 


3136 


175 616 


7.483 3148 


3.825 8624 


.017 857 143 


57 


3 249 


185 193 


7.549 8344 


3.848 5011 


.017 543 860 


58 


3 364 


195 112 


7.615 7731 


3.870 8766 


.017 241 379 


59 


3 481 


205 379 


7.681 1457 


3.892 9965 


.016 949 153 


60 


3 600 


216 000 


7.745 9667 


3.914 8676 


.016 666 667 


61 


3 721 


226 981 


7.810 2497 


3.930 4972 


.016 393 443 


62 


3 844 


238 328 


7.874 0079 


3.957 8915 


.016 129 032 


63 


3 969 


250 047 


7.937 2539 


3.979 0571 


.015 873 016 


64 


4 096 


262 144 


8.000 0000 


4.000 0000 


.015 625 000 


65 


4 225 


274 625 


8.062 2577 


4.020 7256 


.015 384 615 


66 


4 356 


287 496 


8.124 0384 


4.041 2401 


.015 151 515 


67 


4 489 


300 763 


8.185 3528 


4.061 5480 


.014 925 373 


68 


4 624 


314 432 


8.246 2113 


4.081 6551 


.014 705 882 


69 


4 761 


328 509 


8.306 6239 


4.101 5661 


.014 492 754 


70 


4 900 


343 000 


8.366 6003 


4.121 2853 


.014 285 714 


71 


5 041 


357 911 


8.426 1498 


4.140 8178 


.014 084 517 


72 


5184 


373 248 


8.485 2814 


4.160 1676 


.013 888 889 


73 


5 329 


389 017 


8.544 0037 


4.179 3390 


.013 698 630 


74 


5 476 


405 224 


8.602 3253 


4.198 3364 


.013 513 514 


75 


5 625 


421 875 


8.660 2540 


4.217 1633 


.013 333 333 


76 


5 776 


438 976 


8.717 7979 


4.235 8236 


.013 157 895 


77 


5 929 


456 533 


8.774 9644 


4.254 3210 


.012 987 013 


78 


6 084 


474 552 


8.831 7609 


4.272 6586 


.012 820 513 


79 


6 241 


493 039 


8.888 1944 


4.290 8404 


.012 658 228 


80 


6 400 


512 000 


8.944 2719 


4.308 8695 


.012 500 000 


81 


6 561 


531 441 


9.000 0000 


4.326 7487 


.012 345 679 


82 


6 724 


551 368 


9.055 3851 


4.344 4815 


.012 195 122 


83 


6 889 


571 787 


9.110 4336 


4.362 0707 


.012 048 193 


84 


7 056 


592 704 


9.165 1514 


4.379 5191 


.011 904 762 


85 


7 225 


614 125 


9.219 5445 


4.396 8296 


.011 764 706 


86 


7 396 


636 056 


9.273 6185 


4.414 0049 


.011 627 907 


87 


7 569 


658 503 


9.327 3791 


4.431 0476 


.011 494 253 


88 


7 744 


681 472 


9.380 8315 


4.447 9692 


.011 363 636 


89 


7 921 


704 969 


9.433 9811 


4.464 7451 


.011 235 955 


90 


8100 


729 000 


9.486 8330 


4.481 4047 


.011 111 111 


91 


8 281 


753 571 


9.539 3920 


4.497 9414 


.010 989 011 


92 


8 464 


778 688 


9.591 6630 


4.514 3574 


.010 869 565 


93 


8 649 


804 357 


9.643 6508 


4.530 6549 


.010 752 688 


94 


8 836 


830 584 


9.695 3597 


4.546 8359 


.010 638 298 


95 


9 025 


857 375 


9.746 7943 


4.562 9026 


.010 526 316 


96 


9 216 


884 736 


9.797 9590 


4.578 8570 


.010 416 667 


97 


9 409 


912 673 


9.848 8578 


4.594 7009 


.010 309 278 


98 


9 604 


941 192 


9.899 4949 


4.610 4363 


.010 204 082 


99 


9 801 


970 299 


9.949 8744 


4.626 0650 


.010 101 010 


100 


10 000 


1 000 000 


10.000 0000 


4.641 5888 


.010 000 000 


101 


10 201 


1 030 301 


10.049 8756 


4.657 0095 


.009 900 990 


102 


10 404 


1 061 208 


10.099 5049 


4.672 3287 


.009 803 922 


103 


10 609 


1 092 727 


10.148 8916 


4.687 5482 


.009 708 738 


104 


10 816 


1 124 864 


10.198 0390 


4.702 6694 


.009 615 385 







Powers 


and Roots 




27 




Squares, 


Cubes. 




f Roots. 




Number. 


^Koots. 


Reciprocals. 


105 


11 025 


1 157 625 


10.246 9508 


4.717 6940 


.009 523 810 


106 


11236 


1 191 016 


10.295 6301 


4.732 6235 


.009 433 962 


107 


11449 


1 225 043 


10.344 0804 


4.747 4594 


.009 345 794 


108 


11664 


1 259 712 


10.392 3048 


4.762 2032 


.009 259 259 


109 


11881 


1 295 029 


10.440 3065 


4.776 8562 


.009 174 312 


110 


12100 


1 331 000 


10.488 0885 


4.791 4199 


.009 090 909 


111 


12 321 


1 367 631 


10.535 6538 


4.805 8995 


.009 009 009 


112 


12 544 


1 404 928 


10.583 0052 


4.820 2845 


.008 928 571 


113 


12 769 


1 442 897 


10.630 1458 


4.834 5881 


.008 849 558 


114 


12 996 


1 481 544 


10.677 0783 


4.848 8076 


.008 771 930 


115 


13 225 


1 520 875 


10.723 8053 


4.862 9442 


.008 695 652 


116 


13 456 


1 560 896 


10.770 3296 


4.876 9990 


.008 620 690 


117 


13 689 


1 601 613 


10.816 6538 


4.890 9732 


.008 547 009 


118 


13 924 


1 643 032 


10.862 7805 


4.904 8681 


.008 474 576 


119 


14161 


1 685 159 


10.908 7121 


4.918 6847 


.008 403 361 


120 


14 400 


1 728 000 


10.954 4512 


4:932 4242 


.008 333 333 


121 


14 641 


1 771 561 


11.000 0000 


4.946 0874 


.008 264 463 


122 


14 884 


1 815 848 


11.045 3610 


4.959 6757 


.008 196 721 


123 


15 129 


1 860 867 


11.090 5365 


4.973 1898 


.008 130 081 


124 


15 376 


1 906 624 


11.135 5287 


4.986 6310 


.008 064 516 


125 


15 625 


1 953 125 


11.180 3399 


5.000 0000 


.008 000 000 


126 


15 876 


2 000 376 


11.224 9722 


5.013 2979 


.007 936 508 


127 


16 129 


2 048 383 


11.269 4277 


5.026 5257 


.007 874 016 


128 


16 384 


2 097 152 


11.313 7085 


5.039 6842 


.007 812 500 


129 


16 641 


2 146 689 


11.357 8167 


5.052 7743 


.007 751 938 


130 


16 900 


2 197 000 


11.401 7543 


5.065 7970 


.007 692 308 


131 


17 161 


2 248 091 


11.445 5231 


5.078 7531 


.007 633 588 


132 


17 424 


2 299 968 


11.489 1253 


5.091 6434 


.007 575 758 


133 


17 689 


2 352 637 


11.532 5626 


5.104 4687 


.007 518 797 


134 


17 956 


2 406 104 


11.575 8369 


5.117 2299 


.007 462 687 


135 


18 225 


2 460 375 


11.618 9500 


5.129 9278 


.007 407 407 


136 


18 496 


2 515 456 


11.661 9038 


5.142 5632 


.007 352 941 


137 


18 769 


2 571 353 


11.704 6999 


5.155 1367 


.007 299 270 


138 


19 044 


2 628 072 


11.747 3401 


5.167 6493 


.007 246 377 


139 


19 321 


2 685 619 


11.789 8261 


5.180 1015 


.007 194 245 


140 


19 600 


2 74A 000 


11.832 1596 


5.192 4941 


.007 142 857 


141 


19 881 


2 803 221 


11.874 3421 


5.204 8279 


.007 092 199 


142 


20 164 


2 863 288 


11.916 3753 


5.217 1034 


.007 042 254 


143 


20 449 


2 924 207 


11.958 2607 


5.229 3215 


.006 993 007 


144 


20 736 


2 985 984 


12.000 0000 


5.241 4828 


.006 944 444 


145 


21025 


3 048 625 


12.041 5946 


5.253 5879 


.006 896 552 


146 


21316 


3 112 136 


12.083 0460 


5.265 6374 


.006 849 315 


147 


21609 


3 176 523 


12.124 3557 


5.277 6321 


.006 802 721 


148 


21904 


3 241 792 


12.165 5251 


5.289 5725 


.006 756 757 


149 


22 201 


3 307 949 


12.206 5556 


5.301 4592 


.006 711 409 


150 


22 500 


3 375 000 


12.247 4487 


5.313 2928 


.006 666 667 


151 


22 801 


3 442 951 


12.288 2057 


5.325 0740 


.006 622 517 


152 


23 104 


3 511 008 


12.323 8280 


5.336 8033 


.006 578 947 


153 


23 409 


3 581 577 


12.369 3169 


5.348 4812 


.006 535 948 


154 


23 716 


3 652 264 


12.409 6736 


5.360 1084 


.006 493 506 


155 


24 025 


3 723 875 


12.449 8996 


5.371 6854 


.006 451 613 


156 


24 336 


3 796 416 


12.489 9960 


5.383 2126 


.006 410 256 



28 




Powers 


and Roots 








Squares. 


Cubes. 








Number. 


V Boots. 


f Roots. 


Reciprocals. 


157 


24 649 


3 869 893 


12.529 9641 


5.394 6907 


.006 369 427 


158 


24 964 


3 944 312 


12.569 8051 


5.406 1202 


.006 329 114 


159 


25 281 


4 019 679 


12.609 5202 


5.417 5015 


.006 289 308 


160 


25 600 


4 096 000 


12.649 1106 


5.428 8352 


.006 250 000 


161 


25 921 


4 173 281 


12.688 5775 


5.440 1218 


.006 211 180 


162 


26 244 


4 251 528 


12.727 9221 


5.451 3618 


.006 172 840 


163 


26 569 


4 330 747 


12.7671453 


5.462 5556 


.006 134 969 


164 


26 896 


4 410 944 


12.806 2485 


5.473 7037 


.006 097 561 


165 


27 225 


4 492 125 


12.845 2326 


5.484 8066 


.006 060 606 


166 * 


27 556 


4 574 296 


12.884 0987 


5.495 8647 


.006 024 096 


167 


27 889 


4 657 463 


12.922 8480 


5.506 8784 


.005 988 024 


168 


28 224 


4 741 632 


12.961 4814 


5.517 8484 


• .005 952 381 


169 


28 561 


4 826 809 


13.000 0000 


5.528 7748 


.005 917 160 


170 


28 900 


4 913 000 


13.038 4048 


5.539 6583 


.005 882 353 


171 


29 241 


5 000 211 


13.076 6968 


5.550 4991 


.005 847 953 


172 


29 584 


5 088 448 


13.114 8770 


5.561 2978 


.005 813 953 


173 


29 929 


5 177 717 


13.152 9464 


5.572 0546 


.005 780 347 


174 


30 276 


5 268 024 


13.190 9060 


5.582 7702 


.005 747 126 


175 


30 625 


5 359 375 


13.228 7566 


5.593 4447 


.005 714 286 


176 


30 976 


5 451 776 


13.266 4992 


5.604 0787 


.005 681 818 


177 


31329 


5 545 233 


13.304 1347 


5.614 6724 


.005 649 718 


178 


31684 


5 639 752 


13.341 6641 


5.625 2263 


.005 617 978 


179 


32 041 


5 735 339 


13.379 0882 


5.635 7408 


.005 586 592 


180 


32 400 


5 832 000 


13.416 4079 


5.646 2162 


.005 555 556 


181 


32 761 


5 929 741 


13.453 6240 


5.656 6528 


.005 524 862 


182 


33 124 


6 028 568 


13.490 7376 


5.667 0511 


.005 494 505 


183 


33 489 


6 128 487 


13.527 7493 


5.677 4114 


.005 464 481 


184 


33 856 


6 229 504 


13.564 6600 


5.687 7340 


.005 434 783 


185 


34 225 


6 331 625 


13.601 4705 


5.698 0192 


.005 405 405 


186 


34 596 


6 434 856 


13.638 1817 


5.708 2675 


.005 376 344 


187 


34 969 


6 539 203 


13.674 7943 


5.718 4791 


.005 347 594 


188 


35 344 


6 644 672 


13.711 3092 


5.728 6543 


.005 319 149 


189 


35 721 


6 751 269 


13.747 7271 


5.738 7936 


.005 291 005 


190 


36 100 


6 859 000 


13.784 0488 


5.748 8971 


.005 263 158 


191 


36 481 


6 967 871 


13.820 2750 


5.758 9652 


.005 235 602 


192 


36 864 


7 077 888 


13.856 4065 


5.768 9982 


.005 208 333 


193 


37 249 


7 189 517 


13.892 4400 


5.778 9966 


.005 181 347 


194 


37 636 


7 301 384 


13.928 3883 


5.788 9604 


.005 154 639 


195 


38 025 


7 414 875 


13.964 2400 


5.798 8900 


.005 128 205 


196 


38 416 


7 529 536 


14.000 0000 


5.808 7857 


.005 102 041 


197 


38 809 


7 645 373 


14.035 6688 


5.818 6479 


.005 076 142 


198 


39 204 


7 762 392 


14.071 2473 


5.828 4867 


.005 050 505 


199 


39 601 


7 880 599 


14.106 7360 


5.838 2725 


.005 025 126 


200 


40 000 


8 000 000 


14.142 1356 


5.848 0355 


.005 000 000 


201 


40 401 


8 120 601 


14.177 4469 


5.857 7660 


.004 975 124 


202 


40 804 


8 242 408 


14.212 6704 


5.867 4673 


.004 950 495 


203 


41209 


8 365 427 


14.247 8068 


5.877 1307 


.004 926 108 


204 


41616 


8 489 664 


14.282 8569 


5.886 7653 


.004 901 961 


205 


42 025 


8 615 125 


14.317 8211 


5.896 3685 


.004 878 049 


206 


42 436 


8 741 816 


14.352 7001 


5.905 9406 


.004 854 369 


207 


42 849 


8 869 743 


14.387 4946 


5.915 4817 


.004 830 918 


208 


43 264 


8 998 912 


14.422 2051 


5.924 9921 


.004 807 692 







Powers 


and Roots 




29 


Number. 


Squares. 


Cubes. 


V Boots. 


f Roots. 


Reciprocals. 


209 


43 681 


9 129 329 


14.456 8323 


5.934 4721 


.004 784 689 


210 


44 100 


9 261 000 


14.491 3767 


5.943 9220 


.004 761 905 


211 


44 521 


9 393 931 


14.525 8390 


5.953 3418 


.004 739 336 


212 


44 944 


9 528 128 


14 560 2198 


5.962 7320 


.004 716 981 


213 


45 369 


9 663 597 


14.594 5195 


5.972 0926 


.004 694 836 


214 


45 796 


9 800 34 i 


14.628 7388 


5.981 4240 


.004 672 897 


215 


46 225 


9 938 375 


14.662 8783 


5.990 7264 


.004 651 163 


216 


46 656 


10 077 696 


14.696 9385 


6.000 0000 


.004 629 630 


217 


47 089 


10 218 313 


14.730 9199 


6.009 2450 


.004 608 295 


218 


47 524 


10 360 232 


14.764 8231 


6.018 4617 


.004 587 156 


219 


47 961 


10 503 459 


14.798 6486 


6.027 6502 


.004 566 210 


220 


48 400 


10 648 000 


14.832 3970 


6.036 8107 


.004 545 455 


221 


48 841 


10 793 861 


14.866 0687 


6.045 9435 


.004 524 887 


222 


49 284 


10 911 048 


14.899 6644 


6.055 0489 


.004 504 505 


223 


49 729 


11 089 567 


14.933 1845 


6.064 1270 


.004 484 305 


224 


50 176 


11 239 424 


14.966 6295 


6.073 1779 


.004 464 286 


225 


50 625 


11 390 625 


15.000 0000 


6.082 4020 


.004 444 444 


226 


51076 


11 543 176 


15.033 2964 


6.099 1994 


.004 424 779 


227 


51529 


11 697 083 


15.066 5192 


6.100 1702 


.004 405 286 


228 


51984 


11 852 352 


15.099 6689 


6.109 1147 


.004 385 965 


229 


52 441 


12 008 989 


15.132 7460 


6.118 0332 


.004 366 812 


230 


52 900 


12 167 000 


15.165 7509 


6.126 9257 


.004 347 826 


231 


53 361 


12 326 391 


15.198 6842 


6.135 7924 


.004 329 004 


232 


53 824 


12 487 168 


15.231 5462 


6.144 6337 


.004 310 345 


233 


54 289 


12 649 337 


15.264 3375 


6.153 4495 


.004 291 845 


234 


54 756 


12 812 904 


15.297 0585 


6.162 2401 


.004 273 504 


235 


55 225 


12 977 875 


15.329 7097 


6.171 0058 


.004 255 319 


236 


55 696 


13 144 256 


15.362 2915 


6.179 7466 


.004 237 288 


237 


56169 


13 312 053 


15.394 8043 


6.188 4628 


.004 219 409 


238 


56 644 


13 481 272 


15.427 2486 


6.197 1544 


.004 201 681 


239 


57121 


13 651 919 


15.459 6248 


6.205 8218 


.004 184 100 


240 


57 600 


13 824 000 


15.491 9334 


6.214 4650 


.004 166 667 


241 


58 081 


13 997 521 


15.524 1747 


6.223 0843 


.004 149 378 


242 


58 564 


14 172 488 


15.556 3492 


6.231 6797 


.004 132 231 


243 


59 049 


14 348 907 


15.588 4573 


6.240 2515 


.004 115 226 


244 


59 536 


14 526 784 


15.620 4994 


6.248 7998 


.004 098 361 


245 


60 025 


14 706 125 


15.652 4758 


6.257 3248 


.004 081 633 


246 


60 516 


14 886 936 


15.684 3871 


6.265 8266 


.004 065 041 


247 


61009 


15 069 223 


15.716 2336 


6.274 3054 


.004 048 583 


248 


61504 


15 252 992 


15.748 0157 


6.282 7613 


.004 032 258 


249 


62 001 


15 438 249 


15.779 7338 


6.291 1946 


.004 016 064 


250 


62 500 


15 625 000 


15.811 3883 


6.299 6053 


.004 000 000 


251 


63 001 


15 813 251 


15.842 9795 


6.307 9935 


.003 984 064 


252 


63 504 


16 003 008 


15.874 5079 


6.316 3596 


.003 968 254 


253 


64 009 


16 194 277 


15.905 9737 


6.324 7035 


.003 952 569 


254 


64 516 


16 387 064 


15.937 3775 


6.333 0256 


.003 937 008 


255 


65 025 


16 581 375 


15.968 7194 


6.341 3257 


.003 921 569 


256 


65 536 


16 777 216 


16.000 0000 


6.349 6042 


.003 906 250 


257 


66 049 


16 974 593 


16.031 2195 


6.357 8611 


.003 891 051 


258 


66 564 


17 173 512 


16.062 3784 


6.366 0968 


.003 875 969 


259 


67 081 


17 373 979 


.16.093 4769 


6.374 3111 


.003 861 004 


260 


67 600 


17 576 000 


16.124 5155 


6.382 5043 


.003 846 154 



30 




Powers 


and Roots 








Squares. 


Cubes. 








Number. 


^Roots. 


f Roots. 


Reciprocals. 


261 


68121 


17 779 581 


16.155 4944 


6.390 6765 


.033 831 418 


262 


68 644 


17 984 728 


16.186 4141 


6.398 8279 


.003 816 794 


263 


69169 


18 191 447 


16.217 2747 


6.406 9585 


.003 802 281 


264 


69 696 


18 399 744 


16.248 0768 


6.415 0687 


.003 787 879 


265 


70 225 


18 609 625 


16.278 8206 


6.423 1583 


.003 773 585 


266 


70 756 


18 821 096 


16.309 5064 


6.431 2276 


.003 759 398 


267 


71289 


19 034 163 


16.340 1346 


6.439 2767 


.003 745 318 


268 


71824 


19 248 832 


16.370 7055 


6.447 3057 


.003 731 343 


269 


72 361 


19 465 109 


16.401 2195 


6.455 3148 


.003 717 472 


270 


72 900 


19 683 000 


16.431 67G7 


6.463 3041 


.003 703 704 


271 


73 441 


19 902 511 


16.462 0776 


6.471 2736 


.003 690 037 


272 


73 984 


20 123 643 


16.492 4225 


6.479 2236 


.003 676 471 


273 


74 529 


20 346 417 


16.522 7116 


6.487 1541 


.003 663 004 


274 


75 076 


20 570 824 


16.552 9454 


6.495 0653 


.003 649 635 


275 


75 625 


20 796 875 


16.583 1240 


6.502 9572 


.003 636 364 


276 


76176 


21 024 576 


16.613 2477 


6.510 8300 


.003 623 188 


277 


76 729 


21 253 933 


16.643 3170 


6.518 6839 


.003 610 108 


278 


77 284 


21 484 952 


16.673 3320 


6.526 5189 


.003 597 122 


279 


77 841 


21 717 639 


16.703 2931 


6.534 3351 


.003 584 229 


280 


78 400 


21 952 000 


16.733 2005 


6.542 1326 


.003 571 429 


281 


78 961 


22 188 041 


16.763 0546 


6.549 9116 


.003 558 719 


282 


79 524 


22 425 768 


16.792 8556 


6.557 6722 


.003 546 099 


283 


80 089 


22 665 187 


16.822 6038 


6 565 4144 


.003 533 569 


284 


80 656 


22 906 304 


16.852 2995 


6.573 1385 


.003 521 127 


285 


81225 


23 149 125 


16.881 9430 


6.580 8443 


.003 508 772 


286 


81796 


23 393 656 


16.911 5345 


6.588 5323 


.003 496 503 


287 


82 369 


23 639 903 


16.941 0743 


6.596 2023 


.003 484 321 


288 


82 £44 


23 887 872 


16.970 5627 


6.603 8545 


.003 472 222 


289 


83 521 


24 137 569 


17.000 0000 


6.611 4890 


.003 460 208 


290 


84 100 


24 389 000 


17.029 3864 


6.619 1060 


.003 448 276 


291 


84 681 


24 642 171 


17.058 7221 


6.626 7054 


.003 436 426 


292 


85 264 


24 897 088 


17.088 0075 


6.634 2874 


.003 424 658 


293 


85 849 


25 153 757 


17.117 2428 


6.641 8522 


.003 412 969 


294 


86 436 


25 412 184 


17.146 4232 


6.649 3998 


.003 401 361 


295 


87 025 


25 672 375 


17.175 5640 


6.656 9302 


.003 389 831 


296 


87 616 


25 934 836 


17.204 6505 


6.664 4437 


.003 378 378 


297 


88 209 


26 198 073 


17.233 6879 


6.671 9403 


.003 367 003 


298 


88 804 


26 463 592 


17.262 6765 


6.679 4200 


.003 355 705 


299 


89 401 


26 730 899 


17.291 6165 


6.686 8831 


.003 344 482 


300 


90 000 


27 000 000 


17.320 5081 


6.694 3295 


.003 333 333 


301 


90 601 


27 270 901 


17.349 3516 


6.701 7593 


.003 322 259 


302 


91 204 


27 543 608 


17.378 1472 


6.709 1729 


.003 311 258 


303 


91809 


27 818 127 


17.406 8952 


6.716 5700 


.003 301 330 


304 


92 416 


28 094 464 


17.435 5958 


6.723 9508 


.003 289 474 


305 


93 025 


28 372 625 


17.464 2492 


6.731 3155 


.003 278 689 


306 


93 636 


28 652 616 


17.492 8557 


6.738 6641 


.003 267 974 


307 


94 249 


28 934 443 


17.521 4155 


6.745 9967 


.003 257 329 


308 


94 864 


29 218 112 


17.549 92S8 


6.753 3134 


.003 246 753 


309 


95 481 


29 503 609 


17.578 3958 


6.760 6143 


.003 236 246 


310 


96100 


29 791 000 


17.606 8169 


6.767 8995 


.003 225 806 


311 


96 721 


30 080 231 


17.635 1921 


6.775 1690 


.003 215 434 


312 


97 344 


30 371 328 


17.663 5217 


6.782 4229 


.003 205 128 







POWEKS 


and Roots 




31 




Squares. 


Cubes. 








Number. 


l 7 Boots. 


f Boots. 


Beciprocals. 


313 


97 969 


30 664 297 


17.691 8060 


6.789 6613 


.003 194 888 


314 


98 596 


30 959 144 


17.720 0451 


6.796 8844 


.003 184 713 


315 


99 225 


31 255 875 


17.748 2393 


6.804 0921 


.003 174 603 


316 


99 856 


31 554 496 


17.776 3888 


6.811 2847 


.003 164 557 


317 


100 489 


31 855 013 


17.804 4938 


6.818 4620 


.003 154 574 


318 


101 124 


32 157 432 


17.832 5545 


6.825 6242 


.003 144 654 


319 


101 761 


32 461 759 


17.860 5711 


6.832 7714 


.003134 796 


320 


102 400 


32 768 000 


17.888 5438 


6.839 9037 


.003 125 000 


321 


103 041 


33 076 161 


17.916 4729 


6.847 0213 


.003 115 265 


322 


103 684 


33 386 248 


17.944 3584 


6.854 1240 


.003 105 590 


323 


104 329 


33 698 267 


17.972 2008 


6.861 2120 


.003 095 975 


324 


104 976 


34 012 224 


18.000 0000 


6.868 2855 


.003 0S6 420 


325 


105 625 


34 328 125 


18.027 7564 


6 £75 3433 


.003 076 923 


326 


106 276 


34 645 976 


18.C55 47C1 


6.882 3888 


.003 067 485 


327 


106 929 


34 965 783 


18.083 1413 


6.889 4188 


.003 048 104 


328 


107 584 


35 287 552 


18.110 7703 


6.896 4345 


.003 048 780 


329 


108 241 


35 611 289 


18.138 3571 


6.903 4359 


.003 039 514 


330 


108 900 


35 937 000 


18.165 9021 


6.910 4232 


.003 030 303 


331 


109 561 


36 264 691 


18.193 4054 


6.917 3964 


.003 021 148 


332 


110 224 


36 594 368 


18.220 8672 


6.924 3556 


.003 012 048 


333 


110 889 


36 926 037 


18.248 2876 


6.931 3088 


.003 003 003 


334 


111 556 


37 259 704 


18.275 6669 


6.938 2321 


.002 994 012 


335 


112 225 


37 595 375 


18.303 0052 


6.945 1496 


.002 985 075 


336 


112 896 


37 933 056 


18.330 3028 


6.952 0533 


.002 976 190 


337 


113 569 


38 272 753 


18.357 5598 


6.958 9434 


.002 967 359 


338 


114 244 


38 614 472 


18.384 7763 


6.965 8198 


.002 958 580 


339 


114 921 


38 958 219 


18.411 9526 


6.972 6826 


.002 949 853 


340 


115 600 


39 304 000 


18.439 0889 


6 979 5321 


.002 941 176 


341 


116 281 


39 651 821 


18.466 1853 


6.986 3681 


.002 932 551 


342 


116 964 


40 001 688 


18.493 2420 


6.993 1906 


.002 923 977 


343 


117 649 


40 353 607 


18.520 2592 


7.000 0000 


.002 915 452 


344 


118 336 


40 707 584 


18.547 2370 


7.006 7962 


.002 906 977 


345 


119 025 


41 063 625 


18.574 1756 


7.013 5791 


.002 898 551 


346 


119 716 


41 421 736 


18.601 0752 


7.020 3490 


.002 890 173 


347 


120 409 


41 781 923 


18.627 9360 


7.027 1058 


.002 881 844 


348 


121 104 


42 144 192 


18.654 7581 


7.033 8497 


.002 873 563 


349 


121 801 


42 508 549 


18.681 5417 


7.040 5860 


.002 865 330 


350 


122 500 


42 875 000 


18.708 2869 


7.047 2987 


.002 857 143 


351 


123 201 


43 243 551 


18.734 9940 


7.054 0041 


.002 849 003 


352 


123 904 


43 614 208 


18.761 6630 


7.060 6967 


.002 840 909 


353 


124 609 


43 986 977 


18.788 2942 


7.067 3767 


.002 832 861 


354 


125 316 


44 361 864 


18.814 8877 


7.074 0440 


.002 824 859 


355 


126 025 


44 738 875 


18.841 4437 


7.080 6988 


.002 816 901 


356 


126 736 


45 118 016 


18.867 9623 


7.087 3411 


.002 808 989 


357 


127 449 


45 499 293 


18.894 4436 


7.093 9709 


.002 801 120 


358 


128 164 


45 882 712 


18.920 8879 


7.100 5885 


.002 793 296 


359 


128 881 


46 268 279 


18.947 2953 


7.107 1937 


.002 785 515 


360 


129 600 


46 656 000 


18.973 6660 


7.113 7866 


.002 777 778 


361 


130 321 


47 045 831 


19.000 0000 


7.120 3674 


.002 770 083 


362 


131 044 


47 437 928 


19.026 2976 


7.126 9360 


.002 762 431 


363 


131 769 


47 832 147 


19.052 5589 


7.133 4925 


.002 754 821 


364 


132 496 


48 228 544 


19.078 7840 


7.140 0370 


.002 747 253 



32 




Powers 


and Roots 








Squares. 


Cubes. 


I 7 Roots. 






Number. 


f Roots. 


Reciprocals. 


365 


133 225 


48 627 125 


19.104 9732 


7.146 5695 


.002 739 726 


366 


133 956 


49 027 896 


19.131 1265 


7.153 0901 


.002 732 240 


367 


134 689 


49 430 863 


19.157 2441 


7.159 5988 


.002 724 796 


368 


135 424 


49 836 032 


19.183 3261 


7.166 0957 


.002 717 391 


369 


136 161 


50 243 409 


19.209 3727 


7.172 5809 


.002 710 027 


370 


136 900 


50 653 000 


19.235 3841 


7.179 0544 


.002 702 703 


371 


137 641 


51 064 811 


19.261 3603 


7.185 5162 


.002 695 418 


372 


138 384 


51 478 848 


19.287 3015 


7.191 9663 


.002 688 172 


373 


139 129 


51 895 117 


19.313 2079 


7.198 4050 


.002 680 965 


374 


139 876 


52 313 624 


19.339 0796 


7.204 8322 


.002 673 797 


375 


140 625 


52 734 375 


19.364 9167 


7.211 2479 


.002 666 667 


376 


141 376 


53 157 376 


19.390 7194 


7.217 6522 


.002 659 574 


377 


142 129 


53 582 633 


19.416 4878 


7.224 0450 


.002 652 520 


378 


142 884 


54 010 152 


19.442 2221 


7.230 4268 


.002 645 503 


379 


143 641 


54 439 939 


19.467 9223 


7.236 7972 


.002 638 521 


380 


144 400 


54 872 000 


19.493 5887 


7.243 1565 


.002 631 579 


381 


145 161 


55 306 341 


19.519 2213 


7.249 5045 


.002 624 672 


382 


145 924 


55 742 968 


19.544 8203 


7.255 8415 


.002 617 801 


383 


146 689 


56 181 887 


19.570 3858 


7.262 1675 


.002 610 966 


384 


147 456 


56 623 104 


19.595 9179 


7.268 4824 


.002 604 167 


385 


148 225 


57 066 625 


19.621 4169 


7.274 7864 


.002 597 403 


386 


148 996 


57 512 456 


19.646 8827 


7.281 0794 


.002 590 674 


387 


149 769 


57 960 603 


19.672 3156 


7.287 3617 


.002 583 979 


388 


150 544 


58 411 072 


19.697 7156 


7.293 6330 


.002 577 320 


389 


151 321 


58 863 869 


19.723 0829 


7.299 8936 


.002 570 694 


390 


152 100 


'59 319 000 


19.748 4177 


7.306 1436 


.002 564 103 


391 


152 881 


59 776 471 


19.773 7199 


7.312 3828 


.002 557 545 


392 


153 664 


60 236 288 


19.798 9899 


7.318 6114 


.002 551 020 


393 


154 449 


60 698 457 


19.824 2276 


7.324 8295 


.002 544 529 


394 


155 236 


61 162 984 


19.849 4332 


7.331 0369 


.002 538 071 


395 


156 025 


61 629 875 


19.874 6069 


7.337 2339 


.002 531 646 


396 


156 816 


62 099 136 


19.899 7487 


7.343 4205 


.002 525 253 


397 


157 609 


62 570 773 


19.924 8588 


7.349 5966 


.002 518 892 


398 


158 404 


63 044 792 


19.949 9373 


7.355 7624 


.002 512 563 


399 


159 201 


63 521 199 


19.974 9844 


7.361 9178 


.002 506 266 


400 


160 000 


64 000 000 


20.000 0000 


7.368 0630 


.002 500 000 


401 


160 801 


64 481 201 


20.024 9844 


7.374 1979 


.002 493 766 


402 


161 604 


64 964 808 


20.049 9377 


7.380 3227 


.002 487 562 


403 


162 409 


65 450 827 


20.074 8599 


7.386 4373 


.002 481 390 


404 


163 216 


65 939 264 


20.099 7512 


7.392 5418 


.002 475 248 


405 


164 025 


66 430 125 


20.124 6118 


7.398 6363 


.002 469 136 


406 


164 836 


66 923 416 


20.149 4417 


7.404 7206 


.002 463 054 . 


407 


165 649 


67 419 143 


20.174 2410 


7.410 7950 


.002 457 002 


408 


166 464 


67 917 312 


20.199 0099 


7.416 8595 


.002 450 980 


409 


167 281 


68 417 929 


20.223 7484 


7.422 9142 


.002 444 988 


410 


168 100 


68 921 000 


20.248 4567 


7.428 9589 


.002 439 024 


411 


168 921 


69 426 531 


20.273 1349 


7.434 9938 


.002 433 090 


412 


169 741 


69 934 528 


20.297 7831 


7.441 0189 


.002 427 184 


413 


170 569 


70 444 997 


20.322 4014 


7.447 0343 


.002 421308 


414 


171 396 


70 957 91 J 


20.346 9899 


7.453 0399 


.002 415 459 


415 


172 225 


71 473 375 


20.371 5488 


7.459 0359 


.002 409 639 


416 


173 056 


71 991 296 


20.396 0781 


7.465 0223 


.002 406 846 



Poweks and Roots. 



35 



Number. 


Squares. 


Cubes. 








^Roots. 


f Roots. 


Reciprocals. 


521 


271 441 


141 420 761 


22.825 4244 


8.046 6030 


.001 919 386 


522 


272 484 


142 236 648 


22.847 3193 


8.051 7479 


.001 915 709 


523 


273 529 


143 055 667 


22.869 1933 


8.056 8862 


.001 912 046 


524 


274 576 


143 877 824 


22.891 0463 


8.062 0180 


.001 908 397 


525 


275 625 


144 703 125 


22.912 8785 


8.067 1432 


.001 904 762 


526 


276 676 


145 531 576 


22.934 6899 


8.072 2620 


.001 901 141 


527 


277 729 


146 363 183 


22.956 4806 


8.077 3743 


.001 897 533 


528 


278 784 


147 197 952 


22.978 2506 


8.082 4800 


.001 893 939 


529 


279 841 


148 035 889 


23.000 0000 


8.087 5794 


.001 890 359 


530 


280 900 


148 877 001 


23.021 7289 


8.092 6723 


.001 886 792 


531 


281961 


149 721 291 


23.043 4372 


8.097 7589 


.001 883 239 


532 


283 024 


150 568 768 


23.065 1252 


8.102 8390 


.001 879 699 


533 


284 089 


151 419 437 


23.086 7928 


8.107 9128 


.001 876 173 


534 


285156 


152 273 304 


23.108 4400 


8.112 9803 


.001 872 659 


535 


286 225 


153 130 375 


23.130 0670 


8.118 0414 


.001 869 159 


536 


287 296 


153 990 656 


23.151 6738 


8.123 0962 


.001 865 672 


537 


288 369 


154 854 153 


23.173 2605 


8.128 1447 


.001 862 197 


538 


289 444 


155 720 872 


23.194 8270 


8.133 1870 


.001 858 736 


539 


290 521 


156 590 819 


23.216 3735 


8.138 2230 


.001 855 288 


540 


291 600 


157 464 000 


23.237 9001 


8.143 2529 


.001 851 852 


541 


292 681 


158 340 421 


23.259 4067 


8.148 2765 


.001 848 429 


542 


293 764 


159 220 088 


23.280 8935 


8.153 2939 


.001 845 018 


543 


294 849 


160 103 007 


23.302 3604 


8.158 3051 


.001 841 621 


544 


295 936 


160 989 184 


23.323 8076 


8.163 3102 


.001 838 235 


545 


297 025 


161 878 625 


23.345 2351 


8.168 3092 


.001 834 862 


546 


298 116 


162 771 336 


23.366 6429 


8.173 3020 


.001 831 502 


547 


299 209 


163 667 323 


23.388 0311 


8.178 2888 


.001 828 154 


548 


300 304 


164 566 592 


23.409 3998 


8.183 2695 


.001 824 818 


549 


301 401 


165 469 149 


23.430 7490 


8.188 2441 


.001 821 494 


550 


302 500 


166 375 000 


23.452 0788 


8.193 2127 


.001 818 182 


551 


303 601 


167 284 151 


23.473 3892 


8.198 1753 


.001 814 882 


552 


304 704 


168 196 608 


23.494 6802 


8.203 1319 


.001 811 594 


553 


305 809 


169 112 377 


23.515 9520 


8.208 0825 


.001 808 318 


554 


306 916 


170 031 464 


23.537 2046 


8.213 0271 


.001 805 054 


555 


308 025 


170 953 875 


23.558 4380 


8.217 9657 


.001 801 802 


556 


309 136 


171 879 616 


23.579 6522 


8.222 8985 


.001 798 561 


557 


310 249 


172 808 693 


23.600 8474 


8.227 8254 


.001 795 332 


558 


311 364 


173 741 112 


23.622 0236 


8.232 7463 


.001 792 115 


559 


312 481 


174 676 879 


23.643 1808 


8.237 6614 


.001 788 909 


560 


313 600 


175 616 000 


23.664 3191 


8.242 5706 


.001 785 714 


561 


314 721 


176 558 481 


23.685 4386 


8.247 4740 


.001 782 531 


562 


315 844 


177 504 328 


23.706 5392 


8.252 3715 


.001 779 359 


563 


316 969 


178 453 547 


23.727 6210 


8.257 2635 


.001 776 199 


564 


318 096 


179 406 144 


23.748 6842 


8.262 1492 


.001 773 050 


565 


319 225 


180 362 125 


23.769 7286 


8.267 0294 


.001 769 912 


566 


320 356 


181 321 496 


23.790 7545 


8.271 9039 


.001 766 784 


567 


321 489 


182 284 263 


23.811 7618 


8.276 7726 


.001 763 668 


568 


322 624 


183 250 432 


23.832 7506 


8.2816255 


.001 760 563 


569 


323 761 


184 220 009 


23.853 7209 


8.286 4928 


.001 757 469 


570 


324 900 


185 193 000 


23.874 6728 


8.291 3444 


.001 754 386 


571 


326 041 


186 169 411 


23.895 6063 


8.296 1903 


.001 751 313 


572 


327 184 


187 149 248 


23.916 5215 


8.301 0304 


.001 748 252 



36 



Powers and Roots. 



Number. 


Squares. 


Cubes. 


573 


328 329 


188 132 517 


574 


329 476 


189 119 224 


575 


330 625 


190 109 375 


576 


331 776 


191 102 976 


577 


332 927 


192 100 033 


578 


334 084 


193 100 552 


579 


335 241 


194 104 539 


580 


336 400 


195 112 000 


581 


337 561 


196 122 941 


582 


338 724 


197 137 368 


583 


339 889 


198 155 287 


584 


341 056 


199 176 704 


585 


342 225 


200 201 625 


586 


343 396 


201 230 056 


587 


344 569 


202 262 003 


588 


345 744 


203 297 472 


589 


346 921 


204 336 469 


590 


348 100 


205 379 000 


591 


349 281 


206 425 071 


592 


350 464 


207 474 688 


593 


351 649 


208 527 857 


594 


352 836 


209 584 584 


595 


354 025 


210 644 875 


596 


355 216 


211 708 736 


597 


356 409 


212 776 173 


598 


357 604 


213 847 192 


599 


358 801 


214 921 799 


600 


360 000 


216 000 000 


601 


361 201 


217 081 801 


602 


362 404 


218 167 208 


603 


363 609 


219 256 227 


604 


364 816 


220 348 864 


605 


366 025 


221 445 125 


606 


367 236 


222 545 016 


607 


368 449 


223 648 543 


608 


369 664 


224 755 712 


609 


370 881 


225 866 529 


610 


372 100 


226 981 000 


611 


373 321 


228 099 131 


612 


374 544 


229 220 928 


613 


375 769 


230 346 397 


614 


376 996 


231 475 544 


615 


378 225 


232 608 375 


616 


379 456 


233 744 896 


617 


380 689 


234 885 113 


618 


381 924 


236 029 032 


619 


383 161 


237 176 659 


620 


384 400 


238 328 000 


621 


385 641 


239 483 061 


622 


386 884 


240 641 848 


623 


388 129 


241 804 367 


624 


389 376 


242 970 624 



Vltoots. 



Etoots. 



Reciprocals. 



23.937 4184 
23.958 2971 
23.979 1576 
24.000 0000 
24.020 8243 
24.041 6306 
24.062 4188 
24.083 1891 
24.103 9416 
24.124 6762 
24.145 3929 
24.166 0919 
24.186 7732 
24.207 4369 
24.228 0829 
24.248 7113 
24.269 3222 
24.289 9156 
24.310 4996 
24.331 0501 
24.351 5913 
24.372 1152 
24.392 6218 
24.413 1112 
24.433 5834 
24.454 0385 
24.474 4765 
24.494 8974 
24.515 3013 
24.535 6883 
24.556 0583 
24.576 4115 
24.596 7478 
24.617 0673 
24.637 3700 
24.657 6560 
24.677 9254 
24.698 1781 
24.718 4142 
24.738 6338 
24.758 8368 
24.779 0234 
24.799 1935 
24.819 3473 
24.839 4847 
24.859 6058 
24.879 7106 
24.899 7992 
24.919 8716 
24 939 9278 
24.959 9679 
24.979 9920 



8.305 8651 
8.310 6941 
8.315 5175 
8.320 3353 
8.325 1475 
8.329 9542 
8.334 7553 
8.339 5509 
8.344 3410 
8.349 1256 
8.353 9047 
8.358 6784 
8.363 4466 
8.368 2095 
8.372 9668 
8.377 7188 
8.382 4653 
8.387 2065 
8.391 9428 
8.396 6729 
8.401 3981 
8.406 1180 
8.410 8326 
8.415 5419 
8.420 2460 
8.424 9448 
8.429 6383 
8.434 3267 
8.439 0098 
8.443 6877 
8.448 3605 
8.453 0281 
8.457 6906 
8.462 3479 
8.467 0001 
8.471 6471 
8.476 2892 
8.480 9261 
8.485 5579 
8.490 1848 
8.494 8065 
8.499 4233 
8.504 0350 
8.508 6417 
8.513 2435 
8.517 8403 
8.522 4331 
8.527 0189 
8.531 6009 
8.536 1780 
8.540 7501 
8.545 3173 



.001 745 201 
.001 742 160 
.001 739 130 
.001 736 111 
.001 733 102 
.001 730 104 
.001 727 116 
.001 724 138 
.001 721 170 
.001 718 213 
.001 715 266 
.001 712 329 
.001 709 402 
.001 706 485 
.001 703 578 
.001 700 680 
.001 697 793 
.001 694 915 
.001 692 047 
.001 689 189 
.001 686 341 
.001 683 502 
.001 680 672 
.001 677 852 
.001 675 042 
.001 672 241 
.001 669 449 
.001 666 667 
.001 663 894 
.001 661 130 
.001 658 375 
.001 655 629 
.001 652 893 
.001 650 165 
.001 647 446 
.001 644 737 
.001 642 036 
.001 639 344 
.001 636 661 
.001 633 987 
.001 631 321 
.001 628 664 
.001 626 016 
.001 623 377 
.001 620 746 
.001 618 123 
.001 615 509 
.001 612 903 
.001 610 306 
.001 607 717 
.001 605 136 
.001 602 564 







Powers 


and Roots. 




37 


Number. 


Squares. 


Cubes. 


VRoots. 


f Roots. 


Reciprocals. 


625 


390 625 


244 140 625 


25.000 0000 


8.549 8797 


.001 600 000 


626 


391 876 


245 134 376 


25.019 9920 


8.554 4372 


.001 597 444 


627 


393 129 


246 491 883 


25.039 9681 


8.558 9899 


.001 594 896 


628 # 


394 384 


247 673 152 


25.059 9282 


8.563 5377 


.001 592 357 


629 ' 


395 641 


248 858 189 


25.079 8724 


8.568 0807 


.001 589 825 


630 


396 900 


250 047 000 


25.099 8008 


8.572 6189 


.001 587 302 


631 


398 161 


251 239 591 


25.119 7134 


8.577 1523 


.001 584 786 


632 


399 424 


252 435 968 


25.139 6102 


8.581 6809 


.001 582 278 


633 


400 689 


253 636 137 


25.159 4913 


8.586 2247 


.001 579 779 


634 


401 956 


254 840 104 


25.179 3566 


8.590 7238 


.001 577 287 


635 


403 225 


256 047 875 


25.199 2063 


8.595 2380 


.001 574 803 


636 


404 496 


257 259 456 


25.219 0404 


8.599 7476 


.001 572 327 


637 


405 769 


258 474 853 


25.238 8589 


8.604 2525 


.001 569 859 


638 


407 044 


259 694 072 


25.258 6619 


8.608 7526 


.001 567 398 


639 


408 321 


260 917 119 


25.278 4493 


8.613 2480 


.001 564 945 


640 


409 600 


262 144 000 


25.298 2213 


8.617 7388 


.001 562. 500 


641 


410 881 


263 374 721 


25.317 9778 


8.622 2248 


.001 560 062 


642 


412 164 


264 609 288 


25.337 7189 


8.626 7063 


.001 557 632 


643 


413 449 


265 847 707 


25.357 4447 


8.631 1830 


.001 555 210 


644 


414 736 


267 089 984 


25.377 1551 


8.635 6551 


.001 552 795 


645 


416 025 


268 336 125 


25.396 8502 


8.640 1226 


.001 550 388 


646 


417 316 


269 585 136 


25.416 5302 


8.644 5855 


.001 547 988 


647 


418 609 


270 840 023 


25.436 1947 


8.649 0437 


.001 545 595 


648 


419 904 


272 097 792 


25.455 8441 


8.653 4974 


.001 543 210 


649 


421 201 


273 359 449 


25.475 4784 


8.657 9465 


.001 540 832 


650 


422 500 


274 625 000 


25.495 0976 


8.662 3911 


.001 538 462 


651 


423 801 


275 894 451 


25.514 7013 


8.666 8310 


.001 536 098 


652 


425104 


277 167 808 


25.534 2907 


8.671 2665 


.001 533 742 


653 


426 409 


278 445 077 


25.553 8647 


8.675 6974 


.001 531 394 


654 


427 716 


279 726 264 


25.573 4237 


8.680 1237 


.001 529 052 


655 


429 025 


281 Oil 375 


25.592 9678 


8.684 5456 


.001 526 718 


656 


430 336 


282 300 416 


25.612 4969 


8.688 9630 


.001 524 390 


657 


431649 


283 593 393 


25.632 0112 


8.693 3759 


.001 522 070 


658 


432 964 


284 890 312 


25.651 5107 


8.697 7843 


.001 519 757 


659 


434 281 


286 191 179 


25.670 9953 


8.702 1882 


.001 517 451 


660 


435 600 


287 496 000 


25.690 4652 


8.706 5877 


.001 515 152 


661 


436 921 


288 804 781 


25.709 9203 


8.710 9827 


.001 512 859 


662 


438 244 


290 117 528 


25.729 3607 


8.715 3734 


.001 510 574 


663 


439 569 


291 434 247 


25.748 7864 


8.719 7596 


.001 508 296 


664 


440 896 


292 754 944 


25.768 1975 


8.724 1414 


.001 506 024 


665 


442 225 


294 079 625 


25.787 5939 


8.728 5187 


.001 503 759 


666 


443 556 


295 408 296 


25.806 9758 


8.732 8918 


.001 501 502 


667 


444 889 


296 740 963 


25.826 3431 


8.737 2604 


.001 499 250 


668 


446 224 


298 077 632 


25.845 6960 


8.741 6246 


.001 497 006 


669 


447 561 


299 418 309 


25.865 0343 


8.745 9846 


.001 494 768 


670 


448 900 


300 763 000 


25.884 3582 


8.750 3401 


.001 492 537 


671 


450 241 


302 111 711 


25.903 6677 


8.754 6913 


.001 490 313 


672 


451 584 


303 464 448 


25.922 9628 


8.759 0383 


.001 488 095 


673 


452 929 


304 821 217 


25.942 2435 


8.763 3809 


.001 485 884 


674 


454 276 


306 182 024 


25.961 5100 


8.767 7192 


.001 483 680 


675 


455 625 


307 546 875 


25.980 7621 


8.772 0532 


.001 481 481 


676 


456 976 


308 915 776 


26.000 0000 


8.776 3830 


.001 479 290 



38 




Powers 


and Roots 








Squares. 


Cubes. 








Number. 


V Roots. 


f Roots. 


Reciprocals. 


677 


458 329 


310 288 733 


26.019 2237 


8.780 7084 


.001 477 105 


678 


459 684 


311 665 752 


26.038 4331 


8.785 0296 


.001 474 926 


679 


461 041 


313 046 839 


26.057 6284 


8.789 3466 


.001 472 754 


680 


' 462 400 


314 432 000 


26.076 8096 


8.793 6593 


.001 470 588 


681 


463 761 


315 821 241 


26.095 9767 


8.797 9679 


.001 468 429 


682 


465 124 


317 214 568 


26.115 1297 


8.802 2721 


.001 466 276 


683 


466 489 


318 611 987 


26.134 2687 


8.806 5722 


.001 464 129 


684 


467 856 


320 013 504 


26.153 3937 


8.810 8681 


.001 461 988 


685 


469 225 


321 419 125 


26.172 5047 


8.815 1598 


.001 459 854 


686 


470 596 


322 828 856 


26.191 6017 


8.819 4474 


.001 457 726 


687 


471 969 


324 242 703 


26.210 6848 


8.823 7307 


.001 455 604 


688 


473 344 


325 660 672 


26.229 7541 


8.828 0099 


.001 453 488 


689 


474 721 


327 082 769 


26.248 8095 


8.832 2850 


.001 451 379 


690 


476 100 


328 509 000 


26.267 8511 


8.836 5559 


.001 449 275 


691 


477 481 


329 939 371 


26.286 8789 


8.840 8227 


.001 447 178 


692 


478 864 


331 373 888 


26.305 8929 


8.845 0854 


.001 445 087 


693 


480 24-9 


332 812 557 


26.324 8932 


8.849 3440 


.001 443 001 


694 


481 636 


334 255 384 


26.343 8797 


8.853 5985 


.001 440 922 


695 


483 025 


335 702 375 


26.362 8527 


8.857 8489 


.001 438 849 


696 


484 416 


337 153 536 


26.381 8119 


8.862 0952 


.001 436 782 


697 


485 809 


338 608 873 


26.400 7576 


8.866 3375 


.001 434 720 


698 


487 204 


340 068 392 


26.419 6896 


8.870 5757 


.001 432 665 


699 


488 601 


341 532 099 


26.438 6081 


8.874 8099 


.001 430 615 


700 


490 000 


343 000 000 


26.457 5131 


8.879 0400 


.001 428 571 


701 


491 401 


344 472 101 


26.476 4046 


8.883 2661 


.001 426 534 


702 


492 804 


345 948 408 


26.495 2826 


8.887 4882 


.001 424 501 


703 


494 209 


347 428 927 


26.514 1472 


8.891 7063 


.001 422 475 


704 


495 616 


348 913 664 


26.532 9983 


8.895 9204 


.001 420 455 


705 


497 025 


350 402 625 


26.551 8361 


8.900 1304 


.001 418 440 


706 


498 436 


351 895 816 


26.570 6605 


8.904 3366 


.001 416 431 


707 


499 849 


353 393 243 


26.589 4716 


8.908 5387 


.001 414 427 


708 


501264 


354 894 912 


26.608 2694 


8.912 7369 


.001 412 429 


709 


502 681 


356 400 829 


26.627 0539 


8.916 9311 


.001 410 437 


710 


504 100 


357 911 000 


26,645 8252 


8.921 1214 


.001 408 451 


711 


505 521 


359 425 431 


26.664 5833 


8.925 3078 


.001 406 470 


712 


506 944 


360 944 128 


26.683 3281 


8.929 4902 


.001 404 494 


713 


508 369 


362 467 097 


26.702 0598 


8.933 6687 


.001 402 525 


714 


509 796 


363 994 344 


26.720 7784 


8.937 8433 


.001 400 560 


715 


511 225 


-365 525 875 


26.739 4839 


8.942 0140 


.001 398 601 


716 


512 656 


367 061 696 


26.758 1763 


8.946 1809 


.001 396 648 


717 


514 089 


368 601 813 


26.776 8557 


8.950 3438 


.001 394 700 


718 


515 524 


370 146 232 


26.795 5220 


8.954 5029 


.001 392 758 


719 


516 961 


371 694 959 


26.814 1754 


8.958 6581 


.001 390 821 


720 


518 400 


373 248 000 


26.832 8157 


8.962 8095 


.001 388 889 


721 


519 841 


374 805 361 


26.851 4432 


8.966 9570 


.001 386 963 


722 


521284 


376 367 048 


26.870 0577 


8.971 1007 


.001 385 042 


723 


522 729 


377 933 067 


26.888 6593 


8.975 2406 


.001 383 126 


724 


524 176 


379 503 424 


26.907 2481 


8.979 3766 


.001 381 215 


725 


525 625 


381 078 125 


26.925 8240 


8.983 5089 


.001 379 310 


726 


527 076 


382 657 176 


26.944 3872 


8.987 6373 


.001 377 410 


727 


528 529 


384 240 583 


26.962 9375 


8.991 7620 


.001 375 516 


728 


529 984 


385 828 352 


26.981 4751 


8.995 8899 


.001 373 626 







Powers 


and Roots. 




39 




Squares. 


Cubes. 








Number. 


^Roots. 


f Roots. 


Reciprocals. 


729 


531 441 


387 420 489 


27.000 0000 


9.000 0000 


.001 371 742 


730 


532 900 


389 017 000 


27.018 5122 


9.004 1134 


.001 369 863 


731 


534 361 


390 617 891 


27.037 0117 


9.008 2229 


.001 367 989 


732 


535 824 


392 223 168 


27.055 4985 


9.012 3288 


.001 366 120 


733 


537 289 


393 832 837 


27.073 9727 


9.016 4309 


.001 364 256 


734 


538 756 


395 446 904 


27.092 4344 


9.020 5293 


.001 362 398 


735 


540 225 


397 065 375 


27.110 8834 


9.024 6239 


.001 360 544 


736 


541 696 


398 688 256 


27.129 3199 


9.028 7149 


.001 358 696 


737 


543 169 


400 315 553 


27.147 7149 


9.032 8021 


.001 356 8j2 


738 


544 644 


401 947 272 


27.166 1554 


9.036 8857 


.001 355 014 


739 


546121 


403 583 419 


27.184 5544 


9.040 9655 


.001 353 180 


740 


547 600 


405 224 000 


27.202 9140 


9.045 0419 


.001 351 351 


741 


549 081 


406 869 021 


27.221 3152 


9.049 1142 


.001 349 528 


742 


550 564 


408 518 488 


27.239 6769 


9.053 1831 


.001 347 709 


743 


552 049 


410 172 407 


27.258 0263 


9.057 2482 


.001 345 895 


744 


553 536 


411 830 784 


27.276 3634 


9.061 3098 


.001 344 086 


745 


555 025 


413 493 625 


27.294 6881 


9.065 3677 


.001 342 282 


746 


556 516 


415 160 936 


27.313 0006 


9.069 4220 


.001 340 483 


747 


558 009 


416 832 723 


27.331 3007 


9.073 4726 


.001 338 688 


748 


559 504 


418 508 992 


27.349 5887 


9.077 5197 


.001 336 898 


749 


561 001 


420 189 749 


27.367 8644 


9.081 5631 


.001 335 113 


750 


562 500 


421 875 000 


27.386 1279 


9.085 6030 


.001 333 333 


751 


564 001 


423 564 751 


27.404 3792 


9.089 6352 


.001 331 558 


752 


565 504 


425 259 008 


27.422 6184 


9.093 6719 


.001 329 787 


753 


567 009 


426 957 777 


27.440 8455 


9.097 7010 


.001 328 021 


754 


568 516 


428 661 064 


27.459 0604 


9.101 7265 


.001 326 260 


755 


570 025 


430 368 875 


27.477 2633 


9.105 7485 


.001 324 503 


756 


571 536 


432 081 216 


27.495 4542 


9.109 7669 


.001 322 751 


757 


573 049 


433 798 093 


27.513 6330 


9.113 7818 


.001 321 004 


758 


574 564 


435 519 512 


27.531 7998 


9.117 7931 


.001 319 261 


759 


576 081 


437 245 479 


27.549 9546 


9.121 8010 


.001 317 523 


760 


577 600 


438 976 000 


27.568 0975 


9.125 8053 


.001 315 789 


761 


579 121 


440 711 081 


27.586 2284 


9.129 8061 


.001 314 060 


762 


580 644 


442 450 728 


27.604 3475 


9.133 8034 


.001 312 336 


763 


582 169 


444 194 947 


27.622 4546 


9.137 7971 


.001 310 616 


764 


583 696 


445 943 744 


27.640 5499 


9.141 7874 


.001 308 901 


765 


585 225 


447 697 125 


27.658 6334 


9.145 7742 


.001 307 190 


766 


586 756 


449 455 096 


27.676 7050 


9.149 7576 


.001 305 483 


767 


588 289 


451 217 663 


27.694 7648 


9.153 7375 


.001 303 781 


768 


589 824 


452 984 832 


27.712 8129 


9.157 7139 


.001 302 083 


769 


591 361 


454 756 609 


27.730 8492 


9.161 6869 


.001 300 390 


770 


592 900 


456 533 000 


27.748 8739 


9.165 6565 


.001 298 701 


771 


594 441 


458 314 Oil 


27.766 8868 


9.169 6225 


.001 297 017 


772 


595 984 


460 099 648 


. 27.784 8880 


9.173 5852 


.001 295 337 


773 


597 529 


461 889 917 


27.802 8775 


9.177 5445 


.001 293 661 


774 


599 076 


463 684 824 


27.820 8555 


9.181 5003 


.001 291 990 


775 


600 625 


465 484 375 


27.838 8218 


9.185 4527 


.001 290 323 


776 


602 176 


467 288 576 


27.856 7766 


9.189 4018 


.001 288 660 


777 


603 729 


469 097 433 


27.874 7197 


9.193 3474 


.001 287 001 


778 


605 284 


470 910 952 


27.892 6514 


9.197 2897 


.001 285 347 


779 


606 841 


472 729 139 


27.910 5715 


9.201 2286 


.001 283 697 


780 


608 400 


474 552 000 


27.928 4801 


9.205 1641 


.001 282 051 



40 




Powers 


and Roots 








Squares. 


Cubes. 








Number. 


^Koots. 


f Boots. 


Reciprocals. 


781 


609 961 


476 379 541 


27.946 3772 


9.209 0962 


.001 280 410 


782 


611524 


478 211 768 


27.964 2629 


9.213 0250 


.001 278 772 


783 


613 089 


480 048 687 


27.982 1372 


9.216 9505 


.001 277 139 


784 


614 656 


481 890 304 


28.000 0000 


9.220 8726 


.001 275 510 


785 


616 225 


483 736 625 


28.017 8515 


9.224 7914 


.001 273 885 


786 


617 796 


485 587 656 


28.035 6915 


9.228 7068 


.001 272 265 


787 


619 369 


487 443 403 


28.053 5203 


9.232 6189 


.001 270 648 


788 


620 944 


489 303 872 


28.071 3377 


9.236 5277 


.001 269 036 


789 


622 521 


491 169 069 


28.089 1438 


9.240 4333 


.001 267 427 


790 


624 100 


493 039 000 


28.106 9386 


9.244 3355 


.001 265 823 


791 


625 681 


494 913 671 


28.124 7222 


9.248 2344 


.001 264 223 


792 


627 264 


496 793 088 


28.142 4946 


9.252 1300 


.001 262 626 


793 


628 849 


498 677 257 


28.160 2557 


9.256 0224 


.001 261 034 


794 


630 436 


500 566 184 


28.178 0056 


9.259 9114 


.001 259 446 


795 


632 025 


502 459 875 


28.195 7444 


9.263 7973 


.001 257 862 


796 


633 616 


504 358 336 


28.213 4720 


9.267 6798 


.001 256 281 


797 


635 209 


506 261 573 


28.231 1884 


9.271 5592 


.001 254 705 


798 


636 804 


508 169 592 


28.248 8938 


9.275 4352 


.001 253 133 


799 


638 401 


510 082 399 


28.266 5881 


9.279 3081 


.001 251 564 


800 


640 000 


512 000 000 


28.284 2712 


9.283 1777 


.001 250 000 


801 


641 601 


513 922 401 


28.301 9434 


9.287 0444 


.001 248 439 


802 


643 204 


515 849 608 


28.319 6045 


9.290 9072 


.001 246 883 


803 


644 809 


517 781 627 


28.337 2546 


9.294 7671 


.001 245 330 


804 


646 416 


519 718 464 


28.354 8938 


9.298 6239 


.001 243 781 


805 


648 025 


521 660 125 


28.372 5219 


9.302 4775 


.001 242 236 


806 


649 636 


523 606 616 


28.390 1391 


9.306 3278 


.001 240 695 


807 


651 249 


525 557 943 


28.407 7454 


9.310 1750 


.001 239 157 


808 


652 864 


527 514 112 


28.425 3408 


9.314 0190 


.001 237 624 


809 


654 481 


529 475 129 


28.442 9253 


9.317 8599 


.001 236 094 


810 


656 100 


531 441 000 


28.460 4989 


9.321 6975 


.001 234 568 


811 


657 721 


533 411 731 


28.478 0617 


9.325 5320 


.001 233 046 


812 


659 344 


535 387 328 


28.495 6137 


9.329 3634 


.001 231 527 


813 


660 969 


537 367 797 


28.513 1549 


9.333 1916 


.001 230 012 


814 


662 596 


539 353 144 


28.530 6852 


9.337 0167 


.001 228 501 


815 


664 225 


541 343 375 


28.548 2048 


9.340 8386 


.001 226 994 


816 


665 856 


543 338 496 


28.565 7137 


9.344 6575 


.001 225 499 


817 


667 489 


545 338 513 


28.583 2119 


9.348 4731 


.001 223 990 


818 


669 124 


547 343 432 


28.600 6993 


9.352 2857 


.001 222 494 


819 


670 761 


549 353 259 


28.618 1760 


9.356 0952 


.001 221 001 


820 


672 400 


551 368 000 


28.635 6421 


9.359 9016 


.001 219 512 


821 


674 041 


553 387 661 


28.653 0976 


9.363 7049 


.001 218 027 


822 


675 684 


555 412 248 


28.670 5424 


9.367 5051 


.001 216 545 


823 


677 329 


557 441 767 


28.687 9716 


9.371 3022 


.001 215 067 


824 


678 976 


559 476 224 


28.705 4002 


9.375 0963 


.001 213 592 


825 


680 625 


561 515 625 


28.722 8132 


9.378 8873 


.001 212 121 


826 


682 276 


563 559 976 


28.740 2157 


9.382 6752 


.001 210 654 


827 


683 929 


565 609 283 


28.757 6077 


9.386 4600 


.001 209 190 


828 


685 584 


567 663 552 


28.774 9891 


9.390 2419 


.001 207 729 


829 


687 241 


569 722 789 


28.792 3601 


9.394 0206 


.001 206 273 


830 


688 900 


571 787 000 


28.809 7206 


9.397 7964 


.001 204 819 


831 


690 561 


573 856 191 


28.827 0706 


9.401 5691 


.001 203 369 


832 


692 224 


575 930 368 


28.844 4102 


9.405 3387 


.001 201 923 







Powers 


and Roots 




41 




Squares. 


Cubes. 








Number. 


^Roots. 


f Roots. 


Reciprocals. 


833 


693 889 


578 009 537 


28.861 7394 


9.409 1054 


.001 200 480 


834 


695 556 


580 093 704 


28.879 0582 


9.412 8690 


.001 199 041 


835 


697 225 


582 182 875 


28.896 3666 


9.416 6297 


.001 197 605 


836 


698 896 


584 277 056 


28.913 6646 


9.420 3873 


.001 196 172 


837 


700 569 


586 376 253 


28.930 9523 


9.424 1420 


.001 194 743 


838 


702 244 


588 480 472 


28.948 2297 


9.427 8936 


.001 193 317 


839 


703 921 


590 589 719 


28.965 4967 


9.431 6423 


.001 191 895 


840 


705 600 


592 704 000 


28.982 7535 


9.435 3800 


.001 190 476 


841 


707 281 


594 823 321 


29.000 0000 


9.439 1307 


.001 189 061 


842 


708 964 


596 947 688 


29.017 2363 


9.442 8704 


.001 187 648 


843 


710 649 


599 077 107 


29.034 4623 


9.446 6072 


.001 186 240 


844 


712 336 


601 211 584 


29.051 6781 


9.450 3410 


.001 184 834 


845 


714 025 


603 351 125 


29.068 8837 


9.454 0719 


.001 183 432 


846 


715 716 


605 495 736 


29.086 0791 


9.457 7999 


.001 182 033 


847 


717 409 


607 645 423 


29.103 2644 


9.461 5249 


.001 180 638 


848 


719 104 


609 800 192 


29.120 4396 


9.465 2470 


.001 179 245 


849 


720 801 


611 960 049 


29.137 6046 


9.468 9661 


.001 177 856 


850 


722 500 


614 125 000 


29.154 7595 


9.472 6824 


.001 176 471 


851 


724 201 


616 295 051 


29.171 9043 


9.476 3957 


.001 175 088 


852 


725 904 


618 470 208 


29.189 0390 


9.480 1061 


.001 173 709 


853 


727 609 


620 650 477 


29.206 1637 


9.483 8136 


.001 172 333 


854 


729 316 


622 835 864 


29.223 2784 


9.487 5182 


.001 170 960 


855 


731 025 


625 026 375 


29.240 3830 


9.491 2200 


.001 169 591 


856 


732 736 


627 222 016 


29.257 4777 


9.494 9188 


.001 168 224 


857 


734 449 


629 422 793 


29.274 5623 


9.498 6147 


.001 166 861 


858 


736 164 


631 628 712 


29.291 6370 


9.502 3078 


.001 165 501 


859 


737 881 


633 839 779 


29.308 7018 


9.505 9980 


.001 164 144 


860 


739 600 


636 056 000 


29.325 7566 


9.509 6854 


.001 162 791 


861 


741 321 


638 277 381 


29.342 8015 


9.513 3699 


.001 161 440 


862 


743 044 


640 503 928 


29.359 8365 


9.517 0515 


.001 160 093 


863 


744 769 


642 735 647 


29.376 8616 


9.520 7303 


.001 158 749 


864 


746 496 


644 972 544 


29.393 8769 


9.524 4063 


.001 157 407 


865 


748 225 


647 214 625 


29.410 8823 


9.528 0794 


.001 156 069 


866 


749 956 


649 461 896 


29.427 8779 


9.531 7497 


.001 154 734 


867 


751 689 


651 714 363 


29.444 8637 


9.535 4172 


.001 153 403 


868 


753 424 


653 972 032 


29.461 8397 


9.539 0818 


.001 152 074 


869 


755 161 


656 234 909 


29.478 8059 


9.542 7437 


.001 150 748 


870 


756 900 


658 503 000 


29.495 7624 


9.546 4027 


.001 149 425 


871 


758 641 


660 776 311 


29.512 7091 


9.550 0589 


.001 148 106 


872 


760 384 


663 054 848 


29.529 6461 


9.553 7123 


.001 146 789 


873 


762 129 


665 338 617 


29.546 5734 


9.557 3630 


.001 145 475 


874 


763 876 


667 627 624 


29.563 4910 


9.561 0108 


.001 144 165 


875 


765 625 


669 921 875 


29.580 3989 


9.564 6559 


.001 142 857 


876 


767 376 


672 221 376 


29.597 2972 


9.568 2782 


.001 141 553 


877 


769 129 


674 526 133 


29.614 1858 


9.571 9377 


.001 140 251 


878 


770 884 


676 836 152 


29.631 0648 


9.575 5745 


.001 138 952 


879 


772 641 


679 151 439 


29.647 9342 


9.579 2085 


.001 137 656 


880 


774 400 


681 472 000 


29.664 7939 


9.582 8397 


.001 136 364 


881 


776 161 


683 797 841 


29.681 6442 


9.586 4682 


.001 135 074 


882 


777 924 


686 128 968 


29.698 4848 


9.590 0937 


.001 133 787 


883 


779 689 


688 465 387 


29.715 3159 


9.593 7169 


.001 132 503 


884 


781 456 


690 807 104 


29.732 1375 


9.597 3373 


.001 131 222 



42 



Powers and Roots. 





Squares. 


Cubes. 


V Roots. 






Number. 


f Roots. 


Reciprocals. 


885 


783 225 


693 154 125 


29.748 9496 


9.600 9548 


.001129 944 


886 


784 996 


695 506 456 


29.765 7521 


9.604 5696 


.001 128 668 


887 


786 769 


697 864 103 


29.782 5452 


9.608 1817 


.001 127 396 


888 


788 544 


700 227 072 


29.799 3289 


9.611 7911 


.001 126 126 


889 


790 321 


702 595 369 


29.816 1030 


9.615 3977 


.001 124 859 


890 


792100 


704 969 000 


29.832 8678 


9.619 0017 


.001 123 596 


891 


793 881 


707 347 971 


29.849 6231 


9.622 6030 


.001 122 334 


892 


795 664 


707 932 288 


29.866 3690 


9.626 2016 


.001 121 076 


893 


797 449 ■ 


712 121 957 


29.883 1056 


9.629 7975 


.001 119 821 


894 


799 236 


714 516 984 


29.899 8328 


9.633 3907 


.001 118 568 


895 


801 025 


716 917 375 


29.916 5506 


9.636 9812 


.001 117 818 


896 


802 816 


719 323 136 


29.933 2591 


9.640 5690 


.001 116 071 


897 


804 609 


721 734 273 


29.949 9583 


9.644 1542 


.001 114 827 


898 


806 404 


724 150 792 


29.966 6481 


9.647 7367 


.001 113 586 


899 


808 201 


726 572 699 


29.983 3287 


9.651 3166 


.001 112 347 


900 


810 000 


729 000 000 


30.000 0000 


9.654 8938 


.001 111 111 


901 


811 801 


731 432 701 


30.016 6621 


9.658 4684 


.001 109 878 


902 


813 604 


733 870 808 


30.033 3148 


9.662 0403 


.001 108 647 


903 


815 409 


736 314 327 


30.049 9584 


9.665 6096 


.001 107 420 


904 


817 216 


738 763 264 


30.066 5928 


9.669 1762 


.001 106 195 


905 


819 025 . 


741 217 625 


30.083 2179 


9.672 7403 


.001 104 972 


906 


820 836 


743 677 416 


30.099 8339 


9.676 3017 


.001 103 753 


907 


822 649 


746 142 643 


30.116 4407 


9.679 8604 


.001 102 536 


908 


824 464 


748 613 312 


30.133 0383 


9.683 4166 


.001 101 322 


909 


826 281 


751 089 429 


30.149 6269 


9.686 9701 


.001 100 110 


910 


828 100 


753 571 000 


30.166 2063 


9.690 5211 


.001 098 901 


911 


829 921 


756 058 031 


30.182 7765 


9.694 0694 


.001 097 695 


912 


831 744 


758 550 828 


30.199 3377 


9.697 6151 


.001 096 491 


913 


833 569 


761 048 497 


30.215 8899 


9.7011583 


.001 095 290 


914 


835 396 


763 551 944 


30.232 4329 


9.704 6989 


.001 094 092 


915 


837 225 


766 060 875 


30.248 9669 


9.708 2369 


.001 092 896 


916 


839 056 


768 575 296 


30.265 4919 


9.711 7723 


.001 091 703 


917 


840 889 


771 095 213 


30.282 0079 


9.715 3051 


.001 090 513 


918 


842 724 


773 620 632 


30.298 5148 


9.718 8354 


.001 089 325 


919 


844 561 


776 151 559 


30.315 0128 


9.722 3631 


.001 088 139 


920 


846 400 


778 688 000 


30.331 5018 


, 9.725 8883 


.001 086 957 


921 


848 241 


781 229 961 


30.347 9818 


9.729 4109 


.001 085 776 


922 


850 084 


783 777 448 


30.364 4529 


9.732 9309 


.001 084 599 


923 


851 929 


786 330 467 


30.380 9151 


9.736 4484 


.001 083 423 


924 


853 776 


788 889 024 


30.397 3683 


9.739 9634 


.001 082 251 


925 


855 625 


791 453 125 


30.413 8127 


9.743 4758 


.001 081 081 


926 


857 476 


794 022 776 


30.430 2481 


9.746 9857 


.001 079 914 


927 


859 329 


796 597 983 


30.446 6747 


9.750 4930 


.001 078 749 


928 


861 184 


799 178 752 


30.463 0924 


9.753 9979 


.001 077 586 


929 


863 041 


801 765 089 


30.479 5013 


9.757 5002 


.001 076 426 


930 


864 900 


804 357 000 


30.495 9014 


9.761 0001 


.001 075 269 


931 


866 761 


806 954 491 


30.512 2926 


9.764 4974 


.001 074 114 


932 


868 624 


809 557 568 


30.528 6750 


9.767 9922 


.001 072 961 


933 


870 489 


812 166 237 


30.545 0487 


9.771 4845 


.001 071 811 


934 


872 356 


814 780 504 


30.561 4136 


9.774 9743 


.001 070 664 


935 


874 225 


817 400 375 


30.577 7697 


9.778 4616 


.001 069 519 


936 


876 096 


820 025 856 


30.594 1171 


9.781 9466 


.001 068 376 







Powers 


and Roots 




43 




Squares. 


Cubes. 








Number. 


V Boots. 


f Roots. 


Reciprocals. 


937 


877 969 


822 656 953 


30.610.4557 


9.785 4288 


.001 067 236 


938 


879 844 


825 293 672 


30.626 7857 


9.788 9087 


.001 066 098 


939 


881 721 


827 936 019 


30.643 1069 


9.792 3861 


.001 064 963 


940 


883 600 


830 584 000 


30.659 4194 


9.795 8611 


.001 063 830 


941 


885 481 


833 237 621 


30.675 7233 


9.799 3336 


.001 062 699 


942 


887 364 


835 896 888 


30.692 0185 


9.802 8036 


.001061571 


943 


889 249 


838 561 807 


30.708 3051 


9.806 2711 


.001 060 445 


944 


891 136 


841 232 384 


30.724 5830 


9.809 7362 


.001 059 322 


945 


893 025 


843 908 625 


30.740 8523 


9.813 1989 


.001 058 201 


946 


894 916 


846 590 536 


30.757 1130 


9.816 6591 


.001 057 082 


947 


896 809 


849 278 123 


30.773 3651 


9.820 1169 


.001 055 966 


948 


898 704 


851 971 392 


30.789 6086 


9.823 5723 


.001 054 852 


949 


900 601 


854 670 349 


30.805 8436 


9.827 0252 


.001 053 741 


950 


902 500 


857 375 000 


30.822 0700 


9.830 4757 


.001 052 632 


951 


904 401 


860 085 351 


30.838 2879 


9.833 9238 


.001 051 525 


952 


906 304 


862 801 408 


30.854 4972 


9.837 3695 


.001 050 420 


953 


908 209 


865 523 177 


30.870 6981 


9.840 8127 


.001 049 318 


954 


910 116 


868 250 664 


30.886 8904 


9.844 2536 


.001 048 218 


955 


912 025 


870 983 875 


30.903 0743 


9.847 6920 


.001 047 120 


956 


913 936 


873 722 816 


30.919 2477 


9.851 1280 


.001 046 025 


957 


915 849 


876 467 493 


30.935 4166 


9.854 5617 


.001 044 932 


958 


917 764 


879 217 912 


30.951 5751 


9.857 9929 


.001 043 841 


959 


919 681 


881 974 079 


30.967 7251 


9.861 4218 


.001 042 753 


960 


921 600 


884 736 000 


30.983 8668 


9.864 8483 


.001 041 667 


961 


923 521 


887 503 681 


31.000 0000 


9.868 2724 


.001 040 583 


962 


925 444 


890 277 128 


31.016 1248 


9.871 6941 


.001 039 501 


963 


927 369 


893 056 347 


31.032 2413 


9.875 1135 


.001 038 422 


964 


929 296 


895 841 344 


31.048 3494 


9.878 5305 


.001 037 344 


965 


931 225 


898 632 125 


31.064 4491 


9.881 9451 


.001 036 269 


966 


933 156 


901 428 696 


31.080 5405 


9.885 3574 


.001 035 197 


967 


935 089 


904 231 063 


31.096 6236 


9.888 7673 


.001 034 126 


968 


937 024 


907 039 232 


31.112 6984 


9.892 1749 


.001 033 058 


969 


938 961 


909 853 209 


31.128 7648 


9.895 5801 


.001 031 992 


970 


940 900 


912 673 000 


31.144 8230 


9.898 9830 


.001 030 928 


971 


942 841 


915 498 611 


31.160 8729 


9.902 3835 


.001 029 866 


972 


944 784 


918 330 048 


31.176 9145 


9.905 7817 


.001 028 807 


973 


946 729 


921 167 317 


31.192 9479 


9.909 1776 


.001 027 749. 


974 


948 676 


924 010 424 


31.208 9731 


9.912 5712 


.001 026 694 


975 


950 625 


926 859 375 


31.224 9900 


9.915 9624 


.001 025 641 


976 


952 576 


929 714 176 


31.240 9987 


9.919 3513 


.001 024 590 


977 


954 529 


932 574 833 


31.256 9992 


9.922 7379 


.001 023 541 


978 


956 484 


935 441 352 


31.272 9915 


9.926 1222 


.001 022 495 


979 


958 441 


938 313 739 


31.288 9757 


9.929 5042 


.001 021 450 


980 


960 400 


941 192 000 


31.304 9517 


9.932 8839 


.001 020 408 


981 


962 361 


944 076 141 


31.320 9195 


9.936 2613 


.001 019 168 


982 


964 324 


946 966 168 


31.336 8792 


9.939 6363 


.001 018 330 


983 


966 289 


949 862 087 


31.352 8308 


9.943 0092 


.001 017 294 


984 


968 256 


952 763 904 


31.368 7743 


9.946 3797 


.001 016 260 


985 


970 225 


955 671 625 


31.384 7097 


9.949 7479 


.001 015 228 


986 


972 196 


958 585 256 


31.400 6369 


9.953 1138 


.001 014 199 


987 


974 169 


961 504 803 


31.416 5561 


9.956 4775 


.001 013 171 


988 


976 144 


964 430 272 


31.432 4673 


9.959 8389 


.001 012 146 



44 




Powers 


and Roots 








Squares. 


Cubes. 


V Roots. 






Number. 


f Roots. 


Reciprocals. 


989 


978 121 


967 361 669 


31.448 3704 


9.963 1981 


.001 011 122 


990 


980100 


970 299 000 


31.464 2654 


9.966 5549 


.001 010 101 


991 


982 081 


973 242 271 


31.480 1525 


9.969 9055 


.001 009 082 


992 


984 064 


976 191 488 


31.496 0315 


9.973 2619 


.001 008 065 


993 


986 049 


979 146 657 


31.511 9025 


9.976 6120 


.001 007 049 


994 


988 036 


982 107 784 


31.527 7655 


9.979 9599 


.001 006 036 


995 


990 025 


985 074 875 


31.543 6206 


9.983 3055 


.001 005 025 


996 


992 016 


988 047 936 


31.559 4677 


9.986 6488 


.001 004 016 


997 


994 009 


991 026 973 


31.575 3068 


9.989 9900 


.001 003 009 


998 


996 004 


994 Oil 992 


31.591 1380 


9.993 3289 


.001 002 004 


999 


998 001 


997 002 999 


31.606 9613 


9.996 6656 


.001 001 001 


1000 


1000 000 


1 000 000 000 


31.622 7766 


10.000 0000 


.001 000 000 


1001 


1 002 001 


1 003 003 001 


31.638 5840 


10.003 3222 


.000 999 0010 


1002 


1 004 004 


1 006 012 008 


31.654 3866 


10.006 6622 


.000 998 0040 


1003 


1 006 009 


1 009 027 027 


31.670 1752 


10.009 9899 


.000 997 0090 


1004 


1 008 016 


1 012 048 064 


31.685 9590 


10.013 3155 


.000 996 0159 


1005 


1 010 025 


1 015 075 125 


31.701 7349 


10.016 6389 


.000 995 0249 


1006 


1 012 036 


1 018 108 216 


31.717 5030 


10.019 9601 


.000 994 0358 


1007 


1 014 049 


1 021 147 343 


31.733 2633 


10.023 2791 


.000 993 0487 


1008 


1 016 064 


1 024 192 512 


31.749 0157 


10.026 5958 


.000 992 0635 


1009 


1 018 081 


1 027 243 729 


31.764 7603 


10.029 9104 


.000 991 0803 


1010 


1 020 100 


1 030 301 000 


31.780 4972 


10.033 2228 


.000 990 0990 


1011 


1 022 121 


1 033 364 331 


31.796 2262 


10.036 5330 


.000 989 1197 


1012 


1024144 


1 036 433 728 


31.811 9474 


10.039 8410 


.000 988 1423 


1013 


1 026 169 


1 039 509 197 


31.827 6609 


10.043 1469 


.000 987 1668 


1014 


1 028 196 


1 042 590 744 


31.843 3666 


10.046 4506 


.000 986 1933 


1015 


1 030 225 


1 045 678 375 


31.859 0646 


10.049 7521 


.000 985 2217 


1016 


1 032 256 


1 048 772 096 


31.874 7549 


10.053 0514 


.000 984 2520 


1017 


1 034 289 


1 051 871 913 


31.890 4374 


10.056 3485 


.000 983 2842 


1018 


1 036 324 


1 054 977 832 


31.906 1123 


10.059 6435 


.000 982 3183 


1019 


1 038 361 


1 058 089 859 


31.921 7794 


10.062 9364 


.000 981 3543 


1020 


1 040 400 


1 061 208 000 


31.937 4388 


10.066 2271 


.000 980 3922 


1021 


1 042 441 


1 064 332 261 


31.953 0906 


10.069 5156 


.000 979 4319 


1022 


1 044 484 


1 067 462 648 


31.968 7347 


10.072 8020 


.000 978 4736 


1023 


1 046 529 


1 070 599 167 


31.984 3712 


10.076 0863 


.000 977 5171 


1024 


1 048 576 


1 073 741 824 


32.000 0000 


10.079 3684 


.000 976 5625 


1025 


1 050 625 


1 076 890 625 


32.015 6212 


10.082 6484 


.000 975 6098 


1026 


1 052 676 


1 080 045 576 


32.031 2348 


10.085 9262 


.000 974 6589 


1027 


1 054 729 


1 083 206 683 


32.046 8407 


10.089 2019 


.000 973 7098 


1028 


1 056 784 


1 086 373 952 


32.062 4391 


10.092 4755 


.000 972 7626 


1029 


1058.841 


1 089 547 389 


32.078 0298 


10.095 7469 


.000 971 8173 


1030 


1 060 900 


1 092 727 000 


32.093 6131 


10.099 0163 


.000 970 8738 


1031 


1 062 961 


1 095 912 791 


32.109 1887 


10.102 2835 


.000 969 9321 


1032 


1 065 024 


1 099 104 768 


32.124 7568 


10.105 5487 


.000 968 9922 


1033 


1 067 089 


1 102 302 937 


32.140 3173 


10.108 8117 


.000 968 0542 


1034 


1 069 156 


1 105 507 304 


32.155 8704 


10.112 0726 


.000 967 1180 


1035 


1 071 225 


1 108 717 875 


32.171 4159 


10.115 3314 


.000 966 1836 


1036 


1 073 296 


1 111 934 656 


32.186 9539 


10.118 5882 


.000 965 2510 


1037 


1 075 369 


1 115 157 653 


32.202 4844 


10.121 8428 


.000 964 3202 


1038 


1 077 444 


1 118 386 872 


32.218 0074 


10.125 0953 


.000 963 3911 


1039 


1 079 521 


1 121 622 319 


32.233 5229 


10.128 3457 


.000 962 4639 


1040 


1 081 600 


1 124 864 000 


32.249 0310 


10.131 5941 


.000 961 5385 







POWEES 


and Roots 




45 












Number. 


Squares. 


Cubes. 


/Roots. 


f Roots. 


Reciprocals. 


1041 


1 083 681 


1 128 111 921 


32.264.5316 


10.134 8403 


.000 960 6148 


1042 


1 085 764 


1 131 366 088 


32.280 0248 


10.138 0845 


.000 959 6929 


1043 


1 087 849 


1 134 626 507 


32.295 5105 


10.141 3266 


.000 958 7728 


1044 


1 089 936 


1 137 893 184 


32.310 9888 


10.144 5667 


.000 957 8544 


1045 


1 092 025 


1 141 166 125 


32.326 4598 


10.147 8047 


.000 956 9378 


1046 


1 094 116 


1 144 445 336 


32.341 9233 


10.151 0406 


.000 956 0229 


1047 


1 096 209 


1 147 730 823 


32.357 3794 


10.154 2744 


.000 955 1098 


1048 


1 098 304 


1 151 022 592 


32.372 8281 


10.157 5062 


.000 954 1985 


1049 


1 100 401 


1 154 320 649 


32.388 2695 


10.160 7359 


.000 953 2888 


1050 


1 102 500 


1 157 625 000 


32.403 7035 


10.163 9636 


.000 952 3810 


1051 


1 104 601 


1 160 935 651 


32.419 1301 


10.167 1893 


.000 951 4748 


1052 


1 106 704 


1 164 252 608 


32.434 5495 


10.170 4129 


.000 950 5703 


1053 


1 108 809 


1 167 575 877 


32.449 9615 


10.173 6344 


.000 949 6676 


1054 


1 110 916 


1 170 905 464 


32.465 3662 


10.176 8539 


.000 948 7666 


1055 


1 113 025 


1 174 241 375 


32.480 7635 


10.180 0714 


.000 947 8673 


1056 


1 115 136 


1 177 583 616 


32.496 1536 


10.183 2868 


.000 946 9697 


1057 


1 117 249 


1 180 932 193 


32.511 5364 


10.186 5002 


.000 946 0738 


1058 


1 119 364 


1 184 287 112 


32.526 9119 


10.189 7116 


.000 945 1796 


1059 


1 121 481 


1 187 648 379 


32.542 2802 


10.192 9209 


.000 944 2871 


1060 


1 123 600 


1 191 016 000 


32.557 6412 


10.196 1283 


.000 943 3962 


1061 


1 125 721 


1 194 389 981 


32.572 9949 


10.199 3336 


.000 942 5071 


1062 


1 127 844 


1 197 770 328 


32.588 3415 


10.202 5369 


.000 941 6196 


1063 


1 129 969 


1 201 157 047 


32.603 5807 


10.205 7382 


.000 940 7338 


1064 


1 132 096 


1 204 550 144 


32.619 0129 


10.208 9375 


.000 939 8496 


1065 


1 134 225 


1 207 949 625 


32.634 3377 


10.212 1347 


.000 938 9671 


1066 


1 136 356 


1 211 355 496 


32.649 6554 


10.215 3300 


.000 938 0863 


1067 


1 138 489 


1 214 767 763 


32.664 9659 


10.218 5233 


.000 937 2071 


1068 


1 140 624 


1 218 186 432 


32.680 2693 


10.221 7146 


.000 936 3296 


1069 


1 142 761 


1 221 611 509 


32.695 5654 


10.224 9039 


.000 935 4537 


1070 


1 144 900 


1 225 043 000 


32.710 8544 


10.228 0912 


.000 934 5794 


1071 


1 147 041 


1 228 480 911 


32.726 1363 


10.231 2766 


.000 933 7068 


1072 


1 149 184 


1 231 925 248 


32.741 4111 


10.234 4599 


.000 932 8358 


1073 


1 151 329 


1 235 376 017 


32.756 6787 


10.237 6413 


.000 931 9664 


1074 


1 153 476 


1 238 833 224 


32.771 9392 


10.240 8207 


.000 931 0987 


1075 


1 155 625 


1 242 296 875 


32.787 1926 


10.243 9981 


.000 930 2326 


1076 


1 157 776 


1 245 766 976 


32.802 4398 


10.247 1735 


.000 929 3680 


1077 


1 159 929 


1 249 243 533 


32.817 6782 


10.250 3470 


.000 928 5051 


1078 


1 162 084 


1 252 726 552 


32.832 9103 


10.253 5186 


.000 927 6438 


1079 


1 164 241 


1 256 216 039 


32.848 1354 


10.256 6881 


.000 926 7841 


1080 


1 166 400 


1 259 712 000 


32.863 3535 


10.259 8557 


.000 925 9259 


1081 


1 168 561 


1 263 214 441 


32.878 5644 


10.263 0213 


.000 925 0694 


1082 


1 170 724 


1 266 723 368 


32.893 7684 


10.266 1850 


.000 924 2144 


1083 


1 172 889 


1 270 238 787 


32.908 9653 


10.269 3467 


.000 923 3610 


1084 


1 175 056 


1 273 760 704 


32.924 1553 


10.272 5065 


.000 922 5092 


1085 


1 177 225 


1 277 289 125 


32.939 3382 


10.275 6644 


.000 921 6590 


1086 


1 179 396 


1 280 824 056 


32.954 5141 


10.278 8203 


.000 920 8103 


1087 


1 181 569 


1 284 365 503 


32.969 6830 


10.281 9743 


.000 919 9632 


1088 


1 183 744 


1 287 913 472 


32.984 8450 


10.285 1264 


.000 919 1176 


1089 


1 185 921 


1 291 467 969 


33.000 0000 


10.288 2765 


.000 918 2736 


1090 


1 188 100 


1 295 029 000 


33.015 1480 


10.291 4247 


.000 917 4312 


1091 


1190 281 


1 298 596 571 


33.030 2891 


10.294 5709 


.000 916 5903 


1092 


1 192 464 


1 302 170 688 


33.045 4233 


10.297 7153 


.000 915 7509 



46 




Powers 


and Roots 








Squares. 


Cubes. 








Number. 


^ Roots. 


f Roots. 


Reciprocals. 


1093 


1 194 649 


1 305 751 357 


33.060 5505 


10.300 8577 


.000 914 9131 


1094 


1 196 836 


1 309 338 584 


33.075 6708 


10.303 9982 


.000 914 0768 


1095 


1 199 025 


1 312 932 375 


33.090 7842 


10.307 1368 


.000 913 2420 


1096 


1 201 216 


1 316 532 736 


33.105 8907 


10.310 2735 


.000 912 4008 


1097 


1 203 409 


1 320 139 673 


33.120 9903 


10.313 4083 


.000 911 5770 


1098 


1 205 604 


1 323 753 192 


33.136 0830 


10.316 5411 


.000 910 7468 


1099 


1 207 801 


1 327 373 299 


33.151 1689 


10.319 6721 


.000 909 9181 


1100 


1 210 000 


1 331 000 000 


33.166 2479 


10.322 8012 


.000 909 0909 


1101 


1 212 201 


1 334 633 301 


33.181 3200 


10.325 9284 


.000 908 2652 


1102 


1 214 404 


1 338 273 208 


33.196 3853 


10.329 0537 


.000 907 4410 


1103 


1 216 609 


1 341 919 727 


33.211 4438 


10.332 1770 


.000 906 6183 


1104 


1 218 816 


1 345 572 864 


33.226 6955 


10.335 2985 


.000 905 7971 


1105 


1 221 025 


1 349 232 625 


33.241 5403 


10.338 4181 


.000 904 9774 


1106 


1 223 236 


1 352 899 016 


33.256 5783 


10.341 5358 


.000 904 1591 


1107 


1 225 449 


1 356 572 043 


33.271 6095 


10.344 6517 


.000 903 3424 


1108 


1 227 664 


1 360 251 712 


33.286 6339 


10.347 7657 


.000 902 5271 


1109 


1 229 881 


1 363 938 029 


33.301 6516 


10.350 8778 


.000 901 7133 


1110 


1 232 100 


1 367 631 000 


33.316 6625 


10.353 9880 


.000 900 9009 


1111 


1 234 321 


1 371 330 631 


33.331 6666 


10.357 0964 


.000 900 0900 


1112 


1 236 544 


1 375 036 928 


33.346 6640 


10.360 2029 


.000 899 2806 


1113 


1 238 769 


1 378 749 897 


33.361 6546 


10.363 3076 


.000 898 4726 


1114 


1 240 996 


1 382 469 544 


33.376 6385 


10.366 4103 


.000 897 6661 


1115 


1 243 225 


1 386 195 875 


33.391 6157 


10.369 5113 


.000 896 8610 


1116 


1 245 456 


1 389 928 896 


33.406 5862 


10.372 6103 


.000 896 0753 


1117 


1 247 689 


1 393 668 613 


33.421 5499 


10.375 7076 


.000 895 2551 


1118 


1 249 924 


1 397 415 032 


33.436 5070 


10.378 8030 


.000 894 4544 


1119 


1 252 161 


1 401 168 159 


33.451 4573 


10.381 8965 


.000 893 6550 


1120 


1 254 400 


1 404 928 000 


33.466 4011 


10.384 9882 


.000 892 8571 


1121 


1 256 641 


1 408 694 561 


33.481 3381 


10.388 0781 


.000 896 0607 


1122 


1 258 884 


1 412 467 848 


33.496 2684 


10.391 1661 


.000 892 2656 


1123 


1 261 129 


1 416 247 867 


33.511 1921 


10.394 2527 


.000 890 4720 


1124 


1 263 376 


1 420 034 624 


33.526 1092 


10.397 3366 


.000 889 6797 


1125 


1 265 625 


1 423 828 125 


33.541 0196 


10.400 4192 


.000 888 8889 


1126 


1 267 876 


1 427 628 376 


33.555 9234 


10.403 4999 


.000 888 0995 


1127 


1 270 129 


1 431 435 383 


33.570 8206 


10.406 5787 


.000 887 3114 


1128 


1 272 384 ' 


1 435 249 152 


33.585 7112 


10.409 6557 


.000 886 5248 


1129 


1 274 641 


1 439 069 689 


33.600 5952 


10.412 7310 


.000 885 7396 


1130 


1 276 900 


1 442 897 000 


33.615 4726 


10.415 8044 


.000 884 9558 


1131 


1 279 161 


1 446 731 091 


33.630 3434 


10.418 8760 


.000 884 1733 


1132 


1 281 424 


1 450 571 968 


33.645 2077 


10.421 9458 


.000 883 3922 


1133 


1 283 689 


1 454 419 637 


33.660 0653 


10.425 0138 


.000 882 6125 


1134 


1 285 956 


1 458 274 104 


33.674 9165 


10.428 0800 


.000 881 8342 


1135 


1 288 225 


1 462 135 375 


33.689 7610 


10.431 1443 


.000 881 0573 


1136 


1 290 496 


1 466 003 456 


33.704 5991 


10.434 2069 


.000 880 2S17 


1137 


1 292 769 


1 469 878 353 


33.717 4306 


10.437 2677 


.000 879 5075 


1138 


1 295 044 


1 473 760 072 


33.734 0556 


10.440 3677 


.000 878 7346 


1139 


1 297 321 


1 477 648 619 


33.749 0741 


10.443 3839 


.000 877 9631 


1140 


1 299 600 


1 481 544 000 


33.763 8860 


10.446 4393 


.000 877 1930 


1141 


1 301 881 


1 485 446 221 


33.778 6915 


10.449 4929 


.000 876 4242 


1142 


1 304 164 


1 489 355 288 


33.793 4905 


10.452 5448 


.000 875 6567 


1143 


1 306 449 


1 493 271 207 


33.808 2830 


10.455 5948 


.000 874 8906 


1144 


1 308 736 


1 497 193 984 


33.823 0691 


10.458 6431 


.000 874 1259 







Powers 


and Roots 




47 


Number. 


Squares. 


Cubes. 


I 7 Roots. 


f Roots. 


Reciprocals. 


1145 


1 311 025 


1 501 123 625 


33.837 8486 


10.461 6896 


.000 873 3624 


1146 


1 313 316 


1 505 060 136 


33.852 6218 


10.464 7343 


.000 872 6003 


1147 


1 315 609 


1 509 003 523 


33.867 3884 


10.467 7773 


.000 871 8396 


1148 


1 317 904 


1 512 953 792 


33.882 1487 


10.470 8158 


.000 871 0801 


1149 


1 320 201 


1 516 910 949 


33.896 9025 


10.473 8579 


.000 870 3220 


1150 


1 322 500 


1 520 875 000 


33.911 6499 


10.476 8955 


.000 869 5652 


1151 


1 324 801 


1 524 845 951 


33.926 3909 


10.479 9314 


.000 868 8097 


1152 


1 327 104 


1 528 823 808 


33.941 1255 


10.482 9656 


.000 868 0556 


1153 


1 329 409 


1 532 808 577 


33.955 8537 


10.485 9980 


.000 867 3027 


1154 


1 331 716 


1 536 800 264 


33.970 5755 


10.489 0286 


.000 866 5511 


1155 


1 334 025 


1 540 798 875 


33.985 2910 


10.492 0575 


.000 865 8009 


1156 


1 336 336 


1 544 804 416 


34.000 0000 


10.495 0847 


.000 865 0519 


1157 


1 338 649 


1 548 816 893 


34.014 7027 


10.498 1101 


.000 864 3042 


1158 


1 340 964 


1 552 836 312 


34.029 3990 


10.501 1337 


.000 863 5579 


1159 


1 343 281 


1 556 862 679 


34.044 0890 


10.504 1556 


.000 862 8128 


1160 


1 345 600 


1 560 896 000 


34.058 7727 


10.507 1757 


.000 862 0690 


1161 


1 347 921 


1 564 936 281 


34.073 4501 


10.510 1942 


.000 861 3264 


1162 


1 350 244 


1 568 983 528 


34.088 1211 


10.513 2109 


.000 860 5852 


1163 


1 352 569 


1 573 037 747 


34.012 7858 


10.516 2259 


.000 859 8452 


1164 


1 354 896 


1 577 098 944 


34.117 4442 


10.519 2391 


.000 859 1065 


1165 


1 357 225 


1 581 167 125 


34.132 0963 


10.522 2506 


.000 858 3691 


1166 


1 359 556 


1 585 242 296 


34.146 7422 


10.525 2604 


.000 857 6329 


1167 


1 361 889 


1 589 324 463 


34.161 3817 


10.528 2685 


.000 856 8980 


1168 


1 364 224 


1 593 413 632 


34.176 0150 


10.531 2749 


.000 856 1644 


1169 


1 366 561 


1 597 509 809 


34.190 6420 


10.534 2795 


.000 855 4320 


1170 


1 368 900 


1 601 613 000 


34.205 2627 


10.537 2825 


.000 854 7009 


1171 


1 371 241 


1 605 723 211 


34.219 8773 


10.540 2837 


.000 853 9710 


1172 


1 373 584 


1 609 840 448 


34.234 4855 


10.543 2832 


.000 853 2423 


1173 


1 375 929 


1 613 964 717 


34.249 0875 


10.546 2810 


.000 852 5149 


1174 


1 378 276 


1 618 096 024 


34.263 6834 


10.549 2771 


.000 851 7888 


1175 


1 380 625 


1 622 234 375 


34.278 2730 


10.552 2715 


.000 851 0638 


1176 


1 382 976 


1 626 379 776 


34.292 8564 


10.555 2642 


.000 850 3401 


1177 


1 385 329 


1 630 532 233 


34.307 4336 


10.558 2552 


.000 849 6177 


1178 


1 387 684 


1 634 691 752 


34.322 0046 


10.561 2445 


.000 848 8964 


1179 


1 390 041 


1 638 858 339 


34.336 5694 


10.564 2322 


.000 848 1764 


1180 


1 392 400 


1 643 032 000 


34.351 1281 


10.567 2181 


.000 847 1576 


1181 


1 394 761 


1 647 212 741 


34.365 6805 


10.570 2024 


.000 846 7401 


1182 


1 397 124 


1 651 400 568 


34.380 2268 


10.573 1849 


.000 846 0237 


1183 


1 399 489 


1 655 595 487 


34.394 7670 


10.576 1658 


.000 845 3085 


1184 


1 401 856 


1 659 797 504 


34.409 3011 


10.579 1449 


.000 844 5946 


1185 


1 404 225 


1 664 006 625 


34.423 8289 


10.582 1225 


.000 843 8819 


1186 


1 406 596 


1 668 222 856 


34.438 3507 


10.585 0983 


.000 843 1703 


1187 


1 408 969 


1 672 446 203 


34.452 8663 


10.588 0725 


.000 842 4600 


1188 


1 411 344 


1 676 676 672 


34.467 3759 


10.591 0450 


.000 841 7508 


1189 


1 413 721 


1 680 914 629 


34.481 8793 


10.594 0158 


.000 841 0429 


1190 


1 416 100 


1 685 159 000 


34.496 3766 


10.596 9850 


.000 840 3361 


1191 


1 418 481 


1 689 410 871 


34.510 8678 


10.599 9525 


.000 839 6306 


1192 


1 420 864 


1 693 669 888 


34.525 3530 


10.602 9184 


.000 838 9262 


1193 


1 423 249 


1 697 936 057 


34.539 8321 


10.605 8826 


.000 838 2320 


1194 


1 425 636 


1 702 209 384 


34.554 3051 


10.608 8451 


.000 837 5209 


1195 


1 428 025 


1 706 489 875 


34.568 7720 


10.611 8060 


.000 836 8201 


1196 


1 430 416 


1 710 777 536 


34.583 2329 


10.614 7652 


.000 836 1204 



48 




Powers 


and Roots 








Squares. 


Cubes. 








Number. 


V Roots. 


f Roots. 


Reciprocals. 


1197 


1 432 809 


1 715 072 373 


34.597 6879 


10.617 7228 


.000 835 4219 


1198 


1 435 204 


1 719 374 392 


34.612 1366 


10.620 6788 


.000 834 7245 


1199 


1 437 601 


1 723 683 599 


34.626 5794 


10.623 6331 


.000 834 0284 


1200 


1 440 000 


1 728 000 000 


34.641 0162 


10.626 5857 


.000 833 3333 


1201 


1 442 401 


1 732 323 601 


34.655 4469 


10.629 5367 


.000 832 6395 


1202 


1 444 804 


1 736 654 408 


34.669 8716 


10.632 4860 


.000 831 9468 


1203 


1 447 209 


1 740 992 427 


34.684 2904 


10.635 4338 


.000 831 2552 


1204 


1 449 616 


1 745 337 664 


34.698 7031 


10.638 3799 


.000 830 5648 


1205 


1 452 025 


1 749 690 125 


34.713 1099 


10.641 3244 


.000 829 8755 


1206 


1 454 436 


1 754 049 816 


34.727 5107 


10.644 2672 


.000 829 1874 


1207 


1 456 849 


1 758 416 743 


34.741 9055 


10.647 2085 


.000 828 5004 


1208 


1 459 264 


1 762 790 912 


34.756 2944 


10.650 1480 


.000 827 8146 


1209 


1 461 681 


1 767 172 329 


34.770 6773 


10.653 0860 


.000 827 1299 


1210 


1 464 100 


1 771 561 000 


34.785 0543 


10.656 0223 


.000 826 4463 


1211 


1466 521 


1 775 956 931 


34.799 4253 


10.658 9570 


.000 825 7638 


1212 


1 468 944 


1 780 360 128 


34.813 7904 


10.661 8902 


.000 825 0825 


1213 


1 471 369 


1 784 770 597 


34.8281495 


10.664 8217 


.000 824 4023 


1214 


1 473 796 


1 789 188 344 


34.842 5028 


10.667 7516 


.000 823 7232 


1215 


1 476 225 


1 793 613 375 


34.856 8501 


10.670 6799 


.000 823 0453 


1216 


1 478 656 


1 798 045 696 


34.871 1915 


10.673 6066 


.000 822 3684 


1217 


1 481 089 


1 802 485 313 


34.885 5271 


10.676 5317 


.000 821 6927 


1218 


1 483 524 


1 806 932 232 


34.899 8567 


10.679 4552 


.000 821 0181 


1219 


1 485 961 


1 811 386 459 


34.914 1805 


10.682 3771 


.000 820 3445 


1220 


1 488 400 


1 815 848 000 


34.928 4984 


10.685 2973 


.000 819 6721 


1221 


1 490 841 


1 820 316 861 


34.942 8104 


10.688 2160 


.000 819 0008 


1222 


1 493 284 


1 824 793 048 


34.957 1166 


10.691 1331 


.000 818 3306 


1223 


1 495 729 


1 829 276 567 


34.971 4169 


10.694 0486 


.000 817 6615 


1224 


1 498 176 


1 833 764 247 


34.985 7114 


10.696 9625 


.000 816 9935 


1225 


1 500 625 


1 838 265 625 


35.000 0000 


10.699 8748 


.000 816 3265 


1226 


1 503 276 


1842 771176 


35.014 2828 


10.702 7855 


.000 815 6607 


1227 


1 505 529 


1 847 284 083 


35.028 5598 


10.705 6947 


000 814 9959 


1228 


1 507 984 


1 851 804 352 


35.042 8309 


10.708 6023 


.000 814 3322 


1229 


1 510 441 


1 856 331 989 


35.057 0963 


10.711 5083 


.000 813 6696 


1230 


1 512 900 


1 860 867 000 


35.071 3558 


10.714 4127 


.000 813 0081 


1231 


1 515 361 


1 865 409 391 


35.085 6096 


10.717 3155 


.000 812 3477 


1232 


1 517 824 


1 869 959 168 


35.099 8575 


10.720 2168 


.000 811 6883 


1233 


1 520 289 


1 874 516 337 


35.114 0997 


10.723 1165 


.000 811 0300 


1234 


1 522 756 


1 879 080 904 


35.128 3361 


10.726 0146 


.000 810 3728 


1235 


1 525 225 


1 883 652 875 


35.142 5668 


10.728 9112 


.000 809 7166 


1236 


1 527 696 


1 888 232 256 


35.156 7917 


10.731 8062 


.000 809 0615 


1237 


1 530 169 


1892 819 053 


35.171 0108 


10.734 6997 


.000 808 4074 


1238 


1 532 644 


1 897 413 272 


35.185 2242 


10.737 5916 


.000 807 7544 


1239 


1 535 121 


1 902 014 919 


35.199 4318 


10.740 4819 


.000 807 1025 


1240 


1 537 600 


1 906 624 000 


35.213 6337 


10.743 3707 


.000 806 4516 


1241 


1 540 081 


1 911 240 521 


35.227 8299 


10.746 2579 


.000 805 8018 


1242 


1 542 564 


1 915 864 488 


35.212 0204 


10.749 1436 


.000 805 1530 


1243 


1 545 049 


1 920 495 907 


35.256 2051 


10.752 0277 


.000 804 5052 


1244 


1 547 536 


1 925 134 784 


35.270 3842 


10.754 9103^ 


.000 803 8585 


1245 


1 550 025 


1 929 781 125 


35.284 5575 


10.757 7913" 


.000 803 2129 


1246 


1 552 516 


1 934 434 936 


35.298 7252 


10.760 6708 


.000 802 5682 


1247 


1 555 009 


1 939 096 223 


35.312 8872 


10.763 5488 


.000 801 9246 


1248 


1 557 504 


1 943 764 992 


35.327 0435 


10.766 4252 


.000 801 2821 







Powers 


and Roots 




49 












Number. 


Squares. 


Cubes. 


V Roots. 


^ Roots. 


Reciprocals. 


1249 


1 560 001 


1 948 441 249 


35.341 1941 


10.769 3001 


.000 800 6405 


1250 


1 562 500 


1 953 125 000 


35.355 3391 


10.772 1735 


.000 800 0000 


1251 


1 565 001 


1 957 816 251 


35.369 4784 


10.775 0453 


.000 799 3605 


1252 


1 567 504 


1 962 515 008 


35.383 6120 


10.777 9156 


.000 798 7220 


1253 


1 570 009 


1 967 221 277 


35.397 7400 


10.780 7843 


.000 798 0846 


1254 


1 572 516 


1 971 935 064 


35.411 8624 


10.783 6516 


.000 797 4482 


1255 


1 575 025 


1 976 656 375 


35.425 9792 


10.786 5173 


.000 796 8127 


1256 


1 577 536 


1 981 385 216 


35.440 0903 


10.789 3815 


.000 796 1783 


1257 


1 580 049 


1 986 121 593 


35.454 1958 


10.792 2441 


.000 795 5449 


1258 


1 582 564 


1 990 865 512 


35.468 2957 


10.795 1053 


.000 794 9126 


1259 


1 585 081 


1 995 616 979 


35.482 3900 


10.797 9649 


.000 794 2812 


1260 


1 587 600 


2 000 376 000 


35.496 4787 


10.800 8230 


.000 793 6508 


1261 


1 590 121 


2 005 142 581 


35.510 5618 


10.803 6797 


.000 793 0214 


1262 


1 592 644 


2 009 916 728 


35.524 6393 


10.806 5348 


.000 792 3930 


1263 


1 595 169 


2 014 698 447 


35.538 7113 


10.809 3884 


.000 791 7656 


1264 


1 597 696 


2 019 487 744 


35.552 7777 


10.812 2404 


.000 791 1392 


1265 


1 600 225 


2 024 284 625 


35.566 8385 


10.815 0909 


.000 790 5138 


1266 


1 602 756 


2 029 089 096 


35.580 8937 


10.817 9400 


.000 789 8894 


1267 


1 605 289 


2 033 901 163 


35.594 9434 


10.820 7876 


.000 789 2660 


1268 


1 607 824 


2 038 720 832 


35.608 9876 


10.823 6336 


.000 788 6435 


1269 


1 610 361 


2 043 548 109 


35.623 0262 


10.826 4782 


.000 788 0221 


1270 


1 612 900 


2 048 383 000 


35.637 0593 


10.829 3213 


.000 787 4016 


1271 


1 615 441 


2 053 225 511 


35.651 0869 


10.832 1629 


.000 786 7821 


1272 


1 617 984 


2 058 075 648 


35.665 1090 


10.835 0030 


.000 786 1635 


1273 


1 620 529 


2 062 933 417 


35.679 1255 


10.837 8416 


.000 785 5460 


1274 


1 623 076 


2 067 798 824 


35.693 1366 


10.840 6783 


.000 784 9294 


1275 


1 625 625 


2 072 671 875 


35.707 1421 


10.843 5144 


.000 784 3137 


1276 


1 628 176 


2 077 552 576 


35.721 1422 


10.846 3485 


.000 783 6991 


1277 


1 630 729 


2 082 440 933 


35.735 1367 


10.849 1812 


.000 783 0854 


1278 


1 633 284 


2 087 336 952 


35.749 1258 


10.852 0125 


.000 782 4726 


1279 


1635 841 


2 092 240 639 


35.763 1095 


10.854 8422 


.000 781 8608 


1280 


1 638 400 


2 097 152 000 


35.777 0876 


10.857 6704 


.000 781 2500 


1281 


1 640 961 


2 102 071 841 


35.791 0603 


10.860 4972 


.000 780 6401 


1282 


1 643 524 


2 106 997 768 


35.805 0276 


10.863 3225 


.000 780 0312 


1283 


1 646 089 


2 111 932 187 


35.818 9894 


10.866 1454 


.000 779 4232 


1284 


1 648 656 


2 116 874 304 


35.832 9457 


10.868 9687 


.000 778 8162 


1285 


1 651 225 


2 121 824 125 


35.846 8966 


10.871 7897 


.000 778 2101 


1286 


1 653 796 


2 126 781 656 


35.860 8421 


10.874 6091 


.000 777 6050 


1287 


1 656 369 


2 131 746 903 


35.874 7822 


10.877 4271 


.000 777 0008 


1288 


1 658 944 


2 136 719 872 


35.888 7169 


10.880 2436 


.000 776 3975 


1289 


1 661 521 


2 141 700 569 


35.902 6461 


10.883 0587 


.000 775 7952 


1290 


1 664 100 


2 146 689 000 


35.916 5699 


10.885 8723 


.000 775 1938 


1291 


1 666 681 


2 151 685 171 


35 930 4884 


10.888 6845 


.000 774 5933 


1292 


1 669 264 


2 156 689 088 


35.944 4015 


10.891 4952 


.000 773 9938 


1293 


1 671 849 


2 161 700 757 


35.958 3092 


10.894 3044 


.000 773 3952 


1294 


1 674 436 


2 166 720 184 


35.972 2115 


10.897 1123 


.000 772 7975 


1295 


1 677 025 


2 171 747 375 


35.986 1084 


10.899 9186 


.000 772 2008 


1296 


1 679 616 


2 176 782 336 


36.000 0000 


10.902 7235 


.000 771 6049 


1297 


1 682 209 


2 181 825 073 


36.013 8862 


10.905 5269 


.000 771 0100 


1298 


1 684 804 


2 186 875 592 


36.027 7671 


10.908 3290 


.000 770 4160 


1299 


1 687 401 


2 191 933 899 


36.041 6426 


10.911 1296 


.000 769 8229 


1300 


1 690 000 


2 197 000 000 


36.055 5128 


10.913 9287 


.000 769 2308 



50 



Powers and Roots. 





Squares. 


Cubes. 








Number. 


Y Boots. 


f Koots. 


Keciprocals. 


1301 


1 692 601 


2 202 073 901 


36.069 3776 


10.916 7265 


.000 768 6395 


1302 


1 695 204 


2 207 155 608 


36.083 2371 


10.919 5228 


.000 768 0492 


1303 


1 697 809 


2 212 245 127 


36.097 0913 


10.922 3177 


.000 767 4579 


1304 


1 700 416 


2 217 342 464 


36.110 9402 


10.925 1111 


.000 766 8712 


1305 


1 703 025 


2 222 447 625 


36.124 7837 


10.927 9031 


.000 766 2835 


1306 


1 705 636 


2 227 560 616 


36.138 6220 


10.930 6937 


.000 765 6968 


1307 


1 708 249 


2 232 681 443 


36.152 4550 


10.933 4829 


.000 765 1109 


1308 


1 710 864 


2 237 810 112 


36.166 2826 


10.936 2706 


.000 764 5260 


1309 


1 713 481 


2 242 946 629 


36.180 1050 


10.939 0569 


.000 763 9419 


1310 


1 716 100 


2 248 091 000 


36.193 9221 


10.941 8418 


.000 763 3588 


1311 


1 718 721 


2 253 243 231 


36.207 7340 


10.944 6253 


.000 762 7765 


1312 


1 721 344 


2 258 403 328 


36.221 5406 


10.947 5074 


.000 762 1951 


1313 


1 723 969 


2 263 571 297 


36.235 3419 


10.950 1880 


.000 761 6446 


1314 


1 726 596 


2 268 747 144 


36.249 1379 


10.952 9673 


.000 761 0350 


1315 


1 729 225 


2 273 930 875 


36.262 6287 


10.955 7451 


.000 760 4563 


1316 


1 731 856 


2 279 122 496 


36.276 7143 


10.958 5215 


.000 759 8784 


1317 


1 734 489 


2 284 322 013 


36.290 4246 


10.961 2965 


.000 759 3014 


1318 


1 737 124 


2 289 529 432 


36.304 2697 


10.964 0701 


.000 758 7253 


1319 


1 739 761 


2 294 744 759 


36.318 0396 


10.966 8423 


.000 758 1501 


1320 


1 742 400 


2 299 968 000 


36.331 8042 


10.969 6131 


.000 757 5758 


1321 


1 745 041 


2 305 199 161 


36.345 5637 


10.972 3825 


.000 757 0023 


1322 


1 747 684 


2 310 438 248 


36.359 3179 


10.975 1505 


.000 756 4297 


1323 


1 750 329 


2 315 685 267 


36.373 0670 


10.977 9171 


.000 755 8579 


1324 


1 752 976 


2 320 940 224 


36.386 8108 


10.980 6823 


.000 755 2870 


1325 


1 755 625 


2 326 203 125 


36.400 5494 


10.983 4462 


.000 754 7170 


1326 


1 758 276 


2 331 473 976 


36.414 2829 


10.986 2086 


.000 754 1478 


1327 


1 760 929 


2 336 752 783 


36.428 0112 


10.988 9696 


.000 753 5795 


1328 


1 763 584 


2 342 039 552 


36.441 7343 


10.991 7293 


.000 753 0120 


1329 


1 766 241 


2 347 334 289 


36.455 4523 


10.994 4876 


.000 752 4454 


1330 


1 768 900 


2 352 637 000 


36.469 1650 


10.997 2445 


.000 751 8797 


1331 


1 771 561 


2 357 947 691 


36.482 8727 


11.000 0000 


.000 751 3148 


1332 


1 774 224 


2 363 266 368 


36.496 5752 


11.002 7541 


.000 750 7508 


1333 


1 776 889 


2 368 593 037 


36.510 2725 


11.005 5069 


.000 750 1875 


1334 


1 779 556 


2 373 927 704 


36.523 9647 


11.008 2583 


.000 749 6252 


1335 


1 782 225 


2 379 270 375 


36.537 6518 


11.011 0082 


.000 749 0637 


1336 


1 784 896 


2 384 621 056 


36.551 3388 


11.013 7569 


.000 748 5030 


1337 


1 787 569 


2 389 979 753 


36.565 0106 


11.016 5041 


.000 747 9432 


1338 


1 790 244 


2 395 346 472 


36.578 6823 


11.019 2500 


.000 747 3842 


1339 


1 792 921 


2 400 721 219 


36.592 3489 


11.021 9945 


.000 746 8260 


1340 


1 795 600 


2 406 104 000 


36.606 0104 


11.024 7377 


.000 746 2687 


1341 


1 798 281 


2 411 494 821 


36.619 6668 


11.027 4795 


.000 745 7122 


1342 


1 800 964 


2 416 893 688 


36.633 3181 


11.030 2199 


.000 745 1565 


1343 


1 803 649 


2 422 300 607 


36.646 9144 


11.032 9590 


.000 744 6016 


1344 


1 806 336 


2 427 715 584 


36.660 6056 


11.035 6967 


.000 744 047C 


1345 


1 809 025 


2 433 138 625 


36.674 2416 


11.038 4330 


.000 743 4944 


1346 


1 811 716 


2 438 569 736 


36.687 8726 


11.041 1680 


.000 742 9421 


1347 


1 814 409 


2 444 008 923 


36.701 4986 


11.043 9017 


.000 742 3905 


1348 


1 817 104 


2 449456192 


36.715 1195 


11.046 6339 


.000 741 8398 


1349 


1 819 801 


2 454 911549 


36.728 7353 


11.049 3649 


.000 741 2898 


1350 


1 822 500 


2 460 375 000 


36.742 3461 


11.052 0945 


.000 740 7407 


1351 


1 825 201 


2 465 846 551 


36.755 9519 


11.054 8227 


.000 740 1924 


1352 


1 827 904 


2 471 326 208 


36.769 5526 


11.057 5497 


.000 739 6450 







POWEES 


and Roots 




51 




Squares. 


Cubes. 








Number. 


^Koots. 


f Boots. 


Reciprocals. 


1353 


1 830 609 


2 476 813 977 


36.783 1483 


11.060 2752 


.000 739 0983 


1354 


1 833 316 


2 482 309 864 


36.796 7390 


11.062 9994 


.000 738 5524 


1355 


1 836 025 


2 487 813 875 


36.810 3246 


11.065 7222 


.000 738 0074 


1356 


1 838 736 


2 493 326 016 


36.823 9053 


11.068 4437 


.000 737 4631 


1357 


1 841 449 


2 498 846 293 


36.837 4809 


11.071 1639 


.000 736 9197 


1358 


1 844 164 


2 504 374 712 


36.851 0515 


11.073 8828 


.000 736 3770 


1359 


1 846 881 


2 509 911 279 


36.864 6172 


11.076 6003 


.000 735 8352 


1360 


1 849 600 


2 515 456 000 


36.878 1778 


11.079 3165 


.000 735 2941 


1361 


1 852 321 


2 521 008 881 


36.891 7335 


11.082 0314 


.000 734 7539 


1362 


1 855 044 


2 526 569 928 


36.905 2842 


11.084 7449 


.000 734 2144 


1363 


1 857 769 


2 532 139 147 


36.918 8299 


11.087 4571 


.000 733 6757 


1364 


1 860 496 


2 537 716 544 


36.932 3706 


11.090 1679 


.000 733 1378 


1365 


1 863 225 


2 543 302 125 


36.945 9064 


11.092 8775 


.000 732 6007 


1366 


1 865 956 


2 548 895 896 


36.959 4372 


11.095 5857 


.000 732 0644 


1367 


1 868 689 


2 554 497 863 


36.972 9631 


11.098 2926 


.000 731 5289 


1368 


1 871 424 


2 560 108 032 


36.986 4840 


11.100 9982 


.000 730 9942 


1369 


1 874 161 


2 565 726 409 


37.000 0000 


11.103 7025 


.000 730 4602 


1370 


1 876 900 


2 571 353 000 


37.013 5110 


11.106 4054 


.000 729 9270 


1371 


1 879 641 


2 576 987 811 


37.027 0172 


11.109 1070 


.000 729 3946 


1372 


1 882 384 


2 582 630 848 


37.040 5184 


11.111 8073 


.000 728 8630 


1373 


1 885 129 


2 588 232 117 


37.054 0146 


11.114 5064 


.000 728 3321 


1374 


1 887 876 


2 593 941 624 


37.067 5060 


11.117 2041 


.000 727 8020 


1375 


1 890 625 


2 599 609 375 


37.089 9924 


11.119 9004 


.000 727 2727 


1376 


1 893 376 


2 605 285 376 


37.094 4740 


11.122 5955 


.000 726 7442 


1377 


1 896 129 


2 610 969 633 


37.107 9506 


11.125 2893 


.000 726 2164 


1378 


1 898 884 


2 616 662 152 


37.121 4224 


11.127 9817 


.000 725 6894 


1379 


1 901 641 


2 622 362 939 


37.134 8893 


11.130 6729 


.000 725 1632 


1380 


1 904 400 


2 628 072 000 


37.148 3512 


11.133 3628 


.000 724 6377 


1381 


1 907 161 


2 633 789 341 


37.161 8084 


11.136 0514 


.000 724 1130 


1382 


1 909 924 


2 639 514 968 


37.175 2606 


11.138 7386 


.000 723 5890 


1383 


1 912 689 


2 645 248 887 


37.188 7079 


11.141 4246 


.000 723 0658 


1384 


1 915 456 


2 650 991 104 


37.202 1505 


11.144 1093 


.000 722 5434 


1385 


1 918 225 


2 656 741 625 


37.215 5881 


11.146 7926 


.000 722 0217 


1386 


1 920 996 


2 662 500 456 


37.229 0209 


11.149 4747 


.000 721 5007 


1387 


1 923 769 


2 668 267 603 


37.242 4489 


11.152 1555 


.000 720 9805 


1388 


1 926 544 


2 674 043 072 


37.255 8720 


11.154 8350 


.000 720 4611 


1389 


1 929 321 


2 679 826 869 


37.269 2903 


11.157 5133 


.000 719 9424 


1390 


1 932 100 


2 685 619 000 


37.282 7037 


11.160 1903 


.000 719 4245 


1391 


1 934 881 


2 691 419 471 


37.296 1124 


11.162 8659 


.000 718 9073 


1392 


1 937 664 


2 697 228 288 


37.309 5162 


11.165 5403 


.000 718 3908 


1393 


1 940 449 


2 703 045 457 


37.322 9152 


11.168 2134 


.000 717 8751 


1394 


1 943 236 


2 708 870 984 


37.336 3094 


11.170 8852 


.000 717 3601 


1395 


1 946 025 


2 714 704 875 


37.349 6988 


11.173 5558 


.000 716 8459 


1396 


1 948 816 


2 720 547 136 


37.363 0834 


11.176 2250 


.000 716 3324 


1397 


1 951 609 


2 726 397 773 


37.376 4632 


11.178 8930 


.000 715 8196 


1398 


1 954 404 


2 732 256 792 


37.389 8382 


11.181 5598 


.000 715 3076 


1399 


1 957 201 


2 738 124 199 


37.403 2084 


11.184 2252 


.000 714 7963 


1400 


1 960 000 


2 744 000 000 


37.416 5738 


11.186 8894 


.000 714 2857 


1401 


1 962 801 


2 749 884 201 


37.429 9345 


11.189 5523 


.000 713 7759 


1402 


1 965 604 


2 755 776 808 


37.443 2904 


11.192 2139 


.000 713 2668 


1403 


1 968 409 


2 761 677 827 


37.456 6416 


11.194 8743 


.000 712 7584 


1404 


1 971 216 


2 767 587 264 


37.469 9880 


11.197 5334 


.000 712 2507 



52 



Powers and Roots. 





Squares. 


Cubes. 






Reciprocals. 


Number. 


V Roots. 


f Roots. 


1405 


1 974 025 


2 773 505 123 


37.483 3296 


11.200 1913 


.000 711 7438 


1406 


1 976 836 


2 779 431 416 


37.496 6665 


11.202 8479 


.000 7112376 *v 


1407 


1 979 649 


2 785 366 143 


37.509 9987 


11.205 5032 


.000 710 7321 


1408 


1 982 464 


2 791 309 312 


37.523 3261 


11.208 1573 


.000 710 2273 


1409 


1 985 281 


2 797 260 929 


37.536 6487 


11.210 8101 


.000 709 7232 


1410 


1 988 100 


2 803 221 000 


37.549 9667 


11.213 4617 


.000 709 2199 


1411 


1 990 921 


2 809 189 531 


37.563 2799 


11.216 1120 


.000 708 7172 


1412 


1 993 744 


2 815 166 528 


37.576 5885 


11.218 7611 


.000 708 2153 


1413 


1 996 569 


2 821 151 997 


37.589 8922 


11.221 4089 


.000 707 7141 


1414 


1 999 396 


2 827 145 944 


37.603 1913 


11.224 0054 


.000 707 2136 


1415 


2 002 225 


2 833 148 375 


37.616 4857 


11.226 7007 


.000 706 7138 


1416 


2 005 056 


2 839 159 296 


37.629 7754 


11.229 3448 


.000 706 2147 


1417 


2 007 889 


2 845 178 713 


37.643 0604 


11.231 9876 


.000 705 7163 


1418 


2 010 724 


2 851 206 632 


37.656 3407 


11.234 6292 


.000 705 2186 


1419 


2 013 561 


2 857 243 059 


37.669 6164 


11.237 2696 


.000 704 7216 


1420 


2 016 400 


2 863 288 000 


37.682 8874 


11.239 9087 


.000 704 2254 


1421 


2 019 241 


2 869 341 461 


37.696 1536 


11.242 5465 


.000 703 7298 


1422 


2 022 084 


2 875 403 448 


37.709 4153 


11.245 1831 


.000 703 2349 


1423 


2 024 929 


2 881 473 967 


37.722 6722 


11.247 8185 


.000 702 7407 


1424 


2 027 776 


2 887 553 024 


37.735 9245 


11.250 4527 


.000 702 2472 


1425 


2 030 625 


2 893 640 625 


37.749 1722 


11.253 0856 


.000 701 7544 


1426 


2 033 476 


2 899 736 776 


37.762 4152 


11.255 7173 


.000 701 2623 


1427 


2 036 329 


2 905 841 483 


37.775 6535 


11.258 3478 


.000 700 7708 


1428 


2 039 184 


2 911 954 752 


37.788 8873 


11.260 9770 


.000 700 2801 


1429 


2 042 041 


2 918 076 589 


37.802 1163 


11.263 6050 


.000 699 7901 


1430 


2 044 900 


2 924 207 000 


37.815 3408 


11.266 2318 


.000 699 3007 


1431 


2 047 761 


2 930 345 991 


37.828 5606 


11.268 8573 


.000 698 8120 


1432 


2 050 624 


2 936 493 568 


37.841 7759 


11.271 4816 


.000 698 3240 


1433 


2 053 489 


2 942 649 737 


37.854 9864 


11.274 1047 


.000 697 8367 


1434 


2 056 356 


2 948 814 504 


37.868 1924 


11.276 7266 


.000 697 3501 


1435 


2 059 225 


2 954 987 875 


37.881 3938 


11.279 3472 


.000 696 8641 


1436 


2 062 096 


2 961 169 856 


37.894 5906 


11.281 9666 


.000 696 3788 


1437 


2 064 969 


2 967 360 453 


37.907 7828 


11.284 5849 


.000 695 8942 


1438 


2 067 844 


2 973 559 672 


37.920 9704 


11.287 2019 


.000 695 4103 


1439 


2 070 721 


2 979 767 519 


37.934 1535 


11.289 8177 


.000 694 9270 


1440 


2 073 600 


2 985 984 000 


37.947 3319 


11.292 4323 


.000 694 4444 


1441 


2 076 481 


2 992 209 121 


37.960 5058 


11.295 0457 


.000 693 9625 


1442 


2 079 364 


2 998 442 888 


37.973 6751 


11.297 6579 


.000 693 4813 


1443 


2 082 249 


3 004 685 307 


37.986 8398 


11.300 2688 


.000 693 0007 


1444 


2 085 136 


3 010 936 384 


38.000 0000 


11.302 8786 


.000 692 5208 


1445 


2 088 025 


3 017 196 125 


38.013 1556 


11.305 4871 


.000 692 0415 


1446 


2 090 916 


3 023 464 536 


38.026 3067 


11.308 0945 


.000 691 5629 


1447 


2 093 809 


3 029 741 623 


38.039 4532 


11.310 7006 


.000 691 0850 


1448 


2 096 704 


3 036 027 392 


38.052 5952 


11.313 3056 


.000 690 6078 


1449 


2 099 601 


3 042 321 849 


38.065 7326 


11.315 9094 


.000 690 1312 


1450 


2 102 500 


3 048 625 000 


38.078 8655 


11.318 5119 


.000 689 6552 


1451 


2 105 401 


3 054 936 851 


38.091 9939 


11.321 1132 


.000 689 1799 


1452 


2 108 304 


3 061 257 408 


38.105 1178 


11.323 7134 


.000 688 7052 , 


1453 


2 111 209 


3 067 586 777 


38.118 2371 


11.326 3124 


.000 688 2312 


1454 


2 114 116 


3 073 924 664 


38.131 3519 


11.328 9102 


.000 687 7579 


1455 


2 117 025 


3 080 271 375 


38.144 4622 


11.331 5067 


.000 687 2852 


1456 


2 119 936 


3 086 626 816 


38.157 5681 


11.334 1022 


.000 686 8132 







Powers 


and Roots 




53 




Squares. 


Cubes. 








Number. 


V Roots. 


f Roots. 


Reciprocals. 


1457 


2 122 849 


3 092 990 993 


38.170 6693 


11.336 6964 


.000 686 3412 


1458 


2 125 764 


3 099 363 912 


38.183 7662 


11.339 2894 


.000 685 8711 


1459 


2 128 681 


3 105 745 579 


38.196 8585 


11.341 8813 


.000 685 4010 


1460 


2 131 600 


3 112 136 000 


38.209 9463 


11.344 4719 


.000 684 9315 


1461 


2 134 521 


3 118 535 181 


38.223 0297 


11.347 0614 


.000 684 4627 


1462 


2 137 444 


3 124 943 128 


38.236 1085 


11.349 6497 


.000 683 9945 


1463 


2 140 369 


3 131 359 847 


38.249 1829 


11.352 2368 


.000 683 5270 


1464 


2 143 296 


3 137 785 344 


38.262 2529 


11.354 8227 


.000 683 0601 


1465 


2 146 225 


3 144 219 625 


38.275 3184 


11.357 4075 


.000 682 5939 


1466 


2 149 156 


3 150 662 696 


38.288 3794 


11.359 9911 


.000 682 1282 


1467 


2 152 089 


3 157 114 563 


38.301 4360 


11.362 5735 


.000 681 6633 


1468 


2 155 024 


3 163 575 232 


38.314 4881 


11.365 1547 


.000 681 1989 


1469 


2 157 961 


3 170 044 709 


38.327 5358 


11.367 7347 


.000 680 7352 


1470 


2 160 900 


3 176 523 000 


38.340 5790 


11.370 3136 


.000 680 2721 


1471 


2 163 841 


3 183 010 111 


38.353 6178 


11.372 8914 


.000 679 8097 


1472 


2 166 784 


3 189 506 048 


38.366 6522 


11.375 4679 


.000 679 3478 


1473 


2 169 729 


3 196 010 817 


38.379 6821 


11.378 0433 


.000 678 8866 


1474 


2 172 676 


3 202 524 424 


38.392 7076 


11.380 6175 


.000 678 4261 


1475 


2 175 625 


3 209 046 875 


38.405 7287 


11.383 1906 


.000 677 9661 


1476 


2 178 576 


3 215 578 176 


38.418 7454 


11.385 7625 


.000 677 5068 


1477 


2 181 529 


3 222 118 333 


38.431 7577 


11.388 3332 


.000 677 0481 


1478 


2 184 484 


3 228 667 352 


38.444 7656 


11.390 9028 


.000 676 5900 


1479 


2 187 441 


3 235 225 239 


38.457 7691 


11.393 4712 


.000 676 1325 


1480 


2 190 400 


3 241 792 000 


38.470 7681 


11.396 0384 


.000 675 6757 


1481 


2 193 361 


3 248 367 641 


38.483 7627 


11.398 6045 


.000 675 2194 


1482 


2 196 324 


3 254 952 168 


38.496 7530 


11.401 1695 


.000 674 7638 


1483 


2 199 289 


3 261 545 587 


38.509 7390 


11.403 7332 


.000 674 3088 


1484 


2 202 256 


3 268 147 904 


38.522 7206 


11.406 2959 


.000 673 8544 


1485 


2 205 225 


3 274 759 125 


38.535 6977 


11.408 8574 


.000 673 4007 


1486 


2 208 196 


3 281 379 256 


38.548 6705 


11.411 4177 


.000 672 9474 


1487 


2 211 169 


3 288 008 303 


38.561 6389 


11.413 9769 


.000 672 4950 


1488 


2 214 144 


3 294 646 272 


38.574 6030 


11.416 5349 


.000 672 0430 


1489 


2 217 121 


3 301 293 169 


38.587 5627 


11.419 0918 


.000 671 5917 


1490 


2 220 100 


3 307 949 000 


38.600 5181 


11.420 6476 


.000 671 1409 


1491 


2 223 081 


3 314 613 771 


38.613 4691 


11.424 2022 


.000 670 6908 


1492 


2 226 064 


3 321 287 488 


38.626 4158 


11.426 7556 


.000 670 2413 


1493 


2 229 049 


3 327 970 157 


38.639 3582 


11.429 3079 


.000 669 7924 


1494 


2 232 036 


3 334 661 784 


38.652 2962 


11.431 8591 


.000 669 3440 


1495 


2 235 025 


3 341 362 375 


38.665 2299 


11.434 4092 


.000 668 8963 


1496 


2 238 016 


3 348 071 936 


38.678 1593 


11.436 9581 


.000 668 4492 


1497 


2 241 009 


3 354 790 473 


38.691 0843 


11.439 5059 


.000 668 0027 


1498 


2 244 004 


3 361 517 992 


38.704 0050 


11.442 0525 


.000 667 5567 


1499 


2 247 001 


3 368 254 499 


38.716 9214 


11.444 5980 


.000 667 1114 


1500 


2 250 000 


3 375 000 000 


38.729 8335 


11.447 1424 


.000 666 6667 


1501 


2 253 001 


3 381 754 501 


38.742 7412 


11.449 6857 


.000 666 2225 


1502 


2 256 004 


3 388 518 008 


38.755 6447 


11.452 2278 


.000 665 7790 


1503 


2 259 009 


3 395 290 527 


38.768 5439 


11.454 7688 


.000 655 3360 


1504 


2 262 016 


3 402 072 064 


38.781 4389 


11.457 3087 


.000 664 8936 


1505 


2 265 025 


3 408 862 625 


38.794 3294 


11.459 8476 


.000 664 4518 


1506 


2 268 036 


3 415 662 216 


38.807 2158 


11.462 3850 


.000 664 0106 


1507 


2 271 049 


3 422 470 843 


38.820 0978 


11.464 9215 


.000 663 5700 


1508 


2 274 064 


3 429 288 512 


38.832 9757 


11.467 4568 


.000 663 1300 



54 



Poweks and Roots. 



Squares. 



2 277 081 
2 280 100 
2 283 121 
2 286 144 
2 289 169 
2 292 196 
2 295 225 
2 298 256 
2 301 289 
2 304 324 
2 307 361 
2 310 400 
2 313 441 
2 316 484 
2 319 529 
2 322 576 
2 325 625 
2 328 676 
2 331 729 
2 334 784 
2 337 841 
2 340 900 
2 343 961 
2 347 024 
2 350 089 
2 353 156 
2 356 225 
2 359 296 
2 362 369 
2 365 444 
2 368 521 
2 371 600 
2 374 681 
2 377 764 
2 380 849 
2 383 936 
2 387 025 
2 390 116 
2 393 209 
2 396 304 
2 399 401 
2 402 500 
2 405 601 
2 408 704 
2 411 809 
2 414 916 
2 418 025 
2 421 136 
2 424 249 
2 427 364 
2 430 481 
2 433 600 



Cubes. 



3 436 115 229 
3 442 951 000 
3 449 795 831 
3 456 649 728 
3 463 512 697 
3 470 384 744 
3 477 265 875 
3 484 156 096 
3 491 055 413 
3 597 963 832 
3 504 881 359 
3 511 808 000 
3 518 743 761 
3 525 688 648 
3 532 642 667 
3 539 605 824 
3 546 578 125 
3 553 559 576 
3 560 558 183 
3 567 549 552 
3 574 558 889 
3 581 577 000 
3 588 604 291 
3 595 640 768 
3 602 686 437 
3 609 741 304 
3 616 805 375 
3 623 878 656 
3 630 961 153 
3 638 052 872 
3 645 153 819 
3 652 264 000 
3 657 983 421 
3 666 512 088 
3 673 650 007 
3 680 797 184 
3 687 953 625 
3 695 119 336 
3 702 294 323 
3 709 478 592 
3 716 672 149 
3 723 875 000 
3 731 087 151 
3 738 308 608 
3 745 539 377 
3 752 779 464 
3 760 028 875 
3 767 287 616 
3 774 55.", 09:; 
3 78183:; 112 
3 789 119 879 
3 796 416 000 



^Roots. 



38.845 8491 
38.858 7184 
38.871 5834 
38.884 4442 
38.897 3006 
38.910 1529 
38.923 0009 
38.935 8447 
38.948 6841 
38.961 5194 
38.974 3505 
38.987 1774 
39.000 0000 
39.012 8184 
39.025 6326 
39.038 4426 
39.051 2483 
39.064 0499 
39.076 8473 
39.089 6406 
39.102 4296 
39.115 2144 
39.127 9951 
39.140 7716 
39.153 5439 
39.166 3120 
39.179 0760 
39.191 8359 
39.204 5915 
39.217 3431 
39.230 0905 
39.242 8337 
39.255 5728 
39.268 3078 
39.281 0387 
39.293 7654 
39.306 4880 
39.319 2065 
39.331 9208 
39.344 6311 
39.357 3373 
39.370 0394 
39.382 7373 
39.395 4312 
39.408 1210 
39.420 8067 
39.433 4883 
39.446 1658 
39.458 8393 
39.471 5087 
39.484 1740 
39.496 8353 



f Roots. 



11.469 9911 
11.472 5242 
11.475 0562 
11.477 5871 
11.480 1169 
11.482 6455 
11.485 1731 
11.487 6995 
11.490 2249 
11.492 7491 
11.495 2722 
11.497 7942 
11.500 3151 
11.502 8348 
11.505 3535 
11.507 8711 
11.510 3876 
11.512 9030 
11.515 4173 
11.517 9305 
11.520 4425 
11.522 9535 
11.525 4634 
11.527 9722 
11.530 4799 
11.532 9865 
11.535 4920 
11.537 9965 
11.540 4998 
11.543 0021 
11.545 5033 
11.548 0034 
11.550 5025 
11.553 0004 
11.555 4972 
11.557 9931 
11.560 4878 
11.562 9815 
11.565 4740 
11.567 9655 
11.570 4559 
11.572 9453 
11.575 4336 
11.577 9208 
11.580 4069 
11.582 8919 
11.585 3759 
11.587 8588 
11.590 3407 
11.592 8215 
11.595 3013 
11.597 7799 



Reciprocals. 



.000 662 
.000 662 
.000 661 
.000 661 
.000 660 
.000 660 
.000 660 
.000 659 
.000 659 
.000 658 
.000 658 
.000 657 
.000 657 
.000 657 
.000 656 
.000 656 
.000 655 
.000 655 
.000 654 
.000 654 
.000 654 
.000 653 
.000 653 
.000 652 
.000 652 
.000 651 
.000 651 
.000 651 
.000 650 
.000 650 
.000 649 
.000 649 
.000 648 
.000 648 
.000 648 
.000 647 
.000 647 
.000 646 
.000 646 
.000 645 
.000 645 
.000 645 
.000 644 
.000 644 
.000 643 
.000 643 
.000 643 
.000 612 
.000 612 
.000 611 
.000 641 
.000 641 



6905 ; 

2517 ,4 

8134 

3757 

9385 

5020 

0660 

6306 

1958 

7615 

3278 

8947 

4622 

0302 

5988 

1680 

7377 

3080 

8788 

4503 

0222 

5948 

1679 

7415 

3157 

8905 

4658 

0417 

6181 

1951 

7726 

3506 

9293 

5084 

0881 

6684 

2492 

8305 

4124 

9948 

5778 

1613 

7453 

3299 

9150 

5006 

0868 

6735 

2608 

8485 

4368 

0256 







Powers 


and Roots 




55 




Squares. 


Cubes. 








Number. 


V'Boots. 


f Boots. 


Reciprocals. 


1561 


2 436 721 


3 803 721 481 


39.509 4925 


11.600 2576 


.000 640 6150 


1562 


2 439 844 


3 811 036 328 


39.522 1457 


11.602 7342 


.000 640 2049 


1563 


2 442 969 


3 818 360 547 


39.534 7948 


11.605 2097 


.000 639 7953 


1564 


2 446 096 


3 825 641 444 


39.547 4399 


11.607 6841 


.000 639 3862 


1565 


2 449 225 


3 833 037 125 


39.560 0809 


11.610 1575 


.000 638 9776 


1566 


2 452 356 


3 840 389 496 


39.572 7179 


11.612 6299 


- .000 638 5696 


1567 


2 455 489 


3 847 751 263 


39.585 3508 


11.615 1012 


.000 638 1621 


1568 


2 458 624 


3 855 123 432 


39.597 9797 


11.617 5715 


.000 637 7551 


1569 


2 461 761 


3 862 503 009 


39.610 6046 


11.620 0407 


.000 637 3486 


1570 


2 464 900 


3 869 883 000 


39.623 2255 


11.622 5088 


.000 636 9427 


1571 


2 468 041 


3 877 292 411 


39.635 8424 


11.624 9759 


.000 636 5372 


1572 


2 471 184 


3 8S4 701 248 


39.648 4552 


11.627 4420 


.000 636 1323 


1573 


2 474 329 


3 892 119 157 


39.661 0640 


11.629 9070 


.000 635 7279 


1574 


2 477 476 


3 899 547 224 


39.673 6688 


11.632 3710 


.000 635 3240 


1575 


2 480 625 


3 906 984 375 


39.686 2696 


11.634 8339 


.000 634 9206 


1576 


2 483 776 


3 914 430 976 


39.698 8665 


11.637 2957 


.000 634 5178 


1577 


2 486 929 


3 921 887 033 


39.711 4593 


11.639 7566 


.000 634 1154 


1578 


2 490 084 


3 929 352 552 


39.724 0481 


11.642 2164 


.000 633 7136 


1579 


2 493 241 


3 936 827 539 


39.736 6329 


11.644 6751 


.000 633 3122 


1580 


2 496 400 


3 944 312 000 


39.749 2138 


11.647 1329 


.000 632 9114 


1581 


2 499 561 


3 951 805 941 


39.761 7907 


11.649 5895 


.000 632 5111 


1582 


2 502 724 


3 959 309 368 


39.774 3636 


11.652 0452 


.000 632 1113 


1583 


2 505 889 


3 966 822 287 


39.786 9325 


11.654 4998 


.000 631 7119 


1584 


2 509 056 


3 974 344 704 


39.799 4976 


11.656 9534 


.000 631 3131 


1585 


2 512 225 


3 981 876 625 


39.812 0585 


11.659 4059 


.000 630 9148 


1586 


2 515 396 


3 989 418 056 


39.824 6155 


11.661 8574 


.000 630 5170 


1587 


2 518 569 


3 996 969 003 


39.837 1686 


11.664 3079 


.000 630 1197 


1588 


2 521 744 


4 004 529 472 


39.849 7177 


11.666 7574 


.000 629 7229 


1589 


2 524 921 


4 012 099 469 


39.862 2628 


11.669 2058 


.000 629 3266 


1590 


2 528 100 


4 014 679 000 


39.874 8040 


11.671 6532 


.000 628 9308 


1591 


2 531 281 


4 027 268 071 


39.887 3413 


11.674 0996 


.000 628 5355 


1592 


2 534 464 


4 034 866 688 


39.899 8747 


11.676 5449 


.000 628 1407 


1593 


2 537 649 


4 042 474 857 


39.912 4041 


11.678 9892 


.000 627 7464 


1594 


2 540 836 


4 050 092 584 


39.924 9295 


11.681 4325 


.000 627 3526 


1595 


2 544 025 


4 057 719 875 


39.937 4511 


11.683 8748 


.000 626 9592 


1596 


2 547 216 


4 065 356 736 


39.949 9687 


11.686 3161 


.000 626 5664 


1597 


2 550 409 


4 073 003 173 


39.962 4824 


11.688 7563 


.000 626 1741 


1598 


2 553 604 


4 080 659 192 


39.974 9922 


11.691 1955 


.000 625 7822 


1599 


2 556 801 


4 088 324 799 


39.987 4980 


11.693 6337 


.000 625 3909 


1600 


2 560 000 


4 096 000 000 


40.000 0000 


11.696 0709 


.000 625 0000 



The use of the table of powers and roots may be extended far beyond 
its apparent limits by the observance of the following rules : 

Remembering that the extraction of the square root of a number is 
simply the separating it into two equal factors, Ave have: to extract the 
square root of any whole number and decimal, when the whole number is 
within the limits of the table, simply find the square root of the whole 
number in the table and divide the given number and decimal by this 
root. The quotient will be another factor, very nearly equal to the required 
root. Add the divisor and the quotient together and divide by two, and 
the result will be the true root to a very close degree of approximation. 

Thus, let it be required to find the square root of 346.285. 

We find from the table that the square root of 346 is IS. 6010752, or, for 
moderate precision, 18.6011, which is, of course, too small. 

We then have 346.285 -t- 18.6011 = 18.6163, so that we have the number 



56 Interest. 



346.285, composed of the two factors, 18.6011 X 18.6163, which are very nearly- 
equal. Adding them together and dividing by 2, we get 

. 18.6011 + 18.6163 -,o R nft7 

> X 346.285 = \ = 1 8-6087. 

The true root is 18.60873. 

To extract the cube root of a whole number and decimal we proceed in 
a similar manner, remembering that the cube root is one of three equal 
factors, so that we divide twice by the cube root of the whole number and 
then take the mean of the two divisors and the final quotient,— i.e., of the 
three nearly equal factors. — 

Thus, to find the cube root of 346.285, we find in the table F346 = 
7.0203490, or, for moderate precision, = 7.02035. 

We then have 346.285 h- 7.02035 = 49.32588 and 49.32588 -f- 7.02035 = 7.02612, 
and we have 

f ^^ = 7.02035 + 7.02035 + 7.02612 = lmm ^ 

The true root is 7.02226. 

If the square root or the cube root of a number larger than 1600 is 
required, look for the nearest number in the column of squares or cubes, 
as the case may be, and the approximate root will be the corresponding 
number in the first column. By using this as the divisor the given number 
may be resolved into two or three nearly equal factors, and their mean 
will be the required root, very nearly. 

Thus, if it is required to find the square root of 569,245, we look in the 
column of squares and find the nearest number to be 570,025, and the cor- 
responding number in the first column is 755. Taking this as a divisor, we 
have 

569,245 = 753.935, and I 5 ^ + 753.935 = 754#476# 
755 2 

The true root is 754.483. 

INTEREST. 
Simple Interest. 

Interest is money paid for use of money which is lent for a certain 
time. 

Notation. 

c = the amount lent ; 

r = interest on the amount, c ; 

p = per cent, in the certain time. 

Analogy, c : r = 100 : p. 

If p is the per cent, on 100 in one year, then t = time in years for the 
standing capital c and the interest r. 

Analogy, c : r = 100 : pt. 

From this analogy we obtain the equations : 

1. Interest, r = -$&.. 

100 



2. Percent., p = ^. 

ct 



3. Capital, c = ±^-. 
pt 

4. Time in years, t = ^°-^. 

cp 



Now for any question in Simple Interest there is one equation which 
gives the answer. If the time is given in months, weeks, or days, multiply 
the 100 correspondingly by 12, 52, 365. 

Example 1. What is the interest on $3789.35, for 3 years and 5 months, at 
6 per cent, per annum ? 

t == 3 X 12 + 5 = 41 months ; from the Equation 1 we have, 
Interest, r = 3789.35 X 6 X 41 = 7?6 gl dollars 
12 X 100 
Example 2. A capital c = 469.78 dollars, returned interest r = 150.72 
dollars in time I = 4 years and 7 months. Required the per cent, per 
annum ? 

t = 4 X 12 + 7 = 55 months ; from the Equation 2 we have, 



Interest. 



57 



Per cent., p = 12 X 10 ° X 15 - ^- 2 = 7 per cent. 
469.78 X 55 

Example S. What amount is required to return interest r = 345 dollars 
in 6 years, at 5 per cent, per annum? 
From the Equation 3 we have, 

Capital, c = 10 ° X 345 = 1150 dollars. 

5X6 

Example k. An amount c = 2365 dollars is to stand until the interest 
r = 550 dollars, at p = 6 per cent, per annum. How long must the amount 
stand ? 

From the Equation 4 we have, 

. 100 X 550 



Time. 



t = 



= 3.876 years. 



the time t = 3 years, 



2365 X 6 
12 X 0.876 = 10.512 months, 4 X 0.512 = 2.048 weeks ; 
10 months, and 2 weeks. 

Compound Interest. 

Compound Interest is when the interest is added to the capital for 
each year, and the sum is the capital for the following year. 



1. Amount, a = c(l + p) n 

2. Capital, 



3. Per cent., 



■V?- 



-1. 



4. Number of years, n = 



log, a — log, c 



(1 + P) W " log. (1+p) 

4®* In these formulas p must be expressed in hundredths. 

Example 1. A capital c = 8650 standing with compound interest at p = 5 
per cent. What will it amount to in n = 9 years ? 

Amount a = 8650 (1.05) 9 = 13,419 dollars. 

Example 2. A man commenced business with c = 300 dollars : after 
n = 5 years he had a = 6875 dollars. At what rate did his money increase, 
and how soon will he have a fortune of 50,000 dollars ? 

The first question, or the percentage, will be answered by the Formula 3. 

= */ 6875 _ 1== i7 2 2.9166 — 1 = 0.87, or 87 per cent. 
v 300 v 

The time from the commencement of business until the fortune is com- 
pleted will be answered from the Formula 4. 

log. 50,000 — log. 300 = 4.69897 — 2.47712 = 



log. 187 
or 8 years and 2 months. 

Compound Interest Table, 



0.2720048 



= 8.169 years, 



CALCULATED FROM FORMULA 1. 



n 


Compound Interest. 


n 


Compound Interest. 


Years. 


5 per cent. 


6 per cent. 


7 per cent. 


Years. 


5 per cent. 


6 per cent. 


7 per cent. 


1 


1.0500 


1.0600 


1.0700 


17 


2.2920 


2.6928 


3.1588 


2 


1.1025 


1.1236 


1.1449 


18 


2.4066 


2.8543 


3.3799 


3 


1.1576 


1.1910 


1.2250 


19 


2.5269 


3.0256 


3.6165 


4 


1.2155 


1.2625 


1.3108 


20 


2.6533 


3.2071 


3.8697 


5 


1.2770 


1.3382 


1.4025 


21 


2.7859 


3.3995 


4.1406 


6 


1.3400 


1.4185 


1.5007 


22 


2.9252 


3.6035 


4.4304 


7 


1.4071 


1.5036 


1.6058 


23 


3.0715 


3.8197 


4.7405 


8 


1.4774 


1.5938 


1.7182 


24 


3.2251 


4.0487 


5.0724 


9 


1.5513 


1.6895 


1.8385 


25 


3.3864 


4.2919 


5.4274 


10 


1.6289 


1.7908 


1.9671 


30 


4.3219 


5.7435 


7.6123 


11 


1.7103 


1.8983 


2.1048 


35 


5.5166 


7.6861 


10.6766 


12 


1.7958 


2.0122 


2.2522 


40 


7.0400 


10.2858 


14.9745 


13 


1.8856 


2.1329 


2.4098 


45 


8.9850 


13.7646 


21.0025 


14 


1.9799 


2.2609 


2.5785 


50 


11.6792 


18.4190 


29.4570 


15 


2.0789 


2.3965 


2.7599 


60 


18.6792 


32.9878 


57.9466 


16 


2.1829 


2.5403 


2.9522 











58 Weights and Measures. 

This table shows the value of one unit of money at the rates of 5, 6, 
and 7 per cent, per annum, compound interest, up to 60 years. 

Example 1. What is the amount of 864 pounds sterling for 12 years, at 
6 per cent, compound interest ? 

Table, 2.01219 X 864 = 1738.53216, or £1738 10s. 7.7d. 

Example %. What is the amount of 3450 dollars for 18 years, at 5 per 
cent, compound interest? 

Table, 2.40661 X 3450 = 8302.80 dollars. 

When the interest is compounded in more or less than one year, at the 
rate of interest per year, and m — the number of months in which the in- 
terest is compounded ; 

then, instead of p in the formulas, put -^L, and instead of n, put — — . 

12 m 

Example 8. A capital of 500 dollars bears compound interest semi- 
annually at 5 per cent, per annum ; what will it amount to in 10 years? 
m = 6 months, p = ^- = §-X™5 = 0<025 and n = 12_X_10 = 20 
12 12 6 

then, a = c(l + p) n = 500(1 + 0.025) 20 = 8193.11 dollars, the answer, 
log. (1 -f 0.025) = 0.0107239 

20 

0.2144780 

log. 500 = 2.6989700 

Amount, 8193.11= 2.9134480 

WEIGHTS AND MEASURES. 

There are now but two really important systems of weights and meas- 
ures in use in civilized countries,— the English and the metric. Many of 
the older English tables are falling into disuse, volumes of all kinds being 
expressed in cubic feet, solutions in percentages instead of grains per 
gallon, and similar simplifications. 

The metric system is used everywhere in Europe, except in Great Britain, 
and it is also extensively used in America, except in the United States and 
Canada. 

The following tables will be found to cover practically all necessary 
requirements : 

Measures of Length— United States and Great Britain. 

12 inches = 1 foot. 
3 feet = 1 yard = 36 inches. 
by 2 yards = 1 rod = 16% feet = 198 inches. 
40 rods = 1 furlong = 220 yards = 660 feet. 
8 furlongs = 1 mile = 320 rods = 1760 yards = 5280 feet. 
Of the above, the inch and the foot are most frequently used by me- 
chanics. The ordinary two-foot rule has the inches subdivided by the 
system of repeated halving, thus giving %, %, %, and ^ of an inch ; and 
this is sometimes carried as iar as to include 32ds and 64ths. This system, 
however, is now used principally by carpenters, builders, etc., while ma- 
chinists are generally using scales, calipers, and measuring tools which 
have the inch subdivided into lOths, lOOths, and lOOOths. 

The yard is much used by shopkeepers for measuring cloth, carpet, and 
fabrics generally, and is by them also subdivided into halves, quarters, 
and eighths. 

For long distances the mile is universally used, and portions of a mile 
are given either in furlongs and feet or in halves and quarters. 

For engineering measurements steel tapes are much used, — 100 feet long, 
with the feet subdivided into lOths instead of inches, thus giving lOths, 
lOOths, and lOOOths of the length of the tape. 

The mile given in the above table is called the statute mile, and is 
always used on land. The nautical mile, used only at sea, is equal to 6080 
feet, being about 15 per cent, longer than the statute mile. 

A knot is not a distance, but a rate of speed, corresponding to 1 nauti- 



r 



Weights and Measures. 59 

cal mile per hour. The expression "knots per hour" is incorrect, as the 
time element is included in the word knot. 

The only other system of measures of length which is extensively used 
is the Metric System. 

Metrical Measures of Length — Used generally on the 
Continent of Europe. 

The unit is the Metre = 39.37 inches. 

The metre is subdivided decimally and multiplied decimally, as below : 

1 millimetre = y^ metre = 0.03937 inches. 
1 centimetre = -^ metre = 0.3937 inches. 
1 decimetre = ^ metre = 3.937 inches. 
1 metre = 39.37 inches = 3.2808 feet. 
1 dekametre = 10 metres = 32.8087 feet. 
1 hectometre = 100 metres = 328.0869 feet. 
1 kilometre = 1000 metres — 3280.869 feet = 0.621 mile 
In using the metric system it is important to think of the metre as a 
main unit and the subdivisions as decimals of it. In mechanical and 
scientific work the metre and the millimetre are usually employed, and 
sometimes the centimetre, the decimetre more rarely. In the machine 
shop, for instance, measurements are usually given directly in millimetres, 
as 325 mm., not 3 dcm., 2 cm., 5 mm. 

For longer distances the kilometre is used exclusively, and should be 
kept in mind as the unit of out-door measurement, with the metre, its ^ 
part, for all subdivisions, the dekametre and hectometre being hardly 
used at all. It is very desirable that the student should learn the values 
of these measurements directly from the use of a metric scale, and not by 
transformation into English measures. When such transformations must 
be roughly made, however, it will be convenient to remember the follow- 
ing: 

1 millimetre = ^ inch, approximately. 
1 decimetre = 4 inches, approximately. 
1 metre = 3 feet and 3% inches, very closely. 
1 kilometre = % of a mile, nearly. 
An approximate rule to convert metres to feet is to multiply by 3 and 
add 10 per cent. Thus, 100 metres would be 300 + 30 = 330 feet, while it 
really is equal to 328 feet, the error being less than 1 per cent. 

Measures of Weight — United States and British. 

The commercial system is the Avoirdupois ; the unit being the pound 
of 7000 grains. 

The system for weighing gold and silver is,called Troy Weight, of which 
the pound contains 5760 grains. 

For medicines and drugs the Apothecaries' System is used, the grain and 
pound being the same as in Troy Weight, but the subdivisions of the 
pound being different. 

Avoirdupois or Commercial Weight. 

1 dram = 27.34375 grains. 

16 drams = 1 ounce = 437% grains. 

16 ounces = 1 pound = 7000 grains. 

14 pounds a l stone. 

28 pounds = 1 quarter. 

4 quarters = 1 hundredweight = 112 pounds. 

20 hundredweight = 1 ton = 2240 pounds. 
It will be noticed that the "hundredweight" (so called) is 12 pounds 
more than 100 pounds, this having been the allowance for loss in handling 
merchandise in old times. The ton of 2240 pounds is sometimes called the 
long ton in commerce, as distinguished from the short ton of 2000 pounds. 
When no explanation is made, the long ton of 2240 pounds is the legal 
value of the ton, but in engineering calculations, such as the load upon a 
bridge, the pressure of a mass of earthwork, or the lifting capacity of a 
crane, it is customary to use the word ton to mean 2000 pounds. In prac- 



60 Weights and Measures. 

tice a hundredweight (used as one word) means always 112 pounds, while 
a hundred pounds means 100 pounds exactly. 

Troy Weight. 

1 pennyweight = 24 grains. 

20 pennyweights = 1 ounce = 480 grains. 

12 ounces = 1 pound Troy = 5760 grains. 

Apothecaries' Weight. 

1 scruple = 20 grains. 
3 scruples = 1 dram = 60 grains. 
8 drams = 1 ounce = 480 grains. 
12 ounces = 1 pound = 5760 grains. 

Measures of Weight — Metric System. 

The metric unit of weight is the Gramme, which is the weight of a 
cubic centimetre of pure water at a temperature of 4° C, and which is 
equal to 15.432 grains. The gramme is subdivided and multiplied deci- 
mally, as follows : 

1 milligram = yJ^ gramme = 0.015432 grains. 

1 centigram = ^ gramme = 0.15432 grains. 

1 decigram = ^ gramme = 1.5432 grains. 

1 gramme = 1 gramme = 15.432 grains. 

1 dekagram = 10 grammes = 154.32 grains. 

1 hectogram = 100 grammes = 1543.2 grains. 

1 kilogram = 1000 grammes = 2.2046 pounds. 

1 myriagram = 10,000 grammes = 22.046 pounds. 

1 metric ton = 1000 kilograms = 2204.6 pounds. 
In practice many of these subdivisions and multiples are rarely used. 
The gramme and the milligram are used by chemists and physicists all 
over the world. The kilogram is used almost everywhere on the continent 
of Europe except in Russia, and its subdivisions are generally referred to 
as -rV kilo, % kilo, etc., instead of the tabular names, while the multiples 
are similarly named at 10 kilos, 100 kilos, etc. It will be noticed that the 
metric ton, or tonne, as it is written in France, is very nearly the same as 
the English long ton, so nearly that for ordinary commercial purposes they 
may be considered the same. 

Measures of Volume. 

Measures of Volume are not the same in the United States and in Great 
Britain, and hence it should always be stated as to which is meant. 

In the United States the systems for Liquid and for Dry Measures of 
volume are also different from each other, while in England both liquid 
and dry substances are measured by the same system. 

Liquid Measure — U. S. A. only. 

The unit of volume is the Gallon = 231 cubic inches. The gallon is 
subdivided and multiplied as follows : 

4 gills = 1 pint = 28.875 cubic inches. 

2 pints = 1 quart = 57.750 cubic inches. 

4 quarts = 1 gallon = 231 cubic inches. 

63 gallons = 1 hogshead. 

2 hogsheads = 1 pipe or butt. 

2 pipes = 1 tun. 
Of the above measures the pint and quart are most frequently used. 
The barrel is not a standard volume, although in the United States and in 
England a wine barrel is supposed to contain 31% gallons, but in referring 
to a barrel in liquid measure the number of gallons it contains should be 
stated. 

A cylinder 7 inches in diameter and 6 inches high contains almost pre- 
cisely a gallon, and a gallon of pure water at its greatest density weighs 
8.33888 pounds. Ordinarily it may be taken at 8.34 pounds. A cubic foot 
contains 7.48052 United States gallons. 



Monetae? Systems. 61 



Dry Measure — U. S. A. only. 

The unit of dry measure is the Bushel = 2150.42 cubic inches. The 
bushel is subdivided as follows : 

2 pints = 1 quart = 67.2 cubic inches. 
4 quarts = 1 gallon = 268.8 cubic inches. 
2 gallons = 1 peck = 537.6 cubic inches. 
4 pecks = 1 struck bushel = 2150.42 cubic inches. 
The barrel is not a legalized unit in dry measure, and its value should 
always be stated in gallons or in pounds weight of the substance it con- 
tains. A barrel of flour is equal to 196 pounds. 

British Measures of Volume. 

In the British or Imperial system the same measures are used both for 
liquid and for dry measure. The unit of the system is the Imperial Gallon 
= 277.274 cubic inches. This is intended to be equal to 10 pounds avoirdu- 
pois weight of pure water at a temperature of 62° Fahrenheit. 
The imperial gallon is subdivided and multiplied as follows : 

4 gills = 1 pint = 1.25 pounds water. 

2 pints = 1 quart = 2.50 pounds water. 

2 quarts = 1 pottle = 5.00 pounds water. 

2 pottles = 1 gallon = 10.00 pounds water. 

2 gallons = 1 peck = 20.00 pounds water. 

4 pecks = 1 bushel = 80.00 pounds water. 

4 bushels = 1 coomb = 320.00 pounds water. 

2 coombs = 1 quarter = 640.00 pounds water. 
The measures above the gallon are used for dry measures exclusively, 
and it is customary to state all quantities above the bushel in bushels. 

Metric Measures of Volume. 

The unit of volume is the Litre = 1 cubic decimetre. This is subdivided 
and multiplied decimally, as follows : 

Liquid. 

1 millilitre = x^Vtt litre. 1 decalitre = 10 litres. 

1 centilitre = y^ litre. 1 hectolitre = 100 litres. 

1 decilitre = -^ litre. 1 kilolitre = 1000 litres. 

1 litre = 1 litre. 
The principal measure used is the litre itself, and in trade the J^ litre is 
often used, this being a little more than a pint (>^ litre = 1.056 pint), and 
so convenient that the fact of its not being a decimal equivalent is over- 
looked. For chemical and physical measurements the cubic centimetre is 
much used, and called by this name, c.c, and not millilitre, which latter 
it really is. 

The unit of dry measure in the metric system is supposed to be the 
Stere = 1 cubic metre, but in practice the term cubic metre is very gener- 
ally used, and the subdivisions and multiples so named,— i.e., -fr cubic 
metre, 100 cubic metres, etc. 

MONETARY SYSTEMS. 

The various systems used for the money of different countries are too 
numerous to be described here, but a few of the most important will be 
given. 

United States and Canada. 

The unit is the Dollar ($), subdivided and multiplied decimally. The 
dollar is divided into 100 cents, and the other units are as follows : 
1 dime = 10 cents = & dollar. 
1 dollar = 100 cents. 
10 dollars = 1 eagle. 



62 Monetary Systems. 



Besides these decimal units there are coins as follows : 
M dollar = 25 cents. 
% dollar = 50 cents. 
Double eagle = 20 dollars. 
These coins are made for convenience, but are not known by their 
names in reckoning, the quarter- and half-dollar being counted as 25 and 
50 cents, and the double eagle, as well as the eagle, as so many dollars. 

Great Britain. 

The unit is the Pound Sterling, or Sovereign (£), subdivided as follows : 
The penny == ^ pound. 
1 shilling = 12 pence = about 24 cents. 
1 pound = 20 shillings = 240 pence = about $4.86. 
Besides these there are the following coins : 

Half-penny = Y% penny. 
Crown = 5 shillings. 
Half-crown = 2>£ shillings. 
Florin = 2 shillings. 
But the calculations are all made in pounds, shillings, and pence. The 
Guinea, often used in giving prices, is equal to 21 shillings, but it has not 
been coined for many years. 

Latin Monetary Union. 

On the Continent of Europe the following countries have formed them- 
selves into the Latin Monetary Union, and use the same system, — i.e., 
France, Belgium, Switzerland, Italy, and Greece. The unit is the Franc, 
called Lira in Italy and Drachma in Greece. 

The franc is subdivided into 100 centimes,— Centesemi in Italy, Lepta in 
Greece. There are also gold pieces of 20 francs and silver coins = % franc, 
besides minor coins of nickel, but these have no special names, all the 
reckoning being done in francs and lOOths. The equivalent value of the 
franc in United States money is about 19.3 cents. 

Germany. 

The unit is the Mark = about 24 cents, subdivided into 100 pfennigs. 
There are gold coins of 20 marks, but all the reckoning is done in marks 
and lOOths. 

Besides the tables and terms already described, there are many other 
calculations made in trade and commerce which cannot be given here, but 
which must be learned by actual experience. There are many words, such 
as net, gross, rebate, tare, tret, etc., etc., for the meanings of which the 
reader must refer to the dictionary. 

There are two ratios, however, which are of sufficient interest to be 
described here. The " fineness" (so called) of gold or silver is determined 
by the number of parts of pure gold or silver there are in 1000 parts of the 
alloy. The metal is, of course, pure only when it contains no alloy what- 
ever, and is then J$fg fine. The standard alloy for gold for United States 
coinage is 900 parts of pure gold and 100 parts alloy, and hence is j%%°^ fine. 

Of this alloy the gold dollar contains 25.8 grains, the eagle 258 grains, 
and the double eagle 516 grains. 

The standard "fineness" for silver is also T 9 D ° , and the standard dollar 
contains 412.5 grains of this alloy. 



The Metric System. 



63 



THE METRIC SYSTEM. 

The principal advantage of the metric system consists in its use of the 



decimal subdivisions. The attempt to consider the metre as - 



-of 



10,000,000 

a quadrant of the earth's surface has been abandoned, and it is now held 
only to be the length of the standard known as the Metre des Archives, 
copies of which are issued by the Bureau Internationale des Poids et Mesures, 
at Breteuil, near Paris. 

The kilogramme was originally intended to be the weight of a cubic 
decimetre or litre of pure water at the temperature of maximum density, 
but it is really now considered only as the weight of a platinum standard. 
At the same time, this relation between the unit of weight and a standard 
volume of water is sufficiently close for the specific gravity of any sub- 
stance to be considered as equal to the weight of a cubic decimetre of that 
substance. In all hydraulic measurements a cubic metre of water is equal 
in weight to the metric tonne of 1000 kilogrammes, a most convenient fact 
in the determination of the power developed by a given fall and volume 
of water. 

The French Metrical System. 

The French units of weight, measure, and coin are arranged into a per- 
fect decimal system, except those of time and the circle. The division and 
multiplication of the units are expressed by Latin and Greek names, as 
follows : 



Latin, Division. 
Milli = 1000th of the unit. 
Centi = 100th of the unit. 
Deci — 10th of the unit. 
Metre, litre, stere, are, 
gramme. 



franc, 



Greek, Multiplication. 
Deca = 10 times the unit. 
Hecato = 100 times the unit. 
Kilio = 1000 times the unit. 
Myrio = 10000 times the unit. 



French Measure of Length. 



1 millimetre = 0.03937 inch. 
1 centimetre = 0.3937 inch. 
1 decimetre = 3.937 inches. 
1 metre (unit) = 39.37 inches. 
1 sea mile = 1853.25 metres. 
1 kilometre = 0.53959 sea mile. 



1 metre (unit) = 3.28083 feet. 
1 decametre = 32.8083 feet. 

= 328.083 feet. 

= 3280.83 ft. = 0.62137 
mile. 

= 1.60935 kilomets. 

= 49.7096 chains. 



1 hectometre 
1 kilometre 



1 statute mile 
1 kilometre 



French Measure of Surface. 



1 square metre = 10.764 square feet. 
1 are = 100 square metres. 

1 decare = 10 ares. 

1 hectare = 100 ares. 



1 are = 1076.4 square feet. 

1 decare = 107.64 square feet. 

1 hectare = 2.471 Eng. acres. 
1 square mile = 259 hectares. " 



French Measure of Volume. 



1 r e U) Ut>iC } = 1 0^casteres. 

1 stere = 1000 litres. 

1 litre = 1 cubic decimetre. 

1 decistere = 3.5314 cubic feet. 



1 stere = 35.314 Eng. cubic feet. 
1 litre = 61.023 Eng. cub. inches. 

1 gallon == 3.7854 litres. 
1 decistere = 2.838 bushels (nearly). 



French Measure of Weight. 



1 ton 

1 ton 

1 kilogramme 
1 hectogramme 
1 decagramme 
1 gramme 

1 French ton 



= 1 cubic metre dis- 
tilled water. 

= 1000 kilogrammes. 

= 1000 grammes. 

= 100 grammes. 

= 10 grammes. 

= 1 cubic centimetre 
distilled water. 

= 0.9842 Eng. ton. 



1 gramme 
1 decigramme = 
1 centigramme = 
1 kilogramme = 

1 Eng. pound = 

1 gramme 

1 English ton = 



: 10 decigrammes. 
= 10 centigrammes. 
■ 10 milligrammes. 
: 2.20462 pounds av- 
oirdupois. 
= 0.45359 kilograms. 
■■ 15.43 grains trov. 
■■ 1.016 French tons. 



64 



The Metric System. 





Conversion 


of English Inches into Centimetres. 




Inches. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Cm. 


Cm. 


Cm. 


Cm. 


Cm. 


Cm. 


Cm. 


Cm. 


Cm. 


Cm. 





0.000 


2.540 


5.080 


7.620 


10.16 


12.70 


15.24 


17.78 


20.32 


22.86 


10 


25.40 


27.94 


30.48 


33.02 


35.56 


38.10 


40.64 


43.18 


45.72 


48.26 


20 


50.80 


53.34 


55.88 


58.42 


60.96 


63.50 


66.04 


68.58 


71.12 


73.66 


30 


76.20 


78.74 


81.28 


83.82 


86.36 


88.90 


91.44 


93.98 


96.52 


99.06 


40 


101.60 


104.14 


106.68 


109.22 


111.76 


114.30 


116.84 


119.38 


121.92 


124.46 


50 


127.00 


129.54 


132.08 


134.62 


137.16 


139.70 


142.24 


144.78 


147.32 


149.86 


60 


152.40 


154.94 


157.48 


160.02 


162.56 


165.10 


167.64 


170.18 


172.72 


175.26 


70 


177.80 


180.34 


182.88 


185.42 


187.96 


190.50 


193.04 


195.58 


198.12 


200.96 


80 


203.20 


205.74 


208.28 


210.82 


213.36 


215.90 


218.44 


220.98 


223.52 


226.06 


90 


228.60 


231.14 


233.68 


236.22 


238.76 


241.30 


243.84 


246.38 


248.92 


251.46 


100 


254.00 


256.54 


259.08 


261.62 


264.16 


266.70 


269.24 


271.78 


274.32 


276.85 





Conversion 


of Centimetres into English Inches. 




Cm. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 





0.000 


0.394 


0.787 


1.181 


1.575 


1.969 


2.362 


2.756 


3.150 


3.543 


10 


3.937 


4.331 


4.742 


5.118 


5.512 


5.906 


6.299 


6.693 


7.087 


7.480 


20 


7.874 


8.268 


8.662 


9.055 


9.449 


9.843 


10.236 


10.630 


11.024 


11.418 


30 


11.811 


12.205 


12.599 


12.992 


13.386 


13.780 


14.173 


14.567 


14.961 


15.355 


40 


15.748 


16.142 


16.536 


16.929 


17.323 


17.717 


18.111 


18.504 


18.898 


19.292 


50 


19.685 


20.079 


20.473 


20.867 


21.260 


21.654 


22.048 


22.441 


22.835 


23.229 


60 


23.622 


24.016 


24.410 


24.804 


25.197 


25.591 


25.985 


26.378 


26.772 


27.166 


70 


27.560 


27.953 


28.347 


28.741 


29.134 


29.528 


29.922 


30.316 


30.709 


31.103 


80 


31.497 


31.890 


32.284 


32.678 


33.071 


33.465 


33.859 


34.253 


34.646 


35.040 


90 


35.434 


35.827 


36.221 


36.615 


37.009 


37.402 


37.796 


38.190 


38.583 


38.977 


100 


39.370 


39.764 


40.158 


40.552 


40.945 


41.339 


41 .733 


42.126 


42.520 


42.914 





Convers 


ion of English Feet into Metres. 






Feet. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 





0.000 


0.3048 


0.6096 


0.9144 


1.2192 


1.5239 


1.8287 


2.1335 


2.4383 


2.7431 


10 


3.0479 


3.3527 


3.6575 


3.9623 


4.2671 


4.5719 


4.8767 


5.1815 


5.4863 


5.7911 


20 


6.0359 


6.4006 


6.7055 


7.0102 


7.3150 


7.6198 


7.9246 


8.2294 


8.5342 


8.8390 


30 


9.1438 


9.4486 


9.7534 


10.058 


10.363 


10.668 


10.972 


11.277 


11.582 


11.887 


40 


12.192 


12.496 


12.801 


13.106 


13.411 


13.716 


14.020 


14.325 


14.630 


14.935 


50 


15.239 


15.544 


15.849 


16.154 


16.459 


16.763 


17.068 


17.373 


17.678 


17.983 


60 


18.287 


18.592 


18.897 


19.202 


19.507 


19.811 


20.116 


20.421 


20.726 


21.031 


70 


21.335 


21.640 


21.945 


22.250 


22.555 


22.859 


23.164 


23.469 


23.774 


24.079 


80 


24.383 


24.688 


24.993 


25.298 


25.602 


25.907 


26.212 


26.517 


26.822 


27.126 


90 


27.431 


27.736 


28.041 


28.346 


28.651 


28.955 


29.260 


29.565 


29.870 


30.174 


100 


30.479 


30.784 


31.089 


31.394 


31.698 


32.003 


32.308 


32.613 


32.918 


33.222 





Convers 


ion of Metres into English Feet. 






Metres. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 





0.000 


3.2809 


6.5618 


9.8427 


13.123 


16.404 


19.685 


22.966 


26.247 


29.528 


10 


32.809 


36.090 


39.371 


42.651 


45.932 


49.213 


52.494 


55.775 


59.056 


62.337 


20 


65.618 


68.899 


72.179 


75.461 


78.741 


82.022 


85.303 


88.584 


91.865 


95.146 


30 


98.427 


101.71 


104.99 


108.27 


111.55 


114.83 


118.11 


121.39 


124.67 


127.96 


40 


131.24 


134.52 


137.80 


141.08 


144.36 


147.64 


150.92 


154.20 


157.48 


160.76 


50 


164.04 


167.33 


170.61 


173.89 


177.17 


180.45 


183.73 


187.01 


190.29 


193.57 


60 


196.85 


200.13 


203.42 


206.70 


209.98 


213.26 


216.54 


219.82 


223.10 


226.38 


70 


229.66 


232.94 


236.22 


239.51 


242.79 


246.07 


249.35 


252.63 


255.91 


259.19 


80 


262.47 


265.75 


269.03 


272.31 


275.60 


278.88 


282.16 


285.44 


288.72 


292.00 


90 


295.28 


298.56 


301.84 


305.12 


308.40 


311.69 


314.97 


318.25 


321.53 


324.81 


100 


328.09 


331.37 


334.65 


337.93 


341.21 


344.49 


347.78 


351.06 


354.34 


357.62 



The Metric System. 



65 



Conversion of English Statute 


-miles into Kilometres. 




Miles. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 





0.0000 


1.6093 


3.2186 


4.8279 


6.4372 


8.0465 


9.6558 


11.2652 


12.8745 


14.4848 


10 


16.093 


17.702 


19.312 


20.921 


22.530 


24.139 


25.749 


27.358 


28.967 


30.577 


20 


32.186 


33.795 


35.405 


37.014 


38.623 


40.232 


41.842 


43.451 


45.060 


46.670 


30 


48.279 


49.888 


51.498 


53.107 


54.716 


56.325 


57.935 


59.544 


61.153 


62.763 


40 


64.372 


65.981 


67.591 


69.200 


70.809 


72.418 


74.028 


75.637 


77.246 


78.856 


50 


80.465 


82.074 


83.684 


85.293 


86.902 


88.511 


90.121 


91.730 


93.339 


94.949 


60 


96.558 


98.167 


99.777 


101.39 


102.99 


104.60 


106.21 


107.82 


109.43 


111.04 


70 


112.65 


114.26 


115.87 


117.48 


119.08 


120.69 


122.30 


123.91 


125.52 


127.13 


80 


128.74 


130.35 


131.96 


133.57 


135.17 


136.78 


138.39 


140.00 


141.61 


143.22 


90 


144.85 


146.44 


148.05 


149.66 


151.26 


152.87 


154.48 


156.09 


157.70 


159.31 


100 


160.93 


162.53 


164.14 


165.75 


167.35 


168.96 


170.57 


172.18 


173.79 


175.40 



Conversion of Kilometres into English Statute=miles. 




Kilom. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Miles. 


Miles. 


Miles. 


Miles. 


Miles. 


Miles. 


Miles. 


Miles. 


Miles. 


Miles. 





0.0000 


0.6214 


1.2427 


1.8641 


2.4855 


3.1069 


3.7282 


4.3497 


4.9711 


5.5924 


10 


6.2138 


6.8352 


7.4565 


8.0780 


8.6994 


9.3208 


9.9421 


10.562 


11.185 


11.805 


20 


12.427 


13.049 


13.670 


14.292 


14.913 


15.534 


16.156 


16.776 


17.399 


18.019 


30 


18.641 


19.263 


19.884 


20.506 


21.127 


21.748 


22.370 


22.990 


23.613 


24.233 


40 


24.855 


25.477 


26.098 


26.720 


27.341 


27.962 


28.584 


29.204 


29.827 


30.447 


50 


31.069 


31.690 


32.311 


32.933 


33.554 


34.175 


34.797 


35.417 


36.040 


36.660 


60 


37.282 


37.904 


38.525 


39.147 


39.768 


40.389 


41.011 


41.631 


42.254 


42.874 


70 


43.497 


44.118 


44.739 


45.361 


45.982 


46.603 


47.225 


47.845 


48.468 


49.088 


80 


49.711 


50.332 


50.953 


51.575 


52.196 


52.817 


53.439 


54.059 


54.682 


55.302 


90 


55.924 


56.545 


57.166 


57.788 


58.409 


59.030 


59.652 


60.272 


60.895 


61.515 


100 


62.138 


62.759 


63.380 


64.002 


64.623 


65.244 


65.866 


66.486 


67.109 


67.729 





Conversion of Sea=miles into Kilometres. 






Sea-miles. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 





0.0000 


1.8532 


3.7046 


5.5596 


7.4128 


9.2660 


11.119 


12.972 


14.825 


16.788 


10 


18.532 


20.386 


22.237 


24.128 


25.945 


27.798 


29.651 


31.504 


33.357 


35.320 


20 


37.064 


38.918 


40.769 


42.660 


44.477 


46.331 


48.183 


50.036 


51.889 


53.852 


30 


55.596 


57.450 


59.301 


61.192 


63.009 


64.863 


66.715 


68.568 


70.421 


72.384 


40 


74.128 


75.982 


77.833 


79.724 


81.541 


83.396 


85.247 


87.100 


88.953 


90.916 


50 


92.660 


94.514 


96.365 


98.256 


100.07 


101.92 


103.78 


105.63 


107.48 


109.45 


60 


111.19 


113.05 


114.90 


116.79 


118.61 


120.45 


122.21 


124.16 


126.01 


127.98 


70 


129.72 


131.58 


133.43 


135.32 


137.14 


139.98 


140.74 


142.69 


144.54 


146.51 


80 


148.25 


150.11 


151.96 


153.85 


155.67 


157.52 


159.27 


161.22 


163.07 


165.04 


90 


166.78 


168.64 


170.49 


172.38 


174.20 


176.05 


177.80 


179.75 


181.60 


183.57 


100 


185.32 


187.18 


189.03 


190.88 


192.73 


194.58 


196.44 


198.28 


200.14 


201.99 





Co 


nvers 


ion of Kilometres 


into Sea=miles. 






Kilom. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Sea-m. 


Sea-m. 


Sea-m. 


Sea-m. 


Sea-m. 


Sea-m. 


Sea-m. 


Sea-m. 


Sea-m. 


Sea-m. 





0.0000 


0.5396 


1.0792 


1.6188 


2.1584 


2.6880 


3.2375 


3.7771 


4.3167 


4.8563 


10 


5.3959 


5.9356 


6.4751 


7.0147 


7.5543 


8.0839 


8.6334 


9.1730 


9.7126 


10.252 


20 


10.792 


11.331 


11.870 


12.410 


12.950 


13.480 


14.029 


14.568 


15.108 


15.647 


30 


16.188 


16.727 


17.265 


17.806 


18.345 


18.876 


19.424 


19.965 


20.504 


21.044 


40 


21.584 


22.123 


22.661 


23.202 


23.740 


24.271 


24.819 


25.360 


25.900 


26.439 


50 


26.980 


27.519 


28.059 


28.598 


29.135 


29.667 


30.214 


30.757 


31.296 


31.835 


60 


32.375 


32.915 


33.456 


33.994 


34.530 


35.063 


35.609 


36.151 


36.692 


37.231 


70 


37.771 


38.310 


38.852 


39.390 


39.925 


40.459 


41.004 


41.574 


42.088 


42.627 


80 


43.167 


43.705 


44.284 


44.786 


45.320 


45.855 


46.399 


46.943 


47.483 


48.023 


90 


48.563 


49.103 


49.644 


50.182 


50.715 


51.251 


51.794 


52.339 


52.879 


53.419 


100 


53.959 


54.498 


55.038 


55.575 


56.117 


56.658 


57.198 


57.737 


58.275 


58.816 



The Metric System. 



Conversion of Square Inches into Square Centimetres. 




Square in. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Cm2. 


Cm2. 


Cm2. 


Cm2. 


Cm2. 


Cms. 


Cm2. 


Cm2. 


Cm2. 


Cm2. 





0.0000 


6.4515 


12.903 


19.354 


25.806 


32.257 


38.709 


45.160 


51.612 


58.063 


10 


64.515 


70.967 


77.418 


83.869 


90.321 


96.772 


103.22 


109.67 


116.12 


122.57 


20 


129.03 


135.48 


141.93 


148.38 


154.83 


161.29 


167.74 


174.19 


180.64 


187.09 


30 


193.54 


199.99 


206.44 


212.89 


219.34 


225.80 


231.25 


238.70 


245.15 


251.60 


40 


258.06 


264.51 


270.96 


277.41 


283.86 


290.32 


296.77 


303.22 


309.67 


316.12 


50 


322.57 


329.02 


335.47 


341.92 


348.37 


354.83 


361.28 


367.73 


374.18 


380.63 


60 


387.09 


393.54 


399.99 


406.44 


412.89 


419.35 


425.80 


432.25 


438.70 


445.15 


70 


451.60 


458.05 


464.50 


470.95 


477.40 


483.86 


490.31 


496.76 


503.21 


509.66 


80 


516.12 


522.57 


529.02 


535.47 


541.92 


548.38 


554.83 


561.28 


567.73 


574.18 


90 


580.63 


587.08 


593.53 


599.98 


606.43 


612.89 


619.34 


625.79 


632.24 


638.69 


100 


645.15 


651.60 


658.05 


664.50 


670.95 


677.41 


683.86 


690.31 


696.76 


703.21 



Conversion of Square Centimetres into Square Inches. 




Square cm. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




In2. 


In2. 


In2. 


In2. 


In2. 


In2. 


In2. 


In2. 


im, 


In2. 





0.0000 


0.1550 


0.3100 


0.4650 


0.6200 


0.7750 


0.9300 


1.0850 


1.2400 


1.3950 


10 


1.5500 


1.7050 


1.8600 


2.0150 


2.1700 


2.3250 


2.4800 


2.6350 


2.7900 


2.9450 


20 


3.1000 


3.2550 


3.4100 


3.5650 


§.7200 


3.8750 


4.0300 


4.1850 


4.3400 


4.4950 


30 


4.6501 


4.8051 


4.9601 


5.1151 


5.2701 


5.4251 


5.5801 


5.7351 


5.8901 


6.0451 


40 


6.2001 


6.3551 


6.5101 


6.6651 


6.8201 


6.9751 


7.1301 


7.28*1 


7.4401 


7.5951 


50 


7.7501 


7.9051 


8.0601 


8.2151 


8.3701 


8.5251 


8.6801 


8.8351 


8.9901 


9.1451 


60 


9.3002 


9.4552 


9.6102 


9.7652 


9.9202 


10.075 


10.230 


10.385 


10.540 


10.695 


70 


10.850 


11.040 


11.160 


11.315 


11.470 


11.625 


11.780 


11.935 


12.090 


12.245 


80 


12.400 


12.555 


12.710 


12.865 


13.020 


13.175 


13.330 


13.485 


13.640 


13.795 


90 


13.950 


14.105 


14.260 


14.415 


14.570 


14.725 


14.880 


15.035 


15.190 


15.345 


100 


15.500 


15.655 


15.810 


15.965 


16.120 


16.275 


16.430 


16.585 


16.740 


16.895 



Conversion of Cubic Inches into Cubic Centimetres. 




Cubic in. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Cms. 


Cm3. 


Cms. 


Cms. 


Cm3. 


Cm3. 


Cm3. 


Cm3. 


Cm3. 


Cm3. 





0.0000 


16.383 


32.773 


49.160 


65.546 


81.933 


98.320 


114.71 


131.01 


147.48 


10 


163.87 


180.26 


196.64 


213.03 


229.41 


245.80 


262.19 


278.58 


294.88 


311.35 


20 


327.73 


344.12 


360.50 


376.89 


393.27 


409.66 


426.05 


442.44 


458.74 


475.21 


30 


491.60 


507.99 


524.37 


540.76 


557.14 


573.53 


569.92 


606.31 


622.61 


639.08 


40 


655.46 


671.85 


688.23 


704.52 


721.00 


737.39 


753.78 


770.17 


786.47 


802.94 


50 


819.33 


835.72 


851.10 


868.49 


884.87 


901.26 


917.65 


934.04 


950.34 


966.81 


60 


983.20 


999.59 


1016.0 


1032.4 


1048.7 


1065.1 


1081.5 


1097.9 


1114.2 


1130.7 


70 


1147.1 


1163.5 


1179.9 


1196.3 


1212.6 


1229.0 


1245.4 


1261.8 


1278.1 


1294.6 


80 


1310.9 


1327.3 


1343.7 


1360.1 


1376.4 


1392.8 


1409.2 


1425.6 


1441.9 


1458.4 


90 


1474.8 


1491.2 


1507.6 


1524.0 


1540.3 


1556.7 


1573.1 


1589.5 


1605.8 


1622.3 


100 


1638.7 


1655.1 


1671.5 


1687.9 


1704.2 


1720.6 


1737.0 


1753.4 


1769.7 


1786.2 



Conversion of Cubic Centimetres into Cubic Inches. 




Cubic cm. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Iu3. 


Ia3. 


In3. 


In3. 


In*. 


In3. 


Ins. 


In3. 


In3. 


In3. 





0.0000 


0.0610 


0.1221 


0.1831 


0.2441 


0.3051 


0.3661 


0.4272 


0.4882 


0.5492 


10 


0.6102 


0.6712 


0.7323 


0.7933 


0.8543 


0.9153 


0.9763 


1.0374 


1.0984 


1.1594 


20 


1.2205 


1.2815 


1.3426 


1.4036 


1.4646 


1.5256 


1.5866 


1.6477 


1.7087 


1.7697 


30 


1.8308 


1.8918 


1.9529 


2.0139 


2.0749 


2.1359 


2.1969 


2.2580 


2.3190 


2.3800 


40 


2.4410 


2.5020 


2.5631 


2.6241 


2.6851 


2.7461 


2.8071 


2.8682 


2.9292 


2.9902 


50 


3.0513 


3.1123 


3.1734 


3.2344 


3.2954 


3.3564 


3.4174 


3.4785 


3.5395 


3.6005 


60 


3.6615 


3.7225 


3.7836 


3.8446 


3.9056 


3.9666 


4.0276 


4.0887 


4.1497 


4.2107 


70 


4.2718 


4.3328 


4.3939 


4.4549 


4.5159 


4.5769 


4.6379 


4.6990 


4.7600 


4.8210 


80 


4.8820 


4.9430 


5.0041 


5.0651 


5.1261 


5.1871 


5.2481 


5.3092 


5.3702 


5.4312 


90 


5.4923 


5.5533 


5.6144 


5.6754 


5.7364 


5.7974 


5.8584 


5.9195 


5.9805 


6.0415 


100 


6.1025 


6.1635 


6.2246 


6.2856 


6.3466 


6.4076 


6.4686 


6.5297 


6.5907 


6.6517 



The Metric System. 



67 





Conversion of Cubic Yards 


into Cubic Metres. 




Cubic yds. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Met 3 . 


Met 3 . 


Met 3 . 


Met 3 . 


Met 3 . 


Met 3 . 


Met 3 . 


Met 3 . 


Met 3 . 


Met 3 . 





0.0000 


0.7645 


1.5291 


2.2936 


3.0581 


3.8226 


4.5872 


5.3517 


6.1163 


6.8808 


10 


7.6453 


8.4098 


9.1744 


9.9389 


10.703 


11.468 


12.232 


12.997 


13.761 


14.526 


20 


15.291 


16.055 


16.820 


17.585 


18.349 


19.114 


19.878 


20.643 


21.407 


22.172 


30 


22.936 


23.700 


24.455 


25.230 


25.994 


26.759 


27.523 


28.288 


29.052 


29.817 


40 


30.581 


31.345 


32.110 


32.875 


33.639 


34.404 


35.168 


35.933 


36.797 


37.462 


50 


38.226 


38.990 


39.755 


40.520 


41.284 


42.049 


42.813 


43.578 


44.342 


45.107 


60 


45.872 


46.636 


47.401 


48.166 


48.930 


49.695 


50.459 


51.224 


51.988 


52.753 


70 


53.517 


54.281 


55.046 


55.811 


56.575 


57.340 


58.104 


58.869 


59.633 


60.398 


80 


61.163 


61.927 


62.692 


63.457 


64.221 


64.986 


65.750 


66.515 


67.279 


68.044 


90 


68.808 


69.572 


70.337 


71.102 


71.866 


72.631 


73.395 


74.160 


74.924 


75.689 


100 


76.453 


77.217 


77.982 


78.747 


79.511 


80.276 


81.040 


81.805 


82.569 


83.334 





Conversion of Cubic Metres 


into Cubic 


Yards 


;. 




Cubic met. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Yds 3 . 


Yds 3 . 


Yds 3 . 


Yds 3 . 


Yds 3 . 


Yds- 3 . 


Yds 3 . 


Yds 3 . 


Yds 3 . 


Yds 3 . 





0.0000 


1.3080 


2.6160 


3.9240 


5.2329 


6.5399 


7.8479 


9.1559 


10.464 


11.772 


10 


13.080 


14.388 


15.696 


17.004 


18.313 


19.620 


20.928 


22.236 


23.544 


24.852 


20 


26.160 


27.468 


28.776 


30.084 


31.393 


32.700 


34.008 


35.316 


36.624 


37.932 


30 


39.240 


40.548 


41.856 


43.164 


44.473 


45.780 


47.088 


48.396 


49.704 


51.012 


40 


52.319 


53.627 


54.935 


56.243 


57.552 


58.859 


60.167 


61.475 


62,783 


63.091 


50 


65.399 


66.707 


68.015 


69.323 


70.632 


71.939 


73.247 


74.545 


75.863 


77.171 


60 


78.479 


79.787 


81.095 


82.403 


83.712 


85.019 


86.327 


87.535 


88.943 


90.251 


70 


91.559 


92.867 


94.175 


95.483 


96.792 


98.099 


99.407 


100.71 


102.02 


103.33 


80 


104.63 


105.94 


107.25 


108.56 


109.87 


111.17 


112.48 


113.79 


115.10 


116.41 


90 


117.72 


119.03 


120.34 


121.64 


122.95 


124.26 


125.57 


126.88 


128.18 


129.49 


100 


130.80 


132.11 


133.42 


134.72 


136.03 


137.34 


138.65 


139.96 


141.26 


142.57 





Conversion of U. S 


Gallons into Litres. 






Gallons. 





1 


2 


3 


4 


5 


6 


7 


8 


9 



10 
20 
30 
40 
50 
60 
70 
80 
90 
100 


Litres. 

0.0000 
37.853 
75.706 
113.56 
151.42 
189.46 
227.12 
264.97 
302.82 
440.68 
478.53 


Litres. 

3.7853 
41.638 
79.491 
117.34 
155.22 
193.24 
230.90 
268.75 
306.60 
444.46 
482.31 


Litres. 

7.5706 
45.423 
83.276 
121.13 
158.99 
197.03 
234.69 
272.54 
310.39 
448.25 
486.10 


Litres. 

11.356 
49.209 
87.062 
124.92 

162.78 
200.82 
238.48 
276.33 
314.18 
452.04 
789.89 


Litres. 
15.141 
52.994 
90.847 
128.66 
166.56 
204.60 
242.26 
280.11 
317.96 
455.82 
493.67 


Litres. 

18.946 
56.799 
94.652 
132.50 
170.36 
208.40 
246.06 
283.91 
321.76 
459.62 
497.47 


Litres. 
22.712 
60.565 
98.418 
136.27 
174.13 
212.17 
249.83 
286.68 
324.53 
463.39 
501.24 


Litres. 
26.497 
64.350 
102.20 
140.06 
177.92 
215.96 
253.62 
291.47 
329.32 
467.18 
505.03 


Litres. 
30.282 
68.135 
105.99 
143.84 
181.70 
219.74 
257.40 
295.25 
333.10 
470.96 
508.81 


Litres, 

34.068 
71.921 
109.77 
147.63 
185.49 
223.53 
261.19 
299.04 
336.89 
474.75 
512.60 





Conversion of Litres into U. S 


. Gall 


ons. 






Litres. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Gal. 


Gal. 


Gal. 


Gal. 


Gal. 


Gal. 


Gal. 


Gal. 


Gal. 


Gal. 





0.0000 


0.2642 


0.5284 


0.7925 


1.0567 


1.3209 


1.5851 


1.8492 


2.1134 


2.3776 


10 


2.6418 


2.9060 


3.1702 


3.4343 


3.6985 


3.9627 


4.2269 


4.4910 


4.7552 


5.0194 


20 


5.2836 


5.5478 


5.8120 


6.0761 


6.3403 


6.6045 


6.8687 


7.1328 


7.3970 


7.6612 


30 


7.9254 


8.1896 


8.4538 


8.7179 


8.9821 


9.2463 


9.5105 


9.8746 


10.030 


10.303 


40 


10.567 


10.831 


11.095 


11.360 


11.624 


11.888 


12.152 


12.416 


12.680 


12.945 


50 


13.209 


13.473 


13.737 


14.002 


14.266 


14.530 


14.794 


15.058 


15.322 


15.587 


60 


15.851 


16.115 


16.379 


16.644 


16.908 


17.172 


17.436 


17.700 


17.964 


18.229 


70 


18.492 


18.756 


19.020 


19.284 


19.549 


19.813 


20.077 


20.341 


20.605 


20.870 


80 


21.134 


21.-398 


21.662 


21.926 


22.191 


22.455 


22.719 


22.983 


23.247 


23.512 


90 


23.776 


24.040 


24.304 


24.568 


24.832 


25.097 


25.361 


25.625 


25.889 


26.154 


100 


26.418 


26.682 


26.946 


27.210 


27.475 


27.739 


28.003 


28.267 


28.531 


28.796 



68 



The Metric System. 







Conversion of Yards into Metres 








Yards. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 





0.0000 


0.9144 


1.8288 


2.7432 


3.6576 


4.5719 


5.4863 


6.4007 


7.3151 


8.2295 


10 


9.1439 


10.058 


10.973 


11.887 


12.801 


13.716 


14.630 


15.544 


16.458 


17.373 


20 


18.288 


19.202 


20.117 


21.031 


21.945 


22.860 


23.774 


24.689 


25.603 


26.518 


30 


27.432 


28.346 


29.260 


30.174 


31.088 


32.003 


32.917 


33.832 


34.746 


35.661 


40 


36.576 


37.490 


38.404 


39.318 


40.232 


41.147 


42.061 


42.976 


43.890 


44.805 


50 


45.719 


46.634 


47.548 


48.462 


49.376 


50.291 


51.205 


52.120 


53.034 


53.949 


60 


54.863 


55.778 


56.692 


57.606 


58.520 


59.435 


60.349 


61.264 


62.178 


63.093 


70 


64.007 


64.922 


65.836 


66.750 


67.664 


68.578 


69.493 


70.408 


71.322 


72.237 


80 


73.151 


74.066 


74.980 


75.894 


76.808 


77.723 


78.637 


79.552 


80.466 


81.381 


90 


82.295 


83.210 


84.124 


85.038 


85.952 


86.867 


87.781 


88.696 


89.610 


90.525 


100 


91.439 


92.353 


93.267 


94.181 


95.095 


96.010 


96.924 


97.839 


98.753 


99.668 







Conversion of Metres into Yards 








Metres. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Yds. 


Yds. 


Yds. 


Yds. 


Yds. 


Yds. 


Yds. 


Yds. 


Yds. 


Yds. 





0.0000 


1.0936 


2.1872 


3.2809 


4.3745 


5.4681 


6.5617 


7.6553 


8.7490 


9.8426 


10 


10.936 


12.029 


13.122 


14.217 


15.310 


16.404 


17.498 


18.591 


19.685 


20.778 


20 


21.872 


22.966 


24.059 


25.153 


26.247 


27.340 


28.434 


29.527 


30.621 


31.715 


30 


32.809 


33.900 


34.993 


36.090 


37.184 


38.277 


39.371 


40.464 


41.558 


42.652 


40 


43.745 


44.839 


45.932 


47.026 


48.120 


49.213 


50.307 


51.400 


52.544 


53.588 


50 


54.681 


55.775 


56.868 


57.962 


59.056 


60.149 


61.243 


62.336 


63.430 


64.524 


60 


65.617 


66.711 


67.804 


68.898 


69.992 


71.085 


72.179 


73.272 


74.366 


75.460 


70 


76.553 


77.647 


78.740 


79.834 


80.928 


82.021 


83.115 


84.208 


85.302 


86.396 


80 


87.490 


88.584 


89.677 


90.771 


91.865 


92.958 


94.052 


95.145 


96.239 


97.333 


90 


98.426 


99.520 


100.61 


101.71 


102.80 


103.89 


104.99 


106.08 


107.17 


108.27 


100 


109.36 


110.45 


111.55 


112.64 


113.73 


114.83 


115.92 


117.02 


118.11 


119.20 



Conversion of Square Yards 


nto Square 


Metres. 




Sq. yards. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Met2. 


Met 2 . 


Met 2 . 


Met2. 


Met 2 . 


Met 2 . 


Met 2 . 


Met2. 


Met 2 . 


Met 2 . 





0.0000 


0.8361 


1.6722 


2.5803 


3.3444 


4.1805 


5.0167 


5.8528 


6.6889 


7.5250 


10 


8.3611 


9.1972 


10.033 


10.941 


11.706 


12.542 


13.378 


14.214 


15.050 


15.886 


20 


16.722 


17.558 


18.394 


19.102 


20.066 


20.903 


21.739 


22.575 


23.411 


24.247 


30 


25.083 


25.919 


26.755 


27.663 


28.431 


29.264 


30.100 


30.936 


31.772 


32.608 


40 


33.444 


34.280 


35.116 


36.024 


36.788 


37.625 


38.461 


39.297 


40.133 


40.969 


50 


41.805 


42.641 


43.477 


44.385 


45.149 


45.986 


46.822 


47.658 


48.494 


49.330 


60 


50.167 


51.003 


51.839 


52.747 


53.511 


54.348 


55.184 


56.020 


56.856 


57.692 


70 


58.528 


59.364 


60.190 


61.108 


61.872 


62.709 


63.545 


64.381 


65.217 


66.053 


80 


66.889 


67.725 


68.561 


69.469 


70.233 


71.070 


71.906 


72.742 


73.578 


74.414 


90 


75.250 


76.086 


76.922 


77.830 


78.594 


79.431 


80.267 


81.103 


81.939 


82.775 


100 


83.611 


84.447 


85.283 


86.191 


86.955 


87.792 


88.628 


89.464 


90.300 


91.136 



Conversion of Square Metres into Square Yards. 




Sq. metres. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Yds 2 . 


Yds2. 


Yds 2 . 


Yds 2 . 


Yds 2 . 


Yds 2 . 


Yds 2 . 


Yds 2 . 


Yds 2 . 


Yds 2 . 





0.0000 


1.1960 


2.3920 


3.5880 


4.7840 


5.9800 


7.1760 


8.3720 


9.5681 


10.764 


10 


11.960 


13.156 


14.352 


15.548 16.744 


17.940 


19.136 


20.332 


21.528 


22.724 


20 


23.920 


25.116 


26.312 


27. 508 j 28.704 


29.900 


31.096 


32.292 


33.488 


34.684 


30 


35.880 


37.076 


38.272 


39.468! 40.664 


41.860 


43.056 


44.252 


45.448 


46.644 


40 


47.840 


49.036 


50.232! 51.4281 52.624 


53.820 


55.016 


56.212 


57.408 


58.604 


50 


59.800 


60.996 


62.192 63.388 63.584 


65.780 


66.976 68.172 


69.368 70.564 


60 


71.760 


72.956 


74.152 75.348 76.544 


77.740 


78.936 80.132 


81.328 82.524 


70 


83.721 S4.917 


86.113 87.309 88.505 89.701 


90.897i92.093 93.289 94.485 


80 


95.681 96.877 


98.073 99.269 100.46 101.6(5 


102.86] 104.06! 105.25 106.44 


90 


107.(11 108.84 


110.03 111.241 112.44 113.62 


114.81 116.011 117.21! 118.40 


100 


119.60, 120.80 


121.99 123.19j 124.38 125.58 


126.77 127.97 129.17 j 130.36 



The Metkic System. 



69 







Conversion of Hectares into Acres 








Hectares. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Acres. 


Acres. 


Acres. 


Acres. 


Acres. 


Acres. 


Acres. 


Acres. 


Acres. 


Acres. 





0.0000 


2.4711 


4.9422 


7.4133 


9.8844 


12.355 


14.836 


17.298 


19.769 


22.240 


10 


24.711 


27.182 


29.653 


32.124 


34.695 


37.046 


39.547 


42.009 


44.480 


46.951 


20 


49.422 


51.893 


54.364 


56.835 


59.306 


61.757 


64.258 


66.720 


68.191 


71.662 


30 


74.133 


76.604 


79.075 


81.546 


84.017 


86.468 


88.969 


91.431 


93.902 


96.373 


40 


98.844 


101.31 


103.79 


106.26 


108.73 


111.18 


113.68 


116.14 


118.61 


121.08 


50 


123.55 


126.02 


128.49 


130.96 


133.43 


135.88 


138.38 


140.85 


143.32 


145.79 


60 


148.36 


150.83 


153.30 


155.77 


158.24 


160.69 


163.19 


165.66 


168.13 


170.60 


70 


172.95 


175.45 


177.92 


180.39 


182.86 


185.31 


187.81 


190.28 


192.75 


195.22 


80 


197.69 


200.16 


202.63 


205.10 


207.57 


210.02 


212.52 


214.99 


217.46 


219.93 


90 


222.40 


224.87 


227.34 


229.81 


232.28 


234.73 


237.23 


239.70 


242.17 


244.64 


100 


247.11 


249.58 


252.05 


254.52 


256.99 


259.44 


261.94 


264.41 


266.88 


269.35 







Conversion of Acres into Hectares. 






■ Acres. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Hect. 


Hect. 


Hect. 


Hect. 


Hect. 


Hect. 


Hect. 


Hect. 


Hect. 


Hect. 





0.0000 


0.4047 


0.8093 


1.2140 


1.6187 


2.0234 


2.4280 


2.8327 


3.2374 


3.6420 


10 


4.0468 


4.4515 


4.8561 


5.2608 


5.6655 


6.0702 


6.4748 


6.8795 


7.2782 


7.6888 


20 


8.0936 


8.4983 


8.9029 


9.3076 


9.7123 


10.117 


10.521 


10.926 


11.331 


11.735 


30 


12.140 


12.545 


12.949 


13.354 


13.759 


14.163 


14.568 


14.973 


15.377 


15.782 


40 


16.187 


16.592 


16.996 


17.401 


17.806 


18.210 


18.615 


19.020 


19.414 


19.829 


50 


20.234 


20.639 


21.043 


21.448 


21.853 


22.257 


22.662 


23.067 


23.471 


23.876 


60 


24.280 


24.685 


25.089 


25.494 


25.899 


26.303 


26.708 


27.113 


27.517 


27.922 


70 


28.327 


28.732 


29.136 


29.541 


29.946 


30.350 


30.755 


31.160 


31.564 


31.969 


80 


32.374 


32.779 


33.183 


33.588 


33.993 


34.397 


34.802 


35.207 


35.611 


36.016 


90 


36.420 


36.825 


37.229 


37.634 


38.039 


38.443 


38.848 


39.253 


39.657 


40.062 


100 


40.468 


40.873 


41.277 


41.682 


42.087 


42.491 


42.896 


43.301 


43.695 


44.110 



Conversion of Square Miles into Square Kilometres. 




Sq. miles. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Kil2. 


Kil2. 


KiR 


Kil2. 


KiF. 


KiF. 


Kil2. 


Kil2. 


Kil2. 


KiF. 





0.0000 


2.5899 


5.1798 


7.7697 


10.359 


12.929 


15.539 


18.129 


20.718 


23.309 


10 


25.899 


28.490 


31.079 


33.669 


36.259 


38.829 


41.439 


44.029 


46.619 


49.209 


20 


51.798 


54.388 


56.978 


59.568 


62.158 


64.728 


67.338 


69.928 


72.518 


75.108 


30 


77.697 


80.287 


82.877 


85.467 


88.057 


90.627 


93.238 


96.828 


98.417 


101.01 


40 


103.59 


106.18 


108.77 


111.36 


113.95 


116.52 


119.13 


121.72 


124.31 


126.90 


50 


129.29 


131.88 


134.47 


137.06 


139.65 


142.22 


144.83 


147.42 


150.01 


152.50 


60 


155.39 


157.98 


160.57 


163.16 


165.75 


168.32 


170.93 


173.52 


176.11 


178.70 


70 


181.29 


183.88 


186.47 


188.06 


191.65 


194.22 


196.83 


199.42 


202.01 


204.60 


80 


207.19 


209.77 


212.36 


214.95 


217.55 


220.11 


222.73 


225.31 


227.91 


230.50 


90 


233.09 


235.68 


238.27 


240.86 


243.45 


246.02 


248.63 


251.22 


253.81 


256.40 


100 


258.99 


261.58 


264.17 266.76 


269.35 


271.92 


274.53 


277.12 


279.71 


282.20 



Conversion of Square Kilometres into Square Miles. 




Sq. kilom. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Sq. m. 


Sq. m. 


Sq. m. 


Sq. m. 


Sq. m. 


Sq. m. 


Sq. m. 


Sq. m 


Sq. m. 


Sq. m. 





0.0000 


0.3861 


0.7722 


1.1583 


1.5445 


1.9304 


2.3166 


2.7028 


3.0890 


3.4749 


10 


3.8612 


4.2471 


4.6334 


5.0195 


5.4057 


5.7916 


6.1778 


6.5640 


6.9502 


7.3362 


20 


7.7224 


8.1081 


8.4946 


8.8807 


9.2669 


9.6528 


10.039 


10.425 


10.811 


11.197 


30 


11.583 


11.969 


12.355 


12.741 


13.127 


13.513 


13.899 


14.286 


14.672 


15.058 


40 


15.445 


15.830 


16.217 


16.603 


16.989 


17.375 


17.761 


18.146 


18.534 


18.920 


50 


19.304 


19.691 


20.076 


20.462 


20.848 


21.234 


21.620 


22.007 


22.393 


22.779 


60 


23.166 


23.552 


23.938 


24.324 


24.710 


25.096 


25.482 


25.869 


26.245 


26.641 


70 


27.028 


27.413 


27.800 


28.186 


28.572 


28.958 


29.344 


29.731 


30.117 


30.503 


80 


30.890 


31.274 


31.662 


32.048 


32.434 


32.820 


33.206 


33.593 


33.979 


34.365 


90 


34.749 


35.135 


35.521 


35.907 


36.293 


36.679 37.065 


37.452 


37.838 


38.224 


100 


38.612 


38.996 


39.384 


39.770 


40.156 


40.542 40.928 


41.315 


41.701 


42.087 



70 



The Metkic System. 



Conversion of Cubic Feet into Cubic Decimetres. 




Cubic feet. 





1 


2 | 3 


4 


5 


6 


7 


8 


9 




Dm 3 . 


Dm 3 . 


Dm 3 . { Dm 2 . 


Dm 3 . 


Dm 3 . 


Dm3. 


Dm3. 


Dm 3 . 


Dm 3 . 





0.0000 


28.316 


56.632 


84.948 


113.26 


141.58 


169.90 


198.21 


226.53 


254.84 


10 


283.16 


311.47 


339.79 


268.11 


396.42 


424.74 


453.06 


481.37 


509.69 


538.00 


20 


566.32 


594.64 


622.95 


651.27 


679.58 


707.90 


736.22 


764.53 


792.85 


821.16 


30 


849.48 


877.80 


906.11 


934.43 


962.74 


991.06 


1019.4 


1047.7 


1076.0 


1104.3 


40 


1132.6 


1160.8 


1189.2 


1217.5 


1245.9 


1274.2 


1302.5 


1330.8 


1359.1 


1387.4 


50 


1415.8 


1444.0 


1472.4 


1500.7 


1529.1 


1557.4 


1585.7 


1614.0 


1642.3 


1670.6 


60 


1698.9 


1727.2 


1755.5 


1783.8 


1812.2 


1840.5 


1868.8 


1897.1 


1925.4 


1953.7 


70 


1982.1 


2010.3 


2038.7 


2067.0 


2095.4 


2123.7 


2152.0 


2180.3 


2208.6 


2236.9 


80 


2265.3 


2293.5 


2321.9 


2350.2 


2378.6 


2406.9 


2435.2 


2463.5 


2491.8 


2520.1 


90 


2548.4 


2576.6 


2605.0 


2633.3 


2661.6 


2690.0 


2718.3 


2746.6 


2774.9 


2803.2 


100 


2831.6 


2859.8 


2888.2 


2916.5 


2944.9 


2973.2 


3001.5 


3029.8 


3058.1 


3086.4 



Conversion of Cubic Decimetres into Cubic Feet. 




Cubic dm. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Ft 3 . 


Ft3. 


Ft3. 


Ft3. 


Ft3. 


Ftf*. 


Fts. 


Ft 3 . 


Ft 3 . 


Ft 3 . 





0.0000 


0.0353 


0.0706 


0.1059 


0.1413 


0.1766 


0.2119 


0.2472 


0.2825 


0.3178 


10 


0.3531 


0.3884 


0.4237 


0.4590 


0.4944 


0.5297 


0.5540 


0.6003 


0.6356 


0.6709 


20 


0.7063 


0.7416 


0.7766 


0.8122 


0.8476 


0.8829 


0.9182 


0.9535 


0.9888 


1.0241 


30 


1.0594 


1.0947 


1.1300 


1.1653 


1.2007 


1.2360 


1.2713 


1.3066 


1.3419 


1.3772 


40 


1.4126 


1.4479 


1.4832 


1.5185 


1.5539 


1.5892 


1.6245 


1.6608 


1.6951 


1.7304 


50 


1.7658 


1.8011 


1.8364 


1.8717 


1.9071 


1.9424 


1.9777 


2.0130 


2.0483 


2.0836 


60 


2.1189 


2.1542 


2.1895 


2.2248 


2.2602 


2.2955 


2.3308 


2.3661 


2.4014 


2.4367 


70 


2.4721 


2.5074 


2.5427 


2.5780 


2.6134 


2.6487 


2.6840 


2.7193 


2.7546 


2.7899 


80 


2.8252 


2.8605 


2.8958 


2.9311 


2.9665 


3.0018 


3.0371 


3.0724 


3.1077 


3.1430 


90 


3.1784 


3.2137 


3.2490 


3.2843 


3.3197 


3.3550 


3.3903 


3.4256 


3.4609 


3.4962 


100 


3.5315 


3.5668 


3.6021 


3.6374 


3.6728 


3.7081 


3.7434 


3.7787 


3.8140 


3.8493 



Pounds 


per Square Foot into Kilogrammes per 


Square Metre. 


Lb 8. pr ft 2 . 





1 


2 


3 


4 


5 


6 


7 


8 


9 




K.m2. 


K. m2. 


K.m2. 


K.m 2 . 


K.m2. 


K. m2. 


K.m2. 


K.m2. 


K.m2. 


K.m2. 





0.0000 


4.8825 


9.7650 


14.647 


19.530 


24.413 


29.295 


34.177 


39.006 


43.943 


10 


48.825 


53.707 


58.590 


63.472 


68.355 


73.238 


78.120 


83.002 


87.831 


92.768 


20 


97.650 


102.53 


107.41 


112.30 


117.18 


122.06 


126.94 


131.83 


136.66 


141.59 


30 


146.47 


151.35 


156.23 


161.12 


165.90 


170.88 


175.76 


180.65 


185.47 


190.41 


40 


195.30 


200.13 


205.06 


209.95 


214.83 


219.71 


224.59 


229.48 


234.30 


239.24 


50 


244.13 


249.01 


253.89 


258.78 


263.66 


268.54 


273.42 


278.31 


283.13 


288.08 


60 


292.95 


297.83 


302.71 


307.60 


312.48 


317.36 


322.24 


327.13 


331.95 


336.89 


70 


341.77 


346.65 


351.53 


356.42 


361.20 


366.18 


371.06 


375.95 


380.77 


385.71 


80 


390.06 


394.94 


399.82 


404.71 


409.59 


414.47 


419.35 


424.24 


429.06 


434.00 


90 


439.43 


444.31 


449.19 


454.08 


458.96 


463.34 


468.72 


473.61 


478.43 


483.37 


100 


488.25 


493.13 


498.01 


502.90 


507.78 


512.66 


517.54 


522.43 


527.25 


532.19 



Kilogrammes per Square Metre into Pounds per Square Foot. 


K. per m 2 . 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Lb. ft 2 


Lb. ft 2 


Lb. ft 2 


Lb. ft 2 


Lb. ft 2 


Lb. ft 2 


Lb. ft 2 


Lb. ft 2 


Lb. ft 2 


Lb. ft* 





0.0000 


0.2048 


0.4096 


0.6144 


0.8192 


1.0240 


1.2289 


1.4337 


1.6385 


1.8433 


10 


2.0481 


2.2529 


2.4577 


2.6625 


2.8673 


3.0721 


3.2770 


3.4818 


3.6866 


3.8914 


20 


4.0962 


4.3010 


4.5058 


4.7106 


4.9154 


5.1202 


5.3251 


5.5299 


5.7347 


5.9395 


30 


6.1444 


6.3492 


6.5540 


6.7588 


6.9636 


7.1684 


7.3733 


7.5781 


7.7829 


7.9877 


40 


8.1925 


8.3973 


8.6021 


8.8069 


9.0117 


9.2165 


9.4214 


9.6262 


9.8310 


10.036 


50 


10.240 


10.445 


10.649 


10.854 


11.059 


11.264 


11.469 


11.674 


11.878 


12.083 


60 


12.289 


12.494 


12.698 


12.903 


13.108 


13.313 


13.518 


13.723 


13.927 


14.132 


70 


14.337 


14.542 


1 1.710 


14.951 


15.156 


15.361 


15.566 


15.771 


15.975 


16.180 


80 


16.385 


16.590 


16.794 


16.999 


17.204 


17.409 


17.(111 


17.819 


18.023 


18.228 


90 


18.433 


18.638 


18.842 


19.047 


19.252 


19.457 


19.662 


19.867 


20.071 


20.276 


100 


20.481 


20.686 


20.890 


21.095 


21.300 


21.505 


21.710 


21.915 


22.119 


22.324 



The Meteic System. 



71 



Pounds 


per Square 


Inch 


nto Atmospheric 


Pressure. 




Lbs. pr in 2 . 





1 


2 


3 


4 


5 


6 


7 


8 


9 




At. 


At. 


At. 


At. 


At. 


At. 


At. 


At. 


At. 


At. 





0.0000 


0.0630 


0.1361 


0.2041 


0.2722 


0.3402 


0.4082 


0.4763 


0.5443 


0.6124 


10 


0.6804 


0.7484 


0.8165 


0.8845 


0.9526 


1.0206 


1.0886 


1.1567 


1.2247 


1.2928 


20 


1.3608 


1.4288 


1.4969 


1.5649 


1.6330 


1.7010 


1.7690 


1.8371 


1.9051 


1.9732 


30 


2.0413 


2.1093 


2.1774 


2.2454 


2.3135 


2.3814 


2.4495 


2.5176 


2.5856 


2.6537 


40 


2.7217 


2.7897 


2.8578 


2.9258 


2.9939 


3.0619 


3.1299 


3.1980 


3.2660 


3.3341 


50 


3.4021 


3.4701 


3.5382 


3.6062 


3.6743 


3.7423 


3.8103 


3.8784 


3.9464 


4.0145 


60 


4.0825 


4.1505 


4.2186 


4.2866 


4.3547 


4.4227 


4.4907 


4.5588 


4.6268 


4.6949 


70 


4.7630 


4.8310 


4.8991 


4.9671 


5,0352 


5.1031 


5.1712 


5.2393 


5.3073 


5.3754 


80 


5.4434 


5.5114 


5.5795 


5.6475 


5.7156 


5.7836 


5.8516 


5.9197 


5.9877 


6.0558 


90 


6.1238 


6.1918 


6.2599 


6.3279 


6.3960 


6.4640 


6.5320 


6.6001 


6.6681 


6.7362 


100 


6.8042 


6.8722 


6.9403 


7.0083 


7.0764 


7.1444 


7.2124 


7.2805 


7.3485 


7.4166 



Atmospheric Pressure into Pounds per Sq 


uare Inch. 




Atm. pres. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Lb. in 2 


Lb. in 2 


Lb. in 2 


Lb. in 2 


Lb. in 2 


Lb. in 2 


Lb.in 2 


Lb.in 2 


Lb.in 2 


Lb.in* 





0.0000 


14.697 


29.393 


44.090 


58.787 


73.483 


88.180 


102.87 


117.57 


132.27 


10 


146.97 


161.67 


176.36 


191.06 


205.76 


220.45 


235.15 


249.&4 


264.54 


279.24 


20 


293.93 


308.63 


323.32 


338.02 


352.72 


367.41 


382.11 


396.80 


411.50 


426.20 


30 


440.90 


455.60 


470.29 


484.99 


499.69 


514.38 


529.08 


543.77 


558.47 


573.17 


40 


587.87 


602.57 


617.26 


631.96 


646.66 


661.35 


676.05 


690.74 


705.44 


720.14 


50 


734.83 


749.53 


764.22 


778.92 


793.62 


808.31 


823.01 


837.70 


852.40 


867.10 


60 


881.80 


896.50 


911.19 


925.89 


940.59 


955.28 


969.98 


984.67 


999.37 


1014.1 


70 


1028.7 


1043.4 


1058.1 


1072.8 


1087.5 1102.2 


1116.9 


1131.6 


1146.3 


1161.0 


80 


1175.7 


1190.4 


1205.1 


1219.8 


1234.5 1249.2 


1263.9 


1278.6 


1293.3 


1308.0 


90 


1322.7 


1337.4 


1352.1 


1366.8 


1381.5 


1396.2 


1410.9 


1425.6 


1439.3 


1455.0 


100 


1469.7 


1484.4 1499.1 


1513.8 


1528.5 


1543.2 


1557.9 


1572.6 1586.3 


1602.0 



Pounds per Square Inch into Kilogrammes per Square Centimetre 



Lbs. pr in 2 . 





1 2 


3 


4 


5 


6 


7 


8 


9 




K.cm 2 


K.cm 2 K.cm 2 


K.cm 2 


K.cm 2 


K.cm 2 


K.cm 2 


K.cm 2 


K.cm 2 


K.cm2 





0.0000 


0.0703' 0.1406 


0.2109 


0.2812 


0.3515 


0.4218 


0.4921 


0.5625 


0.6328 


10 


0.7031 


0.7734| 0.8437 


0.9140 


0.9843 


1.0546 


1.1249 


1.1952 


1.2655 


1.3358 


20 


1.4062 


1.4765 


1.5468 


1.6171 


1.6874 


1.7577 


1.8280 


1.8983 


1.9686 


2.0389 


30 


2.1092 


2.1795 


2.2498 


2.3202 


2.3905 


2.4608 


2.5311 


2.6014 


2.6717 


2.7420 


40 


2.8123 


2.8826 


2.9529 


3.0232 


3.0935 


3.1639 


3.2342 


3.3045 


3.3748 


3.4451 


50 


3.5154 


3.5857 


3.6560 


3.7263 


3.7966 


3.8669 


3.9372 


4.0075 


4.0779 


4.1482 


60 


4.2185 


4.2888 


4.3591 


4.4294 


4.4997 


4.5700 


4.6403 


4.7106 


4.7809 


4.8512 


70 


4.9216 


4.9919 


5.0622 


5.1325 


5.2028 


5.2731 


5.3434 


5.4137 


5.4840 


5.5543 


80 


5.6246 


5.6949 


5.7652 


5.8356 


5.9059 


5.9762 


6.0465 


6.1168 


6.1871 


6.2574 


90 


6.3277 


6.3980 


6.4683 


6.5386 


6.6089 


6.6793 


6.7496 


6.8199 


6.8902 


6.9605 


100 


7.0308 


7.1011 7.1714 


7.2417 


7.3120 7.3823 


7.4526 


7.5229 


7.5933 


7.6636 



Kilogrammes per Square Centimetre into Pounds per Square Inch. 



K. per cm 2 . 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Lb.in 2 


Lb.in 2 Lb.in 2 


Lb.in 2 


Lb.in 2 


Lb.in 5 


Lb.in 2 


Lb.in 2 


Lb.in 2 


Lb.in* 





0.0000 


14.223 28.446 


42.670 56.893 


71.116 


85.339 


99.562 


113.78 


128.01 


10 


142.23 


156.45 170.68 


184.90 


199.12 


213.35 


227.57 


241.79 


256.02 


270.24 


20 


284.46 


298.69 312.91 


327.13 


341.36 


355.58 


369.80 


384.03 


398.25 


412.47 


30 


426.70 


440.92 455.14 


469.36 


483.59 


497.81 


512.03 


526.26 


540.48 


554.70 


40 


568.93 


583.15! 597.37 


611.60 


625.82 


640.04 


654.27 


668.49 


682.71 


696.94 


50 


711.16 


725.38 


739.61 


753.83 


768.05 


782.28 


796.50 


810.72 


824.94 


839.17 


60 


853.39 


867.61 


881.84 


896.06 


910.28 


924.51 


938.73 


952.95 


967.18 


981.40 


70 


995.62 


1009.8 


1024.1 


1038.3 


1052.5 1066.7 1081.0 1095.2 


1109.4 


1123.6 


80 


1137.8 


1152.1 


1166.3 


1180.5 


1194.7! 1209.0! 1223.2 


1237.4 


1251.6 


1265.9 


90 


1280.1 


1294.3 


1308.5 


1322.7 


1337.0 1351.2] 1365.4 


1379.6 


1393.9 


1408.1 


100 


1422.3 


1436.5 


1450.8 


1465.0 


1479.2 1493.4 1507.7 


1521.9 


1536.1 


1550.3 



72 



The Metric System. 



Conversion of English Pounds into 


Kilogrammes. 




Eng. lbs. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 


Kilo. 





0.000 


0.453 


0.907 


1.361 


1.814 


2.268 


2.722 


3.175 


3.629 


4.082 


10 


4.536 


4.989 


5.443 


5.897 


6.350 


6.804 


7.258 


7.711 


8.165 


8.618 


20 


9.072 


9.525 


9.979 


10.43 


10.89 


11.34 


11.79 


12.25 


12.70 


13.15 


30 


13.61 


14.06 


14.52 


14.97 


15 42 


15.88 


16.33 


16.78 


17.24 


17.69 


40 


18.14 


18.59 


19.05 


19.50 


19.95 


20.41 


20.86 


21.31 


21.77 


22.22 


50 


22.68 


23.13 


23.59 


24.04 


24.49 


24.95 


25.40 


25.85 


26.31 


26.76 


60 


27.22 


27.67 


28.13 


28.58 


29.03 


29.49 


29.94 


30.39 


30.85 


31.30 


70 


31.75 


32.20 


32.66 


33.11 


38.56 


34.02 


34.47 


34.92 


35.38 


35.83 


80 


36.29 


36.74 


37.20 


37.65 


38.10 


38.56 


39.01 


39.46 


39.92 


40.37 


90 


40.82 


41.27 


41.73 


42.18 


42.63 


43.09 


43.54 


43.99 


44.45 


44.90 


100 


45.36 


45.81 


46.27 


46.72 


47.17 


47.63 


48.08 


48.53 


48.99 


49.44 



Conversion of Kilogrammes into English 


Pounds. 




Fr. kilo. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 





0.000 


2.205 


4.410 


6.615 


8.820 


11.02 


13.23 


15.43 


17.64 


19.84 


10 


22.05 


24.25 


26.46 


28.67 


30.87 


33.07 


35.28 


37.48 


39.69 


41.89 


20 


44.10 


46.30 


48.51 


50.72 


52.92 


55.12 


57.33 


59.53 


61.74 


63.94 


30 


66.15 


68.35 


70.56 


72.77 


74.97 


77.17 


79.38 


81.58 


83.79 


85.99 


40 


88.20 


90.40 


92.61 


94.82 


97.02 


99.22 


101.4 


103.6 


105.8 


108.0 


50 


110.2 


112.5 


114.6 


116.8 


119.0 


121.2 


123.4 


125.6 


127.8 


130.0 


60 


132.3 


134.5 


136.7 


138.9 


141.1 


143.3 


145.5 


147.7 


149.9 


152.1 


70 


154.3 


156.5 


158.7 


160.9 


163.1 


165.3 


167.5 


169.7 


171.9 


174.1 


80 


176.4 


178.6 


180.8 


183.0 


185.2 


187.4 


189.6 


191.8 


194.0 


196.2 


90 


198.4 


200.6 


202.8 


205.0 


207.2 


209.4 


211.6 


213.8 


216.0 


218.2 


100 


220.5 


222.7 


224.9 


227.1 


229.3 


231.5 


233.7 


235.9 


238.1 


240.3 





Conversion of English 


Tons 


into Metric Tons 






Eng. tons. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




M.ton 


M.ton 


M. ton 


M. ton 


M. ton 


M.ton 


M. ton 


M.ton 


M.ton 


M.ton 





0.000 


1.016 


2.032 


3.048 


4.064 


5.080 


6.096 


7.112 


8.128 


9.144 


10 


10.16 


11.18 


12.19 


13.21 


14.12 


15.24 


16.26 


17.27 


18.29 


19.30 


20 


20.32 


21.34 


22.35 


23.37 


24.38 


25.40 


26.42 


27.43 


28.45 


29.46 


30 


30.48 


31.50 


32.51 


33.53 


34.54 


35.56 


36.58 


37.59 


38.61 


39.62 


40 


40.64 


41.66 


42.67 


43.69 


44.70 


45.74 


46.74 


47.75 


48.77 


49.78 


50 


50.80 


51.82 


52.83 


53.85 


54.86 


55.88 


56.90 


57.90 


58.93 


59.94 


60 


60.96 


61.97 


62.99 


64.01 


65.02 


66.04 


67.06 


68.07 


69.09 


70.10 


70 


71.12 


72.14 


73.15 


74.17 


75.18 


76.20 


77.22 


78.23 


79.25 


80.26 


80 


81.28 


82.29 


83.31 


84.33 


85.34 


86.36 


87.38 


88.39 


89.41 


90.42 


90 


91.44 


92.46 


93.47 


94.49 


95.50 


96.52 


97.54 


98.55 


99.57 


100.6 


100 


101.6 


102.6 


103.6 


104.6 


105.7 


106.7 


107.7 


108.7 


109.7 


110.7 





Conversion of Metric Tons into E 


nglish Tons 


. 




Fr. m. ton. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




E. ton 


E. ton 


E. ton 


E. ton 


E. ton 


E. ton 


E. ton 


E. ton 


E. ton 


E. ton 





0.000 


0.984 


1.969 


2.953 


3.937 


4.921 


5.906 


6.890 


7.874 


8.858 


10 


9.843 


10.83 


11.81 


12.79 


13.78 


14.76 


15.75 


16.73 


17.72 


18.70 


20 


19.69 


20.67 


21.66 


22.64 


23.63 


24.61 


25.60 


26.58 


27.56 


28.55 


30 


29.53 


30.51 


31.50 


32.48 


33.47 


34.45 


35.44 


36.42 


37.40 


38.39 


40 


39.37 


40.35 


41.34 


42.32 


43.31 


44.29 


45.28 


46.26 


47.24 


48.23 


50 


49.21 


50.19 


51.18 


52.16 


53.15 


54.13 


55.12 


56.10 


57.08 


58.07 


60 


59.06 


60.04 


61.03 


62.01 


63.00 


63.98 


64.97 


65.95 


66.93 


67.92 


70 


68.90 


69.88 


70.87 


71.85 


72.84 


73.82 


74.81 


75.79 


76.77 


77.76 


80 


78.74 


79.72 


80.71 


81.69 


82.68 


83.66 


84.66 


85.63 


86.61 


87.60 


90 


88.58 


89.56 


90.55 


91.53 


92.52 


93.50 


94.49 


95.47 


96.45 


97.44 


100 


98.43 


99.41 


100.4 


101.4 


102.4 


103.3 


104.3 


105.3 


106.3 


107.3 



The Metric System. 



73 



Conversion of 


English Ounces Avoird 


upois 


into French Grammes. 


Eng. ozs. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Grams 


Grams 


Grams 


Grams 


Grams 


Grams 


Grams 


Grams 


Grams 


Grams 





0.0000 


28.348 


56.697 


85.046 


113.39 


141.74 


170.09 


198.44 


226.79 


255.14 


10 


283.48 


311.83 


340.18 


368.52 


396.87 


425.22 


453.57 


481.92 


510.27 


538.62 


20 


566.97 


595.32 


623.67 


652.01 


680.36 


708.71 


737.06 


765.41 


793.76 


822.11 


30 


850.46 


878.81 


907.16 


935.50 


963.85 


992.20 


1020.5 


1048.9 


1077.2 


1105.6 


40 „ 


1133.9 


1162.2 


1190.6 


1218.9 


1247.3 


1275.6 


1304.0 


1332.3 


1360.7 


1389.0 


50 


1417.4 


1445.7 


1474.1 


1502.4 


1530.8 


1559.1 


1587.5 


1615.8 


1644.2 


1672.5 


60 


1700.9 


1729.2 


1756.6 


1785.9 


1814.3 


1842.9 


1871.0 


1899.3 


1927.7 


1956.0 


70 


1984.4 


2012.7 


2041.1 


2079.4 


2097.8 


2126.1 


2154.5 


2182.8 


2211.2 


2239.5 


80 


2267.9 


2296.2 


2324.6 


2352.9 


2381.3 


2409.6 


2438.0 


2466.3 


2494.7 


2523.0 


90 


2551.4 


2579.7 


2608'.1 


2636.4 


2664.8 


2693.1 


2721.5 


2739.8 


2778.2 


2806.5 


100 


2834.8 


2863.1 


2891.5 


2919.8 


2948.2 


2976.5 


3004.9 


3033.2 


3061.6 


3089.9 



Conversion of French Grammes into English Ounces Avoirdupois. 



Fr. grams. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Ozs. 


Ozs. 


Ozs. 


Ozs. 


Ozs. 


Ozs. 


Ozs. 


Ozs. 


Ozs. 


Ozs. 





0.0000 


0.0353 


0.0705 


0.1058 


0.1411 


0.1768 


0.2116 


0.2469 


0.2822 


0.3175 


10 


0.3527 


0.3880 


0.4232 


0.4585 


0.4938 


0.5295 


0.5643 


0.5996 


0.6349 


0.6702 


20 


0.7055 


0.7408 


0.7760 


0.8113 


0.8466 


0.8823 


0.9171 


0.9524 


0.9877 


1.0230 


30 


1.0582 


1.0935 


1.1287 


1.1640 


1.1993 


1.2350 


1.2698 


1.3051 


1.3404 


1.3757 


40 


1.4110 


1.4463 


1.4815 


1.5168 


1.5521 


1.5878 


1.6226 


1.6579 


1.6932 


1.7285 


50 


1.7687 


1.8040 


1.8392 


1.8745 


1.9098 


1.9455 


1.9803 


2.0156 


2.0509 


2.0862 


60 


2.1165 


2.1518 


2.1870 


2.2223 


2.2576 


2.2933 


2.3281 


2.3634 


2.3987 


2.4340 


70 


2.4692 


2.5045 


2.5397 


2.5750 


2.6103 


2.6460 


2.6808 


2.7161 


2.7514 


2.7867 


80 


2.8220 


2.8573 


2.8925 


2.9278 


2.9631 


2.9988 


3.0336 


3.0689 


3.1042 


3.1395 


90 


3.1747 


3.2100 


3.2452 


3.2805 


3.3158 


3.3515 


3.3863 


3.4216 


3.4569 


3.4922 


100 


3.5275 


3.5628 


3.5980 


3.6333 


3.6686 


3.7043 


3.7391 


3.7744 


3.8097 


3.8450 



Conversion 


of English 


Grains Troy into French Grammes. 


Eng. grains 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Grams 


Grams 


Grams 


Grams 


Grams 


Grams 


Grams 


Grams 


Grams 


Grams 





0.0000 


0.0648 


0.1296 


0.1944 


0.2592 


0.3240 


0.3888 


0.4535 


0.5183 


0.5831 


10 


0.6479 


0.7127 


0.7775 


0.8423 


0.9071 


0.9719 


1.0367 


1.1014 


1.1662 


1.2310 


20 


1.2959 


1.3607 


1.4255 


1.4903 


1.5551 


1.6199 


1.6847 


1.7494 


1.8142 


1.8890 


30 


1.9438 


2.0086 


2.0734 


2.1382 


2.2030 


2.2678 


2.3326 


2.3973 


2.4621 


2.5269 


40 


2.5918 


2.6566 


2.7214 


2.7862 


2.8510 


2.9158 


2.9806 


3.0453 


3.1101 


3.1749 


50 


3.2398 


3.3046 


3.3691 


3.4342 


3.4990 


3.5638 


3.6286 


3.6933 


3.7581 


3.8229 


60 


3.8877 


3.9525 


4.0173 


4.0821 


4.1469 


4.2117 


4.2765 


4.3412 


4.4060 


4.4708 


70 


4.5357 


4.6005 


4.6653 


4.7301 


4.7949 


4.8597 


4.9245 


4.9892 


5.0540 


5.1188 


80 


5.1830 


5.2484 


5.3132 


5.3780 


5.4428 


5.5076 


5.5724 


5.6371 


5.7019 


5.7667 


90 


5.8316 


5.8964 


5.9612 


6.0260 


6.0908 


6.1556 


6.2204 


6.2851 


6.3499 


6.4147 


100 


6.4795 


6.5443 


6.6091 


6.6739 


6.7387 


6.8035 


6.8683 


6.9330 


6.9978 


7.0626 



Conversion 


of French Grammes into English Grains Troy. 


Fr. grams. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Grs. 


Grs. 


Grs. 


Grs. 


Grs. 


Grs. 


Grs. 


Grs. 


Grs. 


Grs. 





0.0000 


15.433 


30.866 


46.299 


61.732 


77.165 


92.599 


108.03 


123.46 


138.90 


10 


154.33 


169.76 


185.19 


200.63 


216.06 


231.49 


246.93 


262.36 


277.79 


293.23 


20 


308.66 


324.09 


339.52 


354.96 


370.39 


385.82 


401.26 


416.69 


432.12 


447.56 


30 


462.99 


478.42 


493.86 


509.29 


524.72 


540.15 


555.59 


571.02 


586.45 


601.89 


40 


617.65 


632.75 


648.18 


663.95 


679.38 


694.81 


709.92 


725.35 


740.78 


756.22 


50 


771.65 


787.08 


802.52 


817.95 


833.38 


848.82 


864.25 


879.68 


895.11 


910.55 


60 


925.99 


941.42 


956.85 


972.29 


987.72 


1003.1 


1018.6 


1034.0 


1049.4 


1064.9 


70 


1080.3 


1095.7 


1111.2 


1126.6 


1142.0 


1157.5 


1172.9 


1188.3 


1203.7 


1219.2 


80 


1234.6 


1250.0 


1265.5 


1280.1 


1296.3 


1311.8 


1327.2 


1342.6 


1358.1 


1373.5 


90 


1389.0 


1404.4 


1419.8 


1435.3 


1450.7 


1466.1 


1481.6 


1497.0 


1512.4 


1527.9 


100 


1543.3 


1558.7 


1574.1 


1589.6 


1605.0 


1620.4 


1635.9 


1651.3 


1666.7 


1682.2 



74 



The Metric System. 



Horse=power into Cheval=vapeur. 



H. -power. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




C.-v. 


C.-v. 


C.-v. 


C.-v. 


C.-v. 


C.-v. 


C.-v. 


C.-v. 


C.-v. 


C.-v. 





0.0000 


1.0136 


2.0272 


3.0408 


4.0544 


5.0680 


6.0816 


7.0952 


8.1088 


9.1224 


10 


10.136 


11.150 


12.163 


13.176 


14.190 


15.204 


16.218 


17.231 


18.245 


19.258 


20 


20.272 


21.308 


22.299 


23.313 


24.326 


25.240 


26.354 


27.367 


28.381 


29.394 


30 


30.408 


31.422 


32.435 


33.449 


34.462 


35.476 


36.490 


37.503 


38.517 


39.530 


40 


40.544 


41.557 


42.571 


43.585 


44.598 


45.612 


46.626 


47.639 


48.653 


49.666 


50 


50.680 


51.693 


52.707 


53.721 


54.734 


55.748 


56.762 


57.775 


58.789 


59.802 


60 


60.816 


61.829 


62.843 


63.857 


64.870 


65.884 


66.898 


67.911 


68.925 


69.938 


70 


70.952 


71.965 


72.979 


73.993 


75.006 


76.020 


77.034 


78.047 


79.061 


80.074 


80 


81.088 


82.102 


83.115 


84.129 


85.142 


86.156 


87.170 


88.183 


89.197 


90.210 


90 


91.224 


92.338 


93.251 


94.265 


95.278 


96.292 


97.306 


98.319 


99.333 


100.34 


100 


101.36 


102.37 


103.30 


104.40 


105.41 


106.43 


107.44 


108.45 


109.47 


110.48 







Cheval=vapeur into Horse-power 








Chev.-vap. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




H.-p. 


H.-p. 


H.-p. 


H.-p. 


H.-p. 


H.-p. 


H.-p. 


H.-p. 


H.-p. 


H.-p. 





0.0000 


0.9863 


1.9726 


2.9589 


3.9452 


4.9315 


5.9178 


6.9041 


7.8904 


8.8767 


10 


9.8630 


10.849 


11.835 


12.822 


13.808 


14.794 


15.781 


16.767 


17.753 


18.739 


20 


19.726 


20.712 


21.698 


22.685 


23.671 


24.657 


25.644 


26.630 


27.616 


28.602 


30 


29.589 


30.575 


31.561 


32.548 


33.534 


34.520 


35.507 


36.493 


37.479 


38.465 


40 


39.452 


40.438 


41.424 


42.411 


43.397 


44.383 


45.370 


46.356 


47.342 


48.328 


50 


49.315 


50.301 


51.287 


52.274 


53.260 


54.246 


55.233 


56.219 


57.205 


58.191 


60 


59.178 


60.164 


61.150 


62.137 


63.123 


64.109 


65.096 


66.082 


67.068 


68.054 


70 


69.041 


70.027 


71.013 


71.990 


72.986 


73.972 


74.959 


75.945 


76.941 


77.917 


80 


78.904 


79.890 


80.876 


81.863 


82.849 


83.835 


84.822 


85.808 


86.794 


87.780 


90 


88.767 


89.753 


90.739 


91.726 


92.712 


93.698 


94.785 


95.671 


96.657 


97.643 


100 


98.630 


99.616 


100.60 


101.59 


102.57 


103.56 


104.55 


105.53 


106.52 


107.50 







Foot 


pounds into Kilogram metres 


;. 






Foot-lbs. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Kgm. 


Kgm. 


,Kgm. 


Kgm. 


Kgm. 


Kgm. 


Kgm. 


Kgm. 


Kgm. 


Kgm. 





0.0000 


0.1382 


0.2764 


0.4146 


0.5528 


0.6910 


0.8292 


0.9674 


1.1056 


1.2438 


10 


1.3820 


1.5202 


1.6584 


1.7966 


1.9348 


2.0731 


2.2112 


2.3494 


2.4876 


2.6259 


20 


2.7640 


2.9022 


3.0404 


3.1786 


3.3168 


3.4552 


3.5933 


3.7315 


3.8696 


4.0078 


30 


4.1460 


4.2842 


4.4224 


4.5606 


4.6988 


4.8370 


4.9751 


5.1134 


5.2517 


5.3897 


40 


5.5280 


5.6666 


5.8044 


5.9426 


6.0808 


6.2191 


6.3572 


6.4954 


6.6336 


6.7718 


50 


6.9100 


7.0482 


7.1864 


7.3246 


7.4628 


7.6010 


7.7393 


7.8775 


8.0155 


8.1538 


60 


8.2920 


8.4303 


8.5684 


8.7066 


8.8448 


8.9830 


9.1212 


9.2594 


9.3976 


9.5359 


70 


9.6740 


9.8122 


9.9504 


10.088 


10.227 


10.365 


10.503 


10.641 


10.779 


10.918 


80 


11.056 


11.194 


11.322 


11.570 


11.609 


11.747 


11.885 


12.023 


12.161 


12.300 


90 


12.438 


12.576 


12.714 


12.855 


12.991 


13.129 


13.267 


13.405 


13.544 


13.682 


100 


13.820 


13.958 


14.096 


14.235 


14.373 


14.511 


14.649 


14.787 


14.925 14.064 







Kilogram met res 


into Foot=pounds 








Kgm. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Ft. -lb. 


Ft. -lb. 


Ft. -lb. 


Ft.-lb. 


Ft.-lb. 


Ft.-lb. 


Ft.-lb. 


Ft.-lb. 


Ft.-lb. 


Ft.-lb. 





0.0000 


7.2334 


14.467 


21.700 


28.934 


36.166 


43.400 


50.734 


57.868 


65.100 


10 


72.334 


79.567 


87.101 


94.034 


101.27 


108.50 


115.74 


123.07 


130.20 


137.43 


20 


144.67 


151.90 


158.43 


166.37 


173.60 


180.84 


188.08 


195.40 


202.54 


209.77 


30 


217.00 


224.23 


231.77 


238.70 


245.93 


253.17 


260.41 


267.73 


274.87 


282.10 


40 


289.34 


296.57 


304.11 


311.04 


318.27 


325.50 


332.75 


340.07 


347.21 


354.44 


50 


361.66 


368.89 


376.43 


383.36 


390.59 


397.82 


405.07 


412.39 


419.53 


426.76 


60 


434.00 


441.23 


448.77 


455.70 


462.93 


470.17 


477.41 


484.73 


491.87 


499.10 


70 


507.34 


514.57 


522.11 


529.04 


536.27 


543.50 


550.75 


558.07 


565.21 


572.44 


80 


578.68 


585.91 


593.45 


599.38 


607.61 


614.85 


622.09 


629.41 


636.55 


643.78 


90 


651.00 


658.2:5 


665.77 


672.70 


679.93 


687.17 


694.41 


701.73 


708.87 


716.10 


100 


72:;.:; i 


730.57 


738.11 


745.04 


752.27 


759.51 


766.75 


774.07 


781.21 


788.44 



The Metric System. 



75 





Conversion of Foot-tons into Tonnes=metres 






Foot-tons. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




T.-m. 


T.-m. 


T.-m. 


T.-m. 


T.-m. 


T.-m. 


T.-m. 


T.-m. 


T.-m. 


T.-m. 





0.0000 


0.3097 


0.6194 


0.9291 


1.2382 


1.5484 


1.8581 


2.1678 


2.4775 


2.7872 


10 


3.0969 


3.3166 


3.7163 


4.0260 


4.3356 


4.6453 


4.9550 


5.2667 


5.5744 


5.8841 


20 


6.1938 


6.4135 


6.8132 


7.1229 


7.4325 


7.7422 


8.0519 


8.3636 


8.6713 


8.9810 


30 


9.2906 


9.6003 


9.9100 


10.219 


10.529 


10.839 


11.149 


11.460 


11.768 


12.078 


40 


12.387 


12.697 


13.006 


13.316 


13.626 


13.935 


14.245 


14.557 


14.864 


15.174 


50 


15.484 


15.794 


16.103 


16.413 


16.723 


17.032 


17.342 


17.654 


17.961 


18.271 


60 


18.581 


18.891 


19.200 


19.510 


19.820 


20.129 


20.439 


20.751 


21.058 


21.368 


70 


21.678 


21.988 


22.297 


22 607 


22.917 


23.226 


23.536 


23.848 


24.155 


24.465 


80 


24.775 


25.085 


25.394 


25.704 


26.014 


26.323 


26.633 


26.945 


27.252 


27.562 


90 


27.872 


28.182 


28.491 


28.801 


29.111 


29.420 


29.730 


30.042 


30.349 


30.659 


100 


30.969 


31.279 


31.588 


31.898 


32.208 


32.517 


32.827 


33.139 


33.446 


33.756 





Conversion of Tonnes 


-metres into Foot=tons 






T. -metres. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




F.-tn. 


F.-tn. 


F.-tn. 


F.-tn. 


F.-tn. 


F.-tn. 


F.-tn. 


F.-tn. 


F.-tn. 


F.-tn. 





0.0000 


3.2290 


6.4581 


9.6871 


12.916 


16.145 


19.374 


22.603 


25.832 


29.061 


10 


32.290 


35.519 


38.758 


41.977 


45.206 


48.435 


51.664 


54.893 


58.122 


61.351 


20 


64.581 


67.810 


71.049 


74.268 


77.497 


80.726 


83.955 


87.184 


90.413 


93.642 


30 


96.871 


100.10 


103.34 


106.56 


109.79 


113.01 


116.24 


119.47 


122.70 


125.93 


40 


129.16 


133.39 


135.63 


138.85 


142.07 


145.30 


148.53 


151.76 


154.99 


158.22 


50 


161.45 


164.68 


167.92 


171.14 


174.36 


177.59 


180.82 


184.05 


187.28 


190.51 


60 


193.74 


196.97 


200.21 


203.43 


206.65 


209.88 


213.11 


216.34 


219.57 


222.80 


70 


226.03 


229.26 


232.50 


235.72 


238.94 


242.17 


245.40 


248.63 


251.86 


255.09 


80 


258.32 


261.55 


264.79 


268.01 


271.23 


274.46 


277.69 


280.92 


284.15 


287.38 


90 


290.61 


293.84 


297.08 


300.30 


303.52 


306.75 


309.98 


313.21 


316.44 


319.67 


100 


322.90 


326.13 


329.37 


332.59 


335.81 


339.04 


342.27 


345.50 


348.73 


351.96 





British Thermal Units into French Calories. 






B. T. U. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Cal. 


Cal. 


Cal. 


Cal. 


Cal. 


Cal. 


Cal. 


Cal. 


Cal. 


Cal. 





0.0000 


0.2520 


0.5040 


0.7560 


1.0080 


1.2600 


1.5120 


1.7640 


2.0160 


2.2680 


10 


2 5200 


2.7720 


3.0240 


3.2760 


3.5280 


3.7800 


4.0320 


4.2840 


4.5360 


4.7880 


20 


5.0399 


5.2919 


5.5439 


5.7959 


6.0478 


6.2699 


6.5419 


6.8039 


7.0559 


7.3079 


30 


7.5600 


7.8120 


8.0640 


8.3160 


8.5680 


8.8200 


9.0720 


9.3340 


9.5760 


9.8280 


40 


10.080 


10.332 


10.584 


10.836 


11.088 


11.340 


11.512 


11.844 


12.096 


12.348 


50 


12.600 


12.852 


13.104 


13.356 


13.608 


13.860 


14.112 


14.364 


14.616 


14.868 


60 


15.120 


15.372 


15.624 


15.876 


16.128 


16.380 


16.632 


16.884 


17.136 


17.388 


70 


17.640 


17.892 


18.144 


18.396 


18.648 


18.900 


19.152 


19.404 


19.656 


19.908 


80 


20.160 


20.412 


20.664 


20.916 


21.168 


21.420 


21.672 


21.924 


22.176 


22.428 


90 


22.680 


22.932 


23.184 


23.436 


23.688 


23.940 


24.192 


24.444 


24.696 


24.948 


100 


25.200 


25.452 


25.704 


25.956 


26.208 


26.460 


26.712 


26.964 


27.216 


27.468 





French Calories into British Thermal Units. 






Calories. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




T. U. 


T. U. 


T. U. 


T. U. 


T. U. 


T. U. 


T. U. 


T. U. 


T. TJ. 


T. TJ. 





0.0000 


3.9683 


7.9366 


11.905 


15.873 


19.842 


23.810 


27.778 


31.746 


35.715 


10 


39.683 


43.651 


47.620 


51.598 


55.520 


59.525 


63.493 


67.461 


71.429 


75.398 


20 


79.366 


83.334 


87.303 


91.271 


95.203 


99.208 


103.17 


107.14 


111.11 


115.08 


30 


119.05 


123.02 


126.98 


130.95 


134.89 


138.89 


142.86 146.83 


150.80 


154.77 


40 


158.73 


162.70 


166.66 


170.62 


174.57 


178.57 


182.54 


186.51 


190.48 


194.45 


50 


198.42 


202.39 


206.35 


210.39 


214.26 


218.26 


222.23 


226.20 


230.16 


234.14 


60 


238.10 


242.07 


246.03 


250.00 


253.94 


258.94 


261.91 


265.88 


269.85 


273.82 


70 


277.78 


281.75 


285.72 


289.68 


293.62 


297.62 


301.59 


305.56 


309.53 


313.50 


80 


317.46 


321.43 


325.40 


329.36 


333.29 


337.30 


341.27 


345.24 


349.20 


353.18 


90 


357.15 


361.12 


365.09 


369.05 


372.98 


376.99 


380.96 


384.93 


388.90 


392.87 


100 


396.83 


400.80 


404.77 


408.73 


412.67 


416.67 


420.64 


424.61 


428.58 


432.55 



76 



Algebra. 



ALGEBRA. 

For the detailed operations of Algebra the reader is referred to the 
numerous good text-books upon the subject, and only a few of the more 
important and generally practical matters will here be given in conve- 
nient form for reference. 

Remembering that multiplication is represented in algebra by placing 
the two quantities next each other, without any intermediate sign, we 
have aa = a 2 , aaa = a 3 , etc. ; also a multiplied by b is written ab, a divided 

by b is written — , etc. 

From an examination of these facts we are able to place the rules 
regarding exponents in a form in which they can be conveniently remem- 
bered. 

aaa = a 3 ; dividing this by a we get 
aa = a 2 ; dividing again *by a we get 
a= a. 
In each case we see that dividing any power of a by a is simply sub- 
tracting unity from the exponent. Proceeding, we see that 



a 
a 

_gP_ 
a 

a 



= a° 



= a o-i =^-1=- 



= a -i-i= = , 



etc. 



This shows why a negative exponent to any quantity means the recip- 
rocal of the same power with a positive exponent. 



Binomial Theorem. 

The binomial theorem enables any power of the sum or difference of 
two quantities to be determined. For any value of n we have 



(a ± b) n = a n ± na n ~' [ b -f 



n{n- 



■ 1) __. n(n- 

— *■ a n ~ 2 b 2 ± — * — 



•!)(»- 



^) a »- 3 6 3 + . 



1-2 ~ ~ "^ 1-2-3 

An examination of this will show that the right-hand side consists of 
the quantities a and b arranged according to the ascending and descending 
powers. Thus, when n = 2, we have a 2 + ab + b 2 ; for n 3 we have a 3 -+- 
a 2 b + ab 2 + b'\ and so on. 

The coefficients must be computed for each power, or they may be tabu- 
lated as below. 



Table of Binomial Coefficients. 











Terms. 








Expo- 
















nents. 

































1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


1 




1 
























2 




2 


1 






















3 




3 


3 


1 




















4 




4 


6 


4 


1 


















5 




5 


10 


10 


5 


1 
















6 




6 


15 


20 


15 


6 


1 














7 




7 


21 


35 


35 


21 


7 


1 












8 




8 


28 


56 


70 


56 


28 


8 


1 










9 




9 


36 


84 


126 


126 


84 


36 


9 


1 








10 




10 


45 


120 


210 


252 


210 


120 


45 


10 


1 






1 1 




11 


65 


165 


330 


462 


462 


330 


165 


55 


11 


1 




12 




12 


66 


220 


495 


792 


924 


792 


495 


220 


66 


12 


1 



Arithmetical Progression. 



77 



Thus 



is, 

(a + 6) 4 = a 4 + 4a 3 6 + 6a 2 6 2 + 4a6 3 + 6 4 ; 
(a + 6)7 = tf + 7a 6 6 + 21a 5 6 2 + 35a^3 + 35^54 + 2la?V> + 7a6 6 



■67. 






ARITHMETICAL PROGRESSION. 

Arithmetical Progression is a series of numbers, as 2, 4, 6, 8, 10, 12, 
etc., or 18, 15, 12, 9, 6, 3, in which every successive term is increased or 
diminished by a constant number. 

Letters denote 
a = the first term of the series. 
6 = any other term whose number from a is n. 
n = number of terms within a and 6. 
d = the difference between any two adjacent terms. 
5 = the sum of all the terms. 
In the series 2, 5, 8, 11, a == 2, b = 11, n = 4, d = 3, and 5 = 26. 
4®^ When the series is decreasing, take the first term = 6 and the last 
term = a. 

The accompanying table contains all the formulas or questions in 
Arithmetical Progressions, and the nature of the question will tell which 
formula is to be used. 



1. 

2. 



4. 

5. 



Formulas for Arithmetical Progressions. 



-b — d(n — 1). 
-— — 6 



5 d. 1N 



6 = a + d(n — 1). 

_25 
n 

d 



6 = 



- a. 



^4 +i(7l _i). 



7. n- 



+ 1. 



25 
: a + 6* 



9. d = 



10. d = 



11. d = 



n — 1 

(6 + a)(6 — a) 
25— a — 6 " 
2(S—an) 
71(71 — 1) ' 



12. 


2(6n — 5) 


71(71 — 1) 


13. 


5 = - 2 . 



14. 5 = 



15. 



(a + 6)(6 + d — a) 
2d 

d , 



5 = 7i[a + -(n-l)] 



16. S = n\b 



[■ 



d 



(71-1) 



]■ 



17. a = 



^VFir 



-2d5. 



18. b = • 



^VM) 



+ 2dS. 



19. » = — - 



20. 71 = 



2 + d 






25 



) d' 



GEOMETRICAL PROGRESSION. 

Geometrical Progression is a series of numbers, as 2 : 4 : 8 : 16 : 32 :, 

etc., or 729 : 243 : 81 : 27 : 9 :, etc., in which every successive term is multi- 
plied or divided by a constant factor. 



78 



Special Series. 



Notation. 

a = the first term of the series ; 

6 = any other term whose number from a is n ; 

n = number of terms within a and 6, inclusive ; ■ 

r = ratio, or the factor by which the terms are multiplied or divided ; 

£ = Sum of the terms. 

In the series 1 : 3 : 9 : 27 :, a = 1, 6 = 27, n = 4, r = 3, 5 = 40, inclusive. 

The accompanying table contains all the formulas or questions in Geo- 
metrical Progressions. The nature of the question will tell which formula 
is to be used. 



Formulas for Geometrical Progressions. 



1. a = 



fn — 1 

2. a = S — r(S— b). 

3. a = S r " 



rn — 1' 



4. b = ar n ~ l . 

S— a 



5. 5-5- 

6. b = s(^\)rn~i 

Vr 1 — 1/ 



r 
r — 1' 



7. r = 



■Ti 



8. r 



S—a 



S—b' 

ar* + S — rS — a = 0. 



10. 5 = 

11. S = 

12. 5 = 



13. n = 1 + 

14. n = 1 + 

15. ?i 

16. n = l + 



6r — a 
r — 1' 
a(7™ — 1) 
r — 1 ' 
6(r^ — l) 
(r — l)?™- 1 " 

log. 6 — log. a 



log. b — log. a 



log. (S -a) -log. (5-6)* 
log. [a + 5(r — 1)]— log, a 
log. r 
log. 5 — log, [br — S(r — 1)] 
log. r 



17. S-- 



b ~\/ b — a ~\/ a 

' n — 1 _ n — 1 ' 

|/ b— j/ a 



SPECIAL SERIES. 

Among the great variety of series occurring in practical mathematics 
the following will be found convenient for reference : 

1. 1+2+3+4 + —2&iil 

2. 2 + 4 -f 6 +8+ 2?i = n(n + 1). 

3. 1 +3 +5 +7+ (2ti — l)=7i 2 . 

4 . !. + »+■* + * n2 = n(7^ + lK2n + l) > 

5. 13 + 2 3 + 3 3 + 4 3 n 3 = [^!L_Li)J 2 . 

EQUATIONS. 

Equations of the first degree need not be discussed here. Their solu- 
tion may be found in any elementary algebra 



Logarithms. 



79 



Equations of the second degree may be reduced to one of three forms, 
and solved respectively as follows : 



x 2 + px + q = ; x - 



ax 2 + bx -f c — ; x = 



P , P* 



-ft ± > / &2_4 aC 

2a 



*2n -f pa?* + g == ; x = y — £ ± -J ^- — g. 



3. When a; ± y = « and z?/ = P> we have 



8 + v'g 2 T 4p . _ ^-^t^ 






LOGARITHMS. 

There are four fundamental rules for operations with powers : 

That is, the product of any two powers of a number is equal to the num- 
ber raised to a power whose exponent is the sum of the exponents of the 
two factors. 

a™ 

= a^- n . 

a n 

Or, the quotient of two powers is equal to the number raised to a power 
whose exponent is the difference of the exponents of divisor and dividend. 

(a n ) m = a™*. 

Or, any power may be raised to a higher power by multiplying the two 
exponents. 

n m 

V a m — a n . 

Or, any root of any power may be extracted by dividing the exponent by 
the index of the root. 

If we take any number, such as 2, and use it as the base of a geometrf- 
cal series, we will see that the exponents form an arithmetical series. 
Thus, the exponent of 1 = 0, of 2 = 1, of 4 = 2, of 8 = 3, etc. ; or, proceed- 
ing, we may arrange the following little table : 



Powers. 


Exponents. 


Powers. 


Exponents. 


Powers. 


Exponents. 


1 





1024 


10 


1048576 


20 


2 


1 


2048 


11 


2097152 


21 


4 


2 


4096 


12 


4194304 


22 


8 


3 


8192 


13 


8388608 


23 


16 


4 


16384 


14 


16777216 


24 


32 


5 


32768 


15 






64 


6 


65536 


16 






128 


7 


131072 


17 






256 


8 


262144 


18 






512 


9 


524288 


19 







Suppose now we wish to multiply 128 by 512, we see that 128 = 2 7 and 
512 = 29; hence, 128 X 512 = 2 7 - 9 = 2 16 , and in the table, opposite the 



80 Logarithms. 



exponent 16, we find the power 65536, which is the product of the two 
factors, obtained by the simple addition of the exponents. 

Again, g_ »-»-» -*-«. 

To raise a number to a power, such as 16 to the fifth power, we have 
16 = 24 and (2*)** = 220 = 1048576. 

Again, the seventh root of 2097152 is formed as follows : 

2097152 == 221 and y^ = 2^ = 2* = 8. 

In the small table of the powers of 2 given above there are many gaps, 
because only those powers which have whole exponents are given. For all 
the numbers between 16 and 32, for example, the exponents will be deci- 
mals, and will be greater than 4 and less than 5, etc. In practice, the base 
used is not 2, but 10, and all the intermediate exponents have been com- 
puted to many decimals, these forming a table of logarithms. 

Table of Logarithms of Numbers. 

Pages 82 to 104 give the mantissas, or decimal portions of the logarithms, 
of all whole numbers from 1 to 10009. The characteristics, or whole num- 
bers, which, with these decimals, form the complete logarithms, are found 
as follows : 

The logarithm of 1 = 0, of 10 = 1, of 100 = 2, of 1000 = 3, etc. ; hence, 
the logarithm of any number between 100 and 1000 must lie between 2 and 
3, and be greater than 2 and less than 3, and so for any number. There- 
fore we have the rule that the whole portion of a logarithm of any num- 
ber is one less than there are figures in the number. The decimal portion 
for any number below 10009 is taken directly from the table. Thus, 

log. 365 = 2.56229, 

the decimal portion, 56229, being found directly opposite 365 in the table, 
and the whole portion being 2, or 1 less than the number of places in 365. 
In like manner we have 

log. 36.5 = 1.56229, 

log. 3.65 = 0.56229. 

The mantissa, or decimal portion, is always positive, but the character- 
istic is negative when the number is less than unity. Thus, 

log. 0.365 =1.56229, 
log. 0.0365 =2.56229, 
log. 0.00365 = 3.56229, 

the minus being placed over the characteristic to show that it applies to 
that portion only, and not to the mantissa. 

If the given number has more than three places, the mantissa is found 
in the body of the table. Thus, the logarithm of 1873 = 3.27253, the figures 
0.27 being found opposite 183, and the 253 on the same horizontal line 
under 3. 

If the last three figures of the mantissa are preceded by an asterisk, the 
first two figures are to be taken from the next line below, in the first 
column. Thus, 

log. 3897 = 3.59073, 

in which, opposite 389, we find 59, and then, passing on under 7, we find 
*073, the asterisk indicating that we are to go one line below, taking out 
59, not 58, for the first two figures of the mantissa, giving us 0.59073, as 
above. 

The table, as will be seen, enables the logarithm of any number of four 
places to be taken out at once. If the number of which the logarithm is 
required has more than four places, the logarithm can be found from the 
table, as follows : 

In the column at the extreme right of each page, under the heading 
P. P. (Proportional Parts), will be found in the black figures the differences 
between any logarithm and the next succeeding logarithm for the adjoin- 



- 



Logarithms. 81 



ing portions of the table. The smaller figures in the same column form 
little multiplication tables, in which these differences are multiplied by 
0.1, 0.2, 0.3, etc. 

The use of these proportional parts and their decimal parts is best 
shown by actual example. Suppose it is desired to find the logarithm of 
18702. Opposite 187 and under in the table we find the mantissa, 0.27184. 
The proportional part, or difference at this point between one logarithm 
and the next, is 23, or, in other words, there is a difference of 23 between 
the last two figures of the logarithm of 1870 and 1871. For 0.1 difference 
in the number, the difference in the logarithms would be 2.3; for 0.2, it 
would be 4.6, etc., as shown in the small table under 23 in the column 
P. P. For 2 points additional, therefore, we simply add 4.6 to the loga- 
rithm of 1870, and we have the logarithm of 18702. Thus, 

log. 1870 =0.27184 
p. p. for 2 = 4.6 

log. 18702 = 4.271886, or 
4.27189 

Again, let it be required to find the logarithm of 35.797. 

log. 35.79 = 1.55376 p. p. = 12 
p. p. for 7 = 8.4 

log. 35.797 = 1.553844 

If the given number has six or more figures the method is the same, 
except that the proportional part is reduced one-tenth for each additional 
figure. Thus, the logarithm of 3725.96 is found as follows : 

log. 3725 = 3.57113 p. p. = 11 

p. p. for 9 — 9.9 

p. p. for 6 = 0.66 

log. 3725.96 == 3.5712356, or 3.57124 

The operation of finding the number corresponding to a given loga- 
rithm is the reverse of the preceding. Thus, the number corresponding to 
the logarithm 2.73924 is found as follows : 

In the table the next smaller logarithm is 

73918, and its number = 584500 

The given log. = 73924 
and the difference =6 
The nearest difference in the table = 5.6 = corresponding to 7 

Subtracting 04 corresponding to 5 

Hence, the number is 584575 
Since the characteristic = 2, there must be one more place 
before the decimal point ; hence, 

log. 2.73924 = num. 584.575 



82 



Logarithms of Numbers. 



Num. 100 to 139. Log. 000 to 145. 



N 


L 





1 


2 


3 


4 


5 


6 


7 


8 9 


P. P. 


100 


00 


000 


043 


087 


130 


173 


217 


260 


303 


346 389 


44 


43 


101 




432 


475 


518 


561 


604 


647 


689 


732 


775 817 


1 

2 


4.4 

8.8 


4.3 

8.6 


102 




860 


903 


945 


988 *030 


*072 *115 *157 *199 *242 


103 


01 


284 


326 


368 


410 


452 


494 


536 


578 


620 662 


3 


13.2 


12.9 


104 




703 


745 


787 


828 


870 


912 


953 


995 *036 *078 


4 
5 


17.6 
22.0 


17.2 
21.5 


105 


02 


119 


160 


202 


243 


284 


325 


366 


407 


449 490 


6 

7 

8 


26.4 
30.8 
35.2 


25.8 
30.1 
34.4 


106 




531 


572 


612 


653 


694 


735 


776 


816 


857 898 


107 




938 


979 *019 *060 *100 


*141 *181 *222 *262 *302 


9 


39.6 


38.7 


108 


03 


342 


383 


423 


463 


503 


543 


583 


623 


663 703 




109 




743 


782 


822 


862 


902 


941 


981 *021 *060 *100 


42 


41 


110 


04 


139 


179 


218 


258 


297 


336 


376 


415 


454 493 


1 
2 


4.2 

8.4 


4.1 

8.2 


111 




532 


571 


610 


650 


689 


727 


766 


805 


844 883 


3 


12.6 


12.3 


112 




922 


961 


999 *038 *077 


*115 *154 *192 *231 *269 


4 

5 

6 


16.8 
21.0 
25.2 


16.4 
20.5 
24.6 


113 


05 


308 


346 


385 


423 


461 


500 


538 


576 


614 652 


114 




690 


729 


767 


805 


843 


881 


918 


956 


994 *032 


7 


29.4 


28.7 


115 


06 


070 


108 


145 


183 


221 


258 


296 


333 


371 408 


8 
9 


33.6 
37.8 


32.8 
36.9 


116 




446 


483 


521 


558 


595 


633 


670 


707 


744 781 




117 




819 


856 


893 


930 


967 


*004 *041 *078 *115 *151 


40 


«>y 


118 


07 


188 


225 


262 


298 


335 


372 


408 


445 


482 518 


1 


4.0 


3.9 


119 




555 


591 


628 


664 


700 


737 


773 


809 


846 882 


2 

3 


8.0 
12.0 


7.8 
11.7 


120 




918 


954 


990 *027 *063 


*099 *135 *171 *207 *243 


4 

5 
6 


16.0 
20.0 
24.0 


15.6 
19.5 
23.4 


121 


08 


279 


314 


350 


386 


422 


458 


493 


529 


565 600 


122 




636 


672 


707 


743 


778 


814 


849 


884 


920 955 


7 


28.0 


27.3 


123 




991 *026 *061 *096 *132 


*167 *202 *237 *272 *307 


8 
9 


32.0 
36.0 


31.2 
35.1 


124 


09 


342 


377 


412 


447 


482 


517 


552 


587 


621 656 


125 




691 


726 


760 


795 


830 


864 


899 


934 


968 *003 


38 


37 


126 


10 


037 


072 


106 


140 


175 


209 


243 


278 


312 346 


1 


3.8 


3.7 


127 




380 


415 


449 


483 


517 


551 


585 


619 


653 687 


2 


7.6 
11.4 
15.2 


7.4 


128 




721 


755 


789 


823 


857 


890 


924 


958 


992 *025 


3 

4 


11.1 
14.8 


129 


11 


059 


093 


126 


160 


193 


227 


261 


294 


327 361 


5 


19.0 


18.5 


130 




394 


428 


461 


494 


528 


561 


594 


628 


661 694 


6 

7 


22.8 
26.6 


22.2 
25.9 


131 




727 


760 


793 


826 


860 


893 


926 


959 


992 *024 


8 

9 


30.4 
34.2 


29.6 


132 


12 


057 


090 


123 


166 


189 


222 


254 


287 


320 352 


33.3 


133 




385 


418 


450 


483 


516 


548 


581 


613 


646 678 


36 


35 


134 




710 


743 


775 


808 


840 


872 


905 


937 


969 *001 


1 


3.6 


35 


135 
136 


13 


033 
354 


066 
386 


098 
418 


130 
450 


162 
481 


194 
513 


226 
545 


258 
577 


290 322 
609 640 


2 
3 

4 


7.2 
10.8 
14.4 


7.0 

10.5 
14.0 


137 




672 


704 


735 


767 


799 


830 


862 


893 


925 956 


5 


18.0 


17.5 


138 




988 *019 *051 *082 *1U 


*145 *176 *208 *239 *270 


6 


21.6 


21.0 


139 


14 


301 


333 


364 


395 


426 


457 


489 


520 


551 582 


7 
8 


25.2 

28.8 


24.5 
28.0 


140 




613 


644 


675 


706 


737 


768 


799 


829 


860 891 


9 


32.4 


31.5 


N 


L 





■ 


2 


3 


4 


5 


6 


7 


8 9 


P. P. 



Logarithms of Numbers. 



83 





Num 


. 140 to 


179. 


Log 


. 146 to 255. 








N 


L 


1 


2 


3 


4 


5 


6 


7 8 9 


P. P. 


140 


14 613 


644 


675 


706 


737 


768 


799 


829 860 891 


34 


33 


141 


922 


953 


983 *014 *045 


*076 *106 *137 *168 *198 


1 
2 


3.4 

6.8 


3.3 
6.6 


142 


15 229 


259 


290 


320 


351 


381 


412 


442 473 503 


143 


534 


564 


594 


625 


655 


685 


715 


746 776 806 


3 


10.2 


9.9 


144 


836 


866 


897 


927 


957 


987 *017*047 *077 *107 


4 

5 


13.6 
17.0 


13.2 
16.5 


145 


16 137 


167 


197 


227 


256 


286 


316 


346 376 406 


6 

7 
S 


20.4 
23.8 

27.2 


19.8 
23.1 
26.4 


146 


4S5 


465 


495 


524 


554 


584 


613 


643 673 702 


147 


732 


761 


791 


820 


850 


879 


909 


938 967 997 


9 


30.6 


29.7 


148 


17 026 


056 


085 


114 


143 


173 


202 


231 260 289 


32 


31 


149 


319 


348 


377 


406 


435 


464 


493 


522 551 580 


150 


609 


638 


667 


696 


725 


754 


782 


811 840 869 


1 
2 


3.2 
6.4 


3.1 
6.2 


151 


898 


926 


955 


984 *013 


*041 *070 *099 *127 *156 


3 


9.6 


9.3 


152 


18 184 


213 


241 


270 


298 


327 


355 


384 412 441 


4 
5 

6 


12.8 
16.0 
19.2 


12.4 
15.5 
18.6 


153 


469 


498 


526 


554 


583 


611 


639 


667 696 724 


154 
155 


752 
19 033 


780 
061 


808 
089 


837 
117 


865 
145 


893 
173 


921 
201 


949 977 *005 

229 257 285 


7 
8 
9 


22.4 
25.6 
2S.8 


21.7 
24.8 
27.9 


156 


312 


340 


368 


396 


424 


451 


479 


507 535 562 


30 


29 


157 


590 


618 


645 


673 


700 


728 


756 


783 811 838 


158 


866 


893 


921 


948 


976 


*003 *030 *058 *085 *112 


1 


3.0 


2.9 


159 


20 140 


167 


194 


222 


249 


276 


303 


330 358 385 


2 
3 


6.0 
9.0 


5.8 
8.7 


160 


412 


439 


466 


493 


520 


548 


575 


602 629 656 


4 
5 

6 


12.0 
15.0 
18.0 


11.6 
14.5 
17.4 


161 


683 


710 


737 


763 


790 


817 


844 


871 898 925 


162 


952 


978 *005 *032 *059 


*085 *112 *139 *165 *192 


7 


21.0 


20.3 


163 


21 219 


245 


272 


299 


325 


352 


378 


405 .431 458 


8 
9 


24.0 
27.0 


23.2 
26.1 


164 


484 


511 


537 


564 


590 


617 


643 


669 696 722 


165 


748 


775 


801 


827 


854 


880 


906 


932 958 985 


28 


27 


166 


22 Oil 


037 


063 


089 


115 


141 


167 


194 220 246 


1 


2.8 


2.7 


167 


272 


298 


324 


350 


376 


401 


427 


453 479 505 


2 

3 
4 


5.6 

8.4 

11.2 


5.4 

8.1 

10.8 


168 


531 


557 


583 


608 


634 


660 


686 


712 737 763 


169 
170 


789 
23 045 


814 
070 


840 
096 


866 
121 


891 
147 


917 
172 


943 
198 


968 994 *019 
223 249 274 


5 
6 
7 


14.0 
16.8 
19.6 


13.5 
16.2 
18.9 


171 


300 


325 


350 


376 


401 


426 


452 


477 502 528 


8 

9 


22.4 
25.2 


21.6 
24.3 


172 


553 


578 


603 


629 


654 


679 


704 


729 754 779 


173 


805 


830 


855 


880 


905 


930 


955 


980 *005 *030 


26 


25 


174 


24 055 


080 


105 


130 


155 


180 


204 


229 254 279 


1 


2.6 


2.5 


175 


304 


329 


353 


378 


403 


428 


452 


477 502 527 


2 
3 
4 


5.2 

7.8 
10.4 


5.0 

7.5 

10.0 


176 


551 


576 


601 


625 


650 


674 


699 


724 748 773 


177 


797 


822 


846 


871 


895 


920 


944 


969 993 *018 


5 


13.0 


12.5 


178 


25 042 


066 


091 


115 


139 


164 


188 


212 237 261 


6 
7 

8 


15.6 

18.2 
20.8 


15.0 
17.5 
20.0 


179 


285 


310 


334 


358 


382 


406 


431 


455 479 503 


180 


527 


551 


575 


600 


624 


648 


672 


696 720 744 


9 


23.4 


22.5 


N 


L 


* 


2 


3 


4 


5 


6 


7 8 9 


P. P. 



84 



LOGARITHMS OF NUMBERS. 



Num. 180 to 219. Log. 255 to 342. 



N 


L 


1 


2 


3 


4 


5 


6 


7 8 9 


P. P. 


180 


25 527 


551 


575 


600 


624 


648 


672 


696 720 744 


24 


181 


768 


792 


816 


840 


864 


888 


912 


935 959 983 


1 
2 


2.4 

4.8 


182 


26 007 


031 


055 


079 


102 


126 


150 


174 198 221 


183 


245 


269 


293 


316 


340 


364 


387 


411 435 458 


3 


7.2 


184 


482 


505 


529 


553 


576 


600 


623 


647 670 694 


4 
5 


9.6 
12.0 


185 


717 


741 


764 


788 


811 


834 


858 


881 905 928 


6 

7 
8 


14.4 
168 
19.2 


186 


951 


975 


988 *021 *045 


*068 *091 *114 *138 *161 


187 


27 184 


207 


231 


254 


277 


300 


323 


346*370 393 


9 


21.6 


188 


416 


439 


462 


485 


508 


531 


554 


577 600 623 


23 


189 


646 


669 


692 


715 


738 


761 


784 


807 830 852 


190 


875 


898 


921 


944 


967 


989 *012 *035 *058 *081 


1 

2 


2.3 
4.6 


191 


28 103 


126 


149 


171 


194 


217 


240 


262 285 307 


3 


6.9 


192 


330 


353 


375 


398 


421 


443 


466 


488 511 533 


4 
5 
6 


9.2 

11.5 
13.8 


193 


556 


578 


601 


623 


646 


668 


691 


713 735 758 


194 
195 


780 
29 003 


803 
026 


825 
048 


847 
070 


870 
092 


892 
115 


914 
137 


937 959 981 
159 181 203 


7 
8 
9 


16.1 
18.4 
20.7 


196 


226 


248 


270 


292 


314 


336 


358 


380 403 425 


22 


197 


447 


469 


491 


513 


535 


557 


579 


601 623 645 


198 


667 


688 


710 


732 


754 


776 


798 


820 842 863 


1 


2.2 


199 


885 


907 


929 


951 


973 


994 *016 *038 *060 *081 


2 
3 


4.4 
6.6 


200 


30 103 


125 


146 


168 


190 


211 


233 


255 276 298 


4 
5 
6 


8.8 
11.0 
13.2 


201 


320 


341 


363 


384 


406 


428 


449 


471 492 514 


202 


535 


557 


578 


600 


621 


643 


664 


685 707 728 


7 


15.4 


203 


750 


771 


792 


814 


835 


856 


878 


899 920 942 


8 
9 


17.6 
19.8 


204 


963 


984 *006 *027 *048 


*069 *091 *112 *133 *154 


205 


31 175 


197 


218 


239 


260 


281 


302 


323 345 366 


21 


206 


387 


408 


429 


450 


471 


492 


513 


534 555 576 


1 


2.1 


207 


597 


618 


639 


660 


681 


702 


723 


744 765 785 


2 
3 
4 


4.2 
6.3 

8.4 


208 


806 


827 


848 


869 


890 


911 


931 


952 973 994 


209 
210 


32 015 
222 


035 
243 


056 
263 


077 

284 


098 
305 


118 
325 


139 
346 


160 181 201 
366 387 408 


5 
6 

7 


10.5 
12.6 
14.7 


211 


428 


449 


469 


490 


510 


531 


552 


572 593 613 


8 
9 


16.8 
18.9 


212 


634 


654 


675 


695 


715 


736 


756 


777 797 818 


213 


838 


858 


879 


899 


919 


940 


960 


980 *001 *021 


20 


19 


214 


33 041 


062 


082 


102 


122 


143 


163 


183 203 224 


1 


2.0 


1.9 


215 


244 


264 


284 


304 


325 


345 


365 


385 405 425 


2 
3 
4 


4.0 
6.0 
8.0 


3.8 
5.7 
7.6 


216 


445 


465 


486 


506 


526 


546 


566 


586 606 626 


217 


646 


666 


686 


706 


726 


746 


766 


786 806 826 


5 


10.0 


9.5 


218 


846 


866 


885 


905 


925 


945 


965 


985 *005 *025 


6 

7 
8 


12.0 
14.0 
16.0 


11.4 
13.3 
15.2 


219 


34 044 


064 


084 


104 


124 


143 


163 


183 203 223 


220 


242 


262 


282 


301 


321 


341 


361 


380 400 420 


9 


18.0 


17.1 


N 


L 


1 


2 


3 


4 


5 


6 


7 8 9 


P. P. 



Logarithms op Numbers. 



85 



Num. 220 to 259. Log. 342 to 414. 



N 


L 


1 


2 


3 


4 


5 6 7 8 9 


P 


. P. 


220 

221 


34 242 
439 


262 
459 


282 
479 


301 

498 


321 
518 


341 361 380 400 420 

537 557 577 596 616 


20 


222 


635 


655 


674 


694 


713 


733 753 772 792 811 


1 


2.0 


223 


830 


850 


869 


889 


908 


928 947 967 986 *005 


2 


4.0 


224 


35 025 


044 


064 


083 


102 


122 141 160 180 199 


3 
4 


6.0 

8.0 


225 


218 


238 


257 


276 


295 


315 334 353 372 392 


5 

6 


10.0 
12 


226 


411 


430 


449 


468 


488 


507 526 545 564 583 


7 


14.0 


227 


603 


622 


641 


660 


679 


698 717 736 755 774 


8 


16.0 


228 


793 


813 


832 


851 


870 


889 908 927 946 965 


9 


18.0 


229 


984 *003 *021 *040 *059 


*078 *097 *116 *135 *154 




230 


36 173 


192 


211 


229 


248 


267 286 305 324 342 


19 


231 


361 


380 


399 


418 


436 


455 474 493 511 530 


232 


549 


568 


586 


605 


624 


642 661 680 698 717 


1 


1.9 


233 


736 


754 


773 


791 


810 


829 847 866 884 903 


2 
3 
4 


3.8 
5.7 
7.6 


234 


922 


940 


959 


977 


996 


*014 *033 *051 *070 *088 


235 


37 107 


125 


144 


162 


181 


199 218 236 254 273 


5 
6 


9.5 
11.4 


236 


291 


310 


328 


346 


365 


38 401 420 438 457 


7 


13.3 


237 


475 


493 


511 


530 


548 


566 5 603 621 639 


8 
9 


15.2 
17.1 


238 


658 


676 


694 


712 


731 


749 767 785 803 822 


239 


840 


858 


876 


894 


912 


931 949 967 985 *003 




240 


38 021 


039 


057 


075 


093 


112 130 148 166 184 




241 


202 


220 


238 


256 


274 


292 310 328 346 364 


18 


242 


382 


399 


417 


435 


453 


471 489 507 525 543 


243 


561 


578 


596 


614 


632 


650 668 686 703 721 


1 


1.8 


244 


739 


757 


775 


792 


810 


828 846 863 881 899 


2 
3 


3.6 

5.4 


245 


917 


934 


952 


970 


987 


*005 *023 *041 *058 *076 


4 
5 


7.2 
9 


246 


39 094 


111 


129 


146 


164 


182 199 217 235 252 


6 


10.8 


247 


270 


287 


305 


322 


340 


358 375 393 410 428 


7 


12.6 


248 


445 


463 


480 


498 


515 


533 550 568 585 602 


8 
9 


14.4 
Ifi 9 


249 


620 


637 


655 


672 


690 


707 724 742 759 777 




250 


794 


811 


829 


846 


863 


881 898 915 933 950 




251 


967 


985 *002 *019 *037 


*054 *071 *088 *106 *123 




252 


40 140 


157 


175 


192 


209 


226 243 261 278 295 




253 


312 


329 


346 


364 


381 


398 415 432 449 466 


17 


254 


483 


500 


518 


535 


552 


569 586 603 620 637 


1 


1.7 


255 


654 


671 


688 


705 


722 


739 756 773 790 807 


2 
3 


3.4 
5.1 


256 


824 


841 


858 


875 


892 


909 926 943 960 976 


4 


6.8 


257 


993 *010 *027 *044 *061 


*078 *095 *111 *128 *145 


5 
6 

7 


8.5 
10.2 
11.9 


258 


41 162 


179 


196 


212 


229 


246 263 280 296 313 


259 
260 


330 
497 


347 
514 


363 
531 


380 
547 


397 
564 


414 430 447 464 481 
581 597 614 631 647 


8 
9 


13.6 
15.3 


N 


L 


1 


2 


3 


4 


5 6 7 8 9 


P. P. 



86 



Logarithms of Numbers. 





Num 


. 260 to 299. 


Log. 414 to 476. 






N 


L 


1 


2 


3 


4 


5 6 7 8 9 


P. P. 


260 


41 497 


514 


531 


547 


564 


581 597 614 631 647 




261 


664 


681 


697 


714 


731 


747 764 780 797 814 




262 


830 


847 


863 


880 


896 


913 929 946 963 979 




263 


996 *012 *029 *045 *062 


*078 *095 *111 *127 144 




264 


42 160 


177 


193 


210 


226 


243 259 275 292 308 


17 


265 


325 


341 


357 


374 


390 


406 423 439 455 472 


i i " 


266 


488 


504 


521 


537 


553 


570 586 602 619 635 


l 

2 


JL. / 

3.4 


267 


651 


667 


684 


700 


716 


732 749 765 781 797 


3 


5.1 


268 


813 


830 


846 


862 


878 


894 911 927 943 959 


4 
5 
6 


6.8 

8.5 

10.2 


269 


975 


991 *008 *024 *040 


*056 *072 *088 *104 *120 


270 


43 136 


152 


169 


185 


201 


217 233 249 265 281 


7 
8 


11.9 
13.6 


271 


297 


313 


329 


345 


361 


377 393 409 425 441 


9 


15.3 


272 


457 


473 


489 


505 


521 


537 553 569 584 600 




273 


616 


632 


648 


664 


680 


696 712 727 743 759 




274 


775 


791 


807 


823 


838 


854 870 886 902 917 




275 


933 


949 


965 


981 


996 


*012 *028 *044 *059 *075 




276 


44 091 


107 


122 


138 


154 


170 185 201 217 232 


16 


277 


248 


264 


279 


295 


311 


326 342 358 373 389 


278 


404 


420 


436 


451 


467 


483 498 514 529 545 


1 


1.6 


279 


560 


576 


592 


607 


623 


638 654 669 685 700 


2 
3 


3.2 

4.8 


280 


716 


731 


747 


762 


778 


793 809 824 840 855 


4 
5 
6 


6.4 
8.0 
9.6 


281 


871 


886 


902 


917 


932 


948 963 979 994 *010 


282 


45 025 


040 


056 


071 


086 


102 117 133 148 163 


7 


11.2 


283 


179 


194 


209 


225 


240 


255 271 286 301 317 


8 
9 


12.8 
14.4 


284 


332 


347 


362 


378 


393 


408 423 439 454 469 


285 


484 


500 


515 


530 


545 


561 576 591 606 621 




286 


637 


652 


667 


682 


697 


712 728 743 758 773 




287 


788 


803 


818 


834 


849 


864 879 894 909 924 




288 


939 


954 


969 


984 *000 


*015 *030 *045 *060 *075 




289 


46 090 


105 


120 


135 


150 


165 180 195 210 225 


15 


290 


240 


255 


270 


285 


300 


315 330 345 359 374 


1 


1.5 


291 


389 


404 


419 


434 


449 


464 479 494 509 523 


2 
3 
4 


3.0 
4.5 
6.0 


292 


538 


553 


568 


583 


598 


613 627 642 657 672 


293 


687 


702 


716 


731 


746 


761 776 790 805 820 


5 


7.5 


294 


835 


850 


864 


879 


894 


909 923 938 953 967 


6 

7 


9.0 
10.5 


295 


982 


997 *012 *026 *041 


*056 *070 *085 *100 *114 


8 
9 


12.0 
13.5 


296 


47 129 


144 


159 


173 


188 


202 217 232 246 261 


297 


276 


290 


305 


319 


334 


349 363 378 392 407 




298 


422 


436 


451 


465 


480 


494 509 524 538 553 




299 


567 


582 


596 


611 


625 


640 654 669 683 698 




300 


712 


727 


741 


756 


770 


784 799 813 828 842 




N 


L 


1 


2 


3 


4 


5 6 7 8 9 


P. P. 



LOGARITHMS OF NUMBERS. 



87 





Num. 


300 to 339. 


Log. 477 to 531. 






N 


L 


1 


2 


3 


4 


5 6 7 8 9 


P. P. 


300 


47 712 


727 


741 


756 


770 


784 799 813 828 842 




301 


857 


871 


885 


900 


914 


929 943 958 972 986 




302 


48 001 


015 


029 


044 


058 


073 087 101 116 130 




303 


144 


159 


173 


187 


202 


216 230 244 259 273 




304 


287 


302 


316 


330 


344 


359 373 387 401 416 


14 


305 


430 


444 


458 


473 


487 


501 515 530 544 558 


1 

2 


1.4 

2.8 


306 


572 


586 


601 


615 


629 


643 657 671 686 700 


307 


714 


728 


742 


756 


770 


785 799 813 827 841 


3 


4.2 


308 


855 


869 


883 


897 


911 


926 940 954 968 982 


4 

5 
6 


5.6 
7.0 
8.4 


309 


996 *010 *024 *038 *052 


*066 *080 *094 *108 *122 


310 


49 136 


150 


164 


178 


192 


206 220 234 248 262 


7 
8 


9.8 
11.2 


311 


276 


290 


304 


318 


332 


346 360 374 388 402 


9 


12.6 


312 


415 


429 


443 


457 


471 


485 499 513 527 541 




313 


554 


568 


582 


596 


610 


624 638 651 665 679 




314 


693 


707 


721 


734 


748 


762 776 790 803 817 




315 


831 


845 


859 


872 


886 


900 914 927 941 955 




316 


969 


982 


996 *010 *024 


*037 *051 *065 *079 *092 


13 


317 


50 106 


120 


133 


147 


161 


174 188 202 215 229 


318 


243 


256 


270 


284 


297 


311 325 338 352 365 


1 


1.3 


319 


379 


393 


406 


420 


433 


447 461 474 488 501 


2 
3 


2.6 
3.9 


320 


515 


529 


542 


556 


569 


583 596 610 623 637 


4 
5 
6 


5.2 

6.5 

7.8 


321 


651 


664 


678 


691 


705 


718 732 745 759 772 


322 


786 


799 


813 


826 


840 


853 866 880 893 907 


7 


9.1 


323 


920 


934 


947 


961 


974 


987 *001 *014 *028 *041 


8 
9 


10.4 
11.7 


324 


51 055 


068 


081 


095 


108 


121 135 148 162 175 


325 


188 


202 


215 


228 


242 


255 268 282 295 308 




326 


322 


335 


348 


362 


375 


388 402 415 428 441 




327 


455 


468 


481 


495 


508 


521 534 548 561 574 




328 


587 


601 


614 


627 


640 


654 667 680 693 706 




329 


720 


733 


746 


759 


772' 


786 799 812 825 838 


12 


330 


851 


865 


878 


891 


904 


917 930 943 957 970 


1 


1.2 


331 


983 


996 *009 *022 *035 


*048 *061 *075 *088 *101 


2 
3 
4 


2.4 

3.6 

4.8 


332 


52 114 


127 


140 


153 


166 


179 192 205 218 231 


333 


244 


257 


270 


284 


297 


310 323 336 349 362 


5 


6.0 


334 


375 


388 


401 


414 


427 


440 453 466 479 492 


6 

7 


7.2 
8.4 


335 


504 


517 


530 


543 


556 


569 582 595 608 621 


8 
9 


9.6 
10.8 


336 


634 


647 


660 


673 


686 


699 711 724 737 750 


337 


763 


776 


789 


802 


815 


827 840 853 866 879 




338 


892 


905 


917 


930 


943 


956 969 982 994 *007 




339 


53 020 


033 


046 


058 


071 


084 097 110 122 135 




340 


148 


161 


173 


186 


199 


212 224 237 250 263 




N 


L 


1 


2 


3 


4 


5 6 7 8 9 


P. P. 



88 



Logarithms op Numbers. 





Num 


340 to 379. 


Log 


531 


to 579. 






N 


L 


1 


2 


3 


4 


5 


6 


7 


8 9 


P. P. 


340 


53 148 


161 


173 


186 


199 


212 


224 


237 


250 263 




341 


275 


288 


301 


314 


326 


339 


352 


364 


377 390 




342 


403 


415 


428 


441 


453 


466 


479 


491 


504 517 




343 


529 


542 


555 


567 


580 


593 


605 


618 


631 643 




344 


656 


668 


681 


694 


706 


719 


732 


744 


757 769 


13 


345 


782 


794 


807 


820 


832 


845 


857 


870 


882 895 


1 i o 


346 


908 


920 


933 


945 


958 


970 


983 


995 *008 *020 


1 

2 


x.o 

2.6 


347 


54 033 


045 


058 


070 


083 


095 


108 


120 


133 145 


3 


3.9 


348 


158 


170 


183 


195 


208 


220 


233 


245 


258 270 


4 
5 
6 


5.2 
6.5 

7.8 


349 


283 


295 


307 


320 


332 


345 


357 


370 


382 394 


350 


407 


419 


432 


444 


456 


469 


481 


494 


506 518 


7 
8 


9,1 

10.4 


351 


531 


543 


555 


568 


580 


593 


605 


617 


630 642 


9 


11.7 


352 


654 


667 


679 


691 


704 


716 


728 


741 


753 765 




353 


777 


790 


802 


814 


827 


839 


851 


864 


876 888 




354 


900 


913 


925 


937 


949 


962 


974 


986 


998 *011 




355 


55 023 


035 


047 


060 


072 


084 


096 


108 


121 133 




356 


145 


157 


169 


182 


194 


206 


218 


230 


242 255 


12 


357 


267 


279 


291 


303 


315 


328 


340 


352 


364 376 


358 


388 


400 


413 


425 


437 


449 


461 


473 


485 497 


1 


1.2 


359 


509 


522 


534 


546 


558 


570 


582 


594 


606 618 


2 
3 


2.4 
3.6 


360 


630 


642 


654 


666 


678 


691 


703 


715 


727 739 


4 
5 
6 


4.8 
6.0 
7.2 


361 


751 


763 


775 


787 


799 


811 


823 


835 


847 859 


362 


871 


883 


895 


907 


919 


931 


943 


955 


967 979 


7 


8.4 


363 


991 *003 *015 *027 *038 


*050 *062 *074 *086 *098 


8 
9 


9.6 
10.8 


364 


56 110 


122 


134 


146 


158 


170 


182 


194 


205 217 


365 


229 


241 


253 


265 


277 


289 


301 


312 


324 336 




366 


348 


360 


372 


384 


396 


407 


419 


431 


443 455 




367 


467 


478 


490 


502 


514 


526 


538 


549 


561 573 




368 


585 


597 


608 


620 


632 


644 


656 


667 


679 691 




369 


703 


714 


726 


738 


750 


761 


773 


785 


797 808 


11 


370 


820 


832 


844 


855 


867 


879 


891 


902 


914 926 


1 


1.1 


371 


937 


949 


961 


972 


984 


996 *008 *019 *031 *043 


2 
3 

4 


2.2 
3.3 
4.4 


372 


57 054 


066 


078 


089 


101 


113 


124 


136 


148 159 


373 


171 


183 


194 


206 


217 


229 


241 


252 


264 276 


5 


5.5 


374 


287 


299 


310 


322 


334 


345 


357 


368 


380 392 


6 

7 


6.6 

7.7 


375 


403 


415 


426 


438 


449 


461 


473 


484 


496 507 


8 
9 


8.8 
9.9 


376 


519 


530 


542 


553 


565 


576 


588 


600 


611 623 


377 


634 


646 


657 


669 


680 


692 


703 


715 


726 738 




378 


749 


761 


772 


784 


795 


807 


818 


830 


841 852 




379 


864 


875 


887 


898 


910 


921 


933 


944 


955. 967 




380 


978 


990 *001 *013 *024 


*035 *047 *058 *070 *081 




N 


L 


1 


2 


3 


4 


5 


6 


7 


8 9 


P. P. 



Logarithms of Numbers. 





Num 


380 to 419. 


Log. 


579 to 623. 






N 


L 


1 


2 


3 


4 


5 


6 


7 


8 9 


P. P. 


380 


57 978 


990 *001 *013 *024 


*035 *047 *058 *070 *081 




381 


58 092 


104 


115 


127 


138 


149 


161 


172 


184 195 




382 


206 


218 


229 


240 


252 


263 


274 


286 


297 309 




383 


320 


331 


343 


354 


365 


377 


388 


399 


410 422 




384 


433 


444 


456 


467 


478 


490 


501 


512 


524 535 


11 


385 


546 


557 


569 


580 


591 


602 


614 


625 


636 647 


i i t 


386 


659 


670 


681 


692 


704 


715 


726 


737 


749 760 


l 
2 


JL.J. 

2.2 


387 


771 


782 


794 


805 


816 


827 


838 


850 


861 872 


3 


3.3 


388 


883 


894 


906 


917 


928 


939 


950 


961 


973 984 


4 
5 
6 


4.4 
5.5 
6.6 


389 


995 *006 *017 


*028 *040 


*051 *062 *073 *084 *095 


390 


59 106 


118 


129 


140 


151 


162 


173 


184 


195 207 


7 
8 


7.7 
8.8 


391 


218 


229 


240 


251 


262 


273 


284 


295 


306 318 


9 


9.9 


392 


329 


340 


351 


362 


373 


384 


395 


406 


417 428 




393 


439 


450 


461 


472 


483 


494 


506 


517 


528 539 




394 


550 


561 


572 


583 


594 


605 


616 


627 


638 649 




395 


660 


671 


682 


693 


704 


715 


726 


737 


748 759 




396 


770 


780 


791 


802 


813 


824 


835 


846 


857 868 


10 


397 


879 


890 


901 


912 


923 


934 


945 


956 


966 977 


398 


988 


999 *010 *021 *032 


*043 *054 *065 *076 *086 


1 


1.0 


399 


60 097 
206 


108 

217 


119 
228 


130 


141 


152 


163 


173 

282 


184 195 
293 304 


2 
3 

4 
5 
6 


2.0 
3.0 
4.0 
5.0 
6.0 


400 


239 


249 


260 


271 


401 


314 


325 


336 


347 


358 


369 


379 


390 


401 412 


402 


423 


433 


444 


455 


466 


477 


487 


498 


509 520 


7 


7.0 


403 


531 


541 


552 


563 


574 


584 


595 


606 


617 627 


8 
9 


8.0 
9.0 


404 


638 


649 


660 


670 


681 


692 


703 


713 


724 735 


405 


746 


756 


767 


778 


788 


799 


810 


821 


831 842 




406 


853 


863 


874 


885 


895 


906 


917 


927 


938 949 




407 


959 


970 


981 


991 *002 


*013 *023 *034 *045 *055 




408 


61 066 


077 


087 


098 


109 


119 


130 


140 


151 162 




409 


172 


183 


194 


204 


215 


225 


236 


247 


257 268 




410 


278 


289 


300 


310 


321 


331 


342 


352 


363 374 




411 


384 


395 


405 


416 


426 


437 


448 


458 


469 479 




412 


490 


500 


511 


521 


532 


542 


553 


563 


574 584 




413 


595 


606 


616 


627 


637 


648 


658 


669 


679 690 




414 


700 


711 


721 


731 


742 


752 


763 


773 


784 794 




415 


805 


815 


826 


836 


847 


857 


868 


878 


888 899 




416 


909 


920 


930 


941 


951 


962 


972 


982 


993 *003 




417 


62 014 


024 


034 


045 


055 


066 


076 


086 


097 107 




418 


118 


128 


138 


149 


159 


170 


180 


190 


201 211 




419 


221 


232 


242 


252 


263 


273 


284 


294 


304 315 




420 


325 


335 


346 


356 


366 


377 


387 


397 


408 418 




N 


L 


1 


2 


3 


4 


5 


6 


7 


8 9 


P. P. 



90 



Logarithms of Numbers. 



Num. 420 to 459. Log. 623 to 662. 



N 


L 


1 


2 


3 


4 


5 


6 7 8 9 


P. P. 


420 


62 325 


335 


346 


356 


366 


377 


387 397 408 418 




421 


428 


439 


449 


459 


469 


480 


490 500 511 521 




422 


531 


542 


552 


562 


572 


583 


593 603 613 624 




423 


634 


644 


655 


665 


675 


685 


696 706 716 726 




424 


737 


747 


757 


767 


778 


788 


798 808 818 829 




425 


839 


849 


859 


870 


880 


890 


900 910 921 931 




426 


941 


951 


961 


972 


982 


992 *002 *012 *022 *033 




427 


63 043 


053 


063 


073 


083 


094 


104 114 124 134 




428 


144 


155 


165 


175 


185 


195 


205 215 225 236 


10 


429 


246 


256 


266 


276 


286 


296 


306 317 327 337 


430 


347 


357 


367 


377 


387 


397 


407 417 428 438 


1 

2 


1.0 
2.0 


431 


448 


458 


468 


478 


488 


498 


508 518 528 538 


3 


3.0 


432 


548 


558 


568 


579 


589 


599 


609 619 629 639 


4 
5 
6 


4.0 
5.0 
6.0 


433 


649 


659 


669 


679 


689 


699 


709 719 729 739 


434 
435 


749 
849 


759 
859 


769 
869 


779 

879 


789 
889 


799 
899 


809 819 829 839 
909 919 929 939 


7 
8 
9 


7.0 
8.0 
9.0 


436 


949 


959 


969 


979 


988 


998 *008 *018 *028 *038 




437 


64 048 


058 


068 


078 


088 


098 


108 118 128 137 




438 


147 


157 


167 


177 


187 


197 


207 217 227 237 




439 


246 


256 


266 


276 


286 


296 


306 316 326 335 




440 


345 


355 


365 


375 


385 


395 


404 414 424 434 




441 


444 


454 


464 


473 


483 


493 


503 513 523 532 




442 


542 


552 


562 


572 


582 


591 


601 611 621 631 




443 


640 


650 


660 


670 


680 


689 


699 709 719 729 




444 


738 


748 


758 


768 


777 


787 


797 807 816 826 




445 


836 


846 


856 


865 


875 


885 


895 904 914 924 


9 


446 


933 


943 


953 


963 


972 


982 


992 *002 *011 *021 


1 


0.9 


447 


65 031 


040 


050 


060 


070 


079 


089 099 108 118 


2 
3 

4 


1.8 
2.7 
3.6 


448 


128 


137 


147 


157 


167 


176 


186 196 205 215 


449 
450 


225 
321 


234 
331 


244 
341 


254 
350 


263 
360 


273 
369 


283 292 302 312 
379 389 398 408 


5 
6 

7 


4.5 
5.4 
6.3 


451 


418 


427 


437 


447 


456 


466 


475 485 495 504 


8 
9 


7.2 
8.1 


452 


514 


523 


533 


543 


552 


562 


571 581 591 600 


453 


610 


619 


629 


639 


648 


658 


667 677 686 696 




454 


706 


715 


725 


734 


744 


753 


763 772 782 792 




455 


801 


811 


820 


830 


839 


849 


858 868 877 887 




456 


896 


906 


916 


925 


935 


944 


954 963 973 982 




457 


992 *001 *011 *020 *030 


*039 *049 *058 *068 *077 




458 


66 087 


096 


106 


115 


124 


134 


143 153 162 172 




459 


181 


191 


200 


210 


219 


229 


238 247 257 266 




460 


276 


285 


295 


304 


314 


323 


332 342 351 361 




N 


L 


1 


2 


3 


4 


5 


6 7 8 9 


P. P. 



Logarithms of Numbers. 



91 





Num 


460 to 499. 


Log. 


662 to 698. 






N 


L 


i 


2 


3 


4 


5 


6 


7 8 9 


P. P. 


460 


66 276 


285 


295 


304 


314 


323 


332 


342 351 361 




461 


370 


380 


389 


398 


408 


417 


427 


436 445 455 




462 


464 


474 


483 


492 


502 


511 


521 


530 539 549 




463 


558 


567 


577 


586 


596 


605 


614 


624 633 642 




464 


652 


661 


671 


680 


689 


699 


708 


717 727 736 




465 


745 


755 


764 


773 


783 


792 


801 


811 820 829 




466 


839 


848 


857 


867 


876 


885 


894 


904 913 922 




467 


932 


941 


950 


960 


969 


978 


987 


997 *006 *015 




468 


67 025 


034 


043 


052 


062 


071 


080 


089 099 108 


10 


469 


117 


127 


136 


145 


154 


164 


173 


182 191 201 


470 


210 


219 


228 


237 


247 


256 


265 


274 284 293 


1 

2 


1.0 
2.0 


471 


302 


311 


321 


330 


339 


348 


357 


367 376 385 


3 


3.0 


472 


394 


403 


413 


422 


431 


440 


449 


459 468 477 


4 
5 
6 


4.0 
5.0 
6.0 


473 


486 


495 


504 


514 


523 


532 


541 


550 560 569 


474 


578 


587 


596 


605 


614 


624 


633 


642 651 660 


7 
8 


7.0 

8.0 


475 


669 


679 


688 


697 


706 


715 


724 


733 742 752 


9 


9.0 


476 


761 


770 


779 


788 


797 


806 


815 


825 834 843 




477 


852 


861 


870 


879 


888 


897 


906 


916 925 934 




478 


943 


952 


961 


970 


979 


988 


997 *006 *015 *024 




479 


68 034 


043 


052 


061 


070 


079 


088 


097 106 115 




480 


124 


133 


142 


151 


160 


169 


178 


187 196 205 




481 


215 


224 


233 


242 


251 


260 


269 


278 287 296 




482 


305 


314 


323 


332 


341 


350 


359 


368 377 386 




483 


395 


404 


413 


422 


431 


440 


449 


458 467 476 




484 


485 


494 


502 


511 


520 


529 


538 


547 556 565 




485 


574 


583 


592 


601 


610 


619 


628 


637 646 655 


9 


486 


664 


673 


681 


690 


699 


708 


717 


726 735 744 


1 


0.9 


487 


753 


762 


771 


780 


789 


797 


806 


815 824 833 


2 
3 
4 


1.8 
2.7 
3.6 


488 


842 


851 


860 


869 


878 


886 


895 


904 913 922 


489 


931 


940 


949 


958 


966 


975 


984 


993 *002 *011 


5 

6 


4.5 
5.4 


490 


69 020 


028 


037 


046 


055 


064 


073 


082 090 099 


7 


6.3 


491 


108 


117 


126 


135 


144 


152 


162 


170 179 188 


8 
9 


7.2 
8.1 


492 


197 


205 


214 


223 


232 


241 


249 


258 267 276 


493 


285 


294 


302 


311 


320 


329 


338 


346 355 364 




494 


373 


381 


390 


399 


408 


417 


425 


434 443 452 




495 


461 


469 


478 


487 


496 


504 


513 


522 531 539 




496 


548 


557 


566 


574 


583 


592 


601 


609 618 627 




497 


636 


644 


653 


662 


671 


679 


688 


697 705 714 




498 


723 


732 


740 


749 


758 


767 


775 


784 793 801 




499 


810 


819 


827 


836 


845 


854 


862 


871 880 888 




500 


897 


906 


914 


923 


932 


940 


949 


958 966 975 




N 


L 


1 


2 


3 


4 


5 


6 


7 8 9 


P. P. 



92 



Logarithms op Numbers. 





Num 


500 to 539. 


Log 


698 to 732. 






N 


L 


1 


2 


3 


4 


5 


6 


7 


8 9 


P. P. 


500 


69 897 


906 


914 


922 


932 


940 


949 


958 


966 975 




501 


984 


992 *001 *010 *018 


*027 *036 *044 *053 *062 




502 


70 070 


079 


088 


096 


105 


114 


122 


131 


140 148 




503 


157 


165 


174 


183 


191 


200 


209 


217 


226 234 




504 


243 


252 


260 


269 


278 


286 


295 


303 


312 321 




505 


329 


338 


346 


355 


364 


372 


381 


389 


398 406 




506 


415 


424 


432 


441 


449 


458 


467 


475 


484 492 




507 


501 


509 


518 


526 


535 


544 


552 


561 


569 578 




508 


586 


595 


603 


612 


621 


629 


638 


646 


655 663 


9 


509 


672 


680 


689 


697 


706 


714 


723 


731 


740 749 


510 


757 


766 


774 


783 


791 


800 


808 


817 


825 834 


1 

2 


0.9 
1.8 


511 


842 


851 


859 


868 


876 


885 


893 


902 


910 919 


3 


2.7 


512 


927 


935 


944 


952 


961 


969 


978 


986 


995 *003 


4 
5 
6 


3.6 
4.5 
5.4 


513 


71 012 


020 


029 


037 


046 


054 


063 


071 


079 088 


514 
515 


096 
181 


105 
189 


113 
198 


122 
206 


130 
214 


139 
223 


147 
231 


155 

240 


164 172 
248 257 


7 
8 
9 


6.3 
7.2 
8.1 


516 


265 


273 


282 


290 


299 


307 


315 


324 


332 341 




517 


349 


357 


366 


374 


383 


391 


399 


408 


416 425 




518 


433 


441 


450 


458 


466 


475 


483 


492 


500 508 




519 


517 


525 


533 


542 


550 


559 


567 


575 


584 592 




520 


600 


609 


617 


625 


634 


642 


650 


659 


667 675 




521 


684 


692 


700 


709 


717 


725 


734 


742 


750 759 




522 


767 


775 


784 


792 


800 


809 


817 


825 


834 842 




523 


850 


858 


867 


875 


883 


892 


900 


908 


917 925 




524 


933 


941 


950 


958 


966 


975 


983 


991 


999 *008 




525 


72 016 


024 


032 


041 


049 


057 


066 


074 


082 090 


8 


526 


099 


107 


115 


123 


132 


140 


148 


156 


165 173 


1 


0.8 


527 


181 


189 


198 


206 


214 


222 


230 


239 


247 255 


2 
3 

4 


1.6 

2.4 
3.2 


528 


263 


272 


280 


288 


296 


304 


313 


321 


329 337 


529 
530 


346 

428 


354 

436 


362 

444 


370 

452 


378 
460 


387 
469 


395 
477 


403 

485 


411 419 
493 501 


5 
6 

7 


4.0 

4.8 
5.6 


531 


509 


518 


526 


534 


542 


550 


558 


567 


575 583 


8 
9 


6.4 
7.2 


532 


591 


599 


607 


616 


624 


632 


640 


648 


656 665 


533 


673 


681 


689 


697 


705 


713 


722 


730 


738 746 




534 


754 


762 


770 


779 


787 


795 


803 


811 


819 827 




535 


835 


843 


852 


860 


868 


876 


884 


892 


900 908 




536 


916 


925 


933 


941 


949 


957 


965 


973 


981 989 




537 


997 *006 *014 *022 *030 


*038 *046 *054 


062 *070 




538 


73 078 


086 


094 


102 


111 


119 


127 


135 


143 151 




539 


159 


167 


175 


183 


191 


199 


207 


215 


223 231 




540 


239 


247 


255 


263 


272 


280 


288 


296 


304 312 




N 


L 


1 


2 


3 


4 


5 


6 


7 


8 9 


P. P. 



Logarithms of Numbers. 



93 



Num. 540 to 579. Log. 732 to 763. 



P. P. 



540 

541 

542 
543 
544 

545 
546 
547 

548 
549 

550 

551 
552 
553 
554 

555 
556 
557 
558 
559 

560 

561 
562 
563 
564 

565 
566 
567 
568 
569 

570 

571 
572 
573 
574 

575 
576 
577 
578 
579 

580 



73 239 247 255 263 272 
320 328 336 344 352 
400 408 416 424 432 
480 488 496 504 512 
560 568 576 584 592 

640 648 656 664 672 

719 727 735 743 751 

799 807 815 823 830 

878 886 894 902 910 

957 965 973 981 989 

74 036 044 052 060 068 
115 123 131 139 147 
194 202 210 218 225 
273 280 288 296 304 
351 359 367 374 382 

429 437 445 453 461 

507 515 523 531 539 

586 593 601 609 617 

663 671 679 687 695 
741 749 757 764 772 

819 827 834 842 850 

896 904 912 920 927 

974 981 989 997 *005 

75 051 059 066 074 082 
128 136 143 151 159 

205 213 220 228 236 

282 289 297 305 312 

358 366 374 381 389 

435 442 450 458 465 

511 519 526 534 542 

587 595 603 610 618 

664 671 679 686 694 
740 747 755 762 770 
815 823 831 838 846 
891 899 906 914 921 

967 974 982 989 997 

76 042 050 057 065 072 
118 125 133 140 148 
193 200 208 215 223 
268 275 283 290 298 

343 350 358 365 373 



280 288 296 304 312 

360 368 376 384 392 

440 448 456 464 472 

520 528 536 544 552 

600 608 616 624 632 

679 687 695 703 711 

759 767 775 783 791 

838 846 854 862 870 

918 926 933 941 949 

997 *005 *013 *020 *028 

076 084 092 099 107 

155 162 170 178 186 

233 241 249 257 265 

312 320 327 335 343 

390 398 406 414 421 

468 476 484 492 500 

547 554 562 570 578 

624 632 640 648 656 

702 710 718 726 733 

780 788 796 803 811 

858 865 873 881 889 

935 943 950 958 966 

*012 *020 *028 *035 *043 

089 097 105 113 120 

166 174 182 189 197 

243 251 259 266 274 



320 


328 


335 


343 


351 


1 


0.7 


397 


404 


412 


420 


427 


2 
3 

4 


1.4 
2.1 

28 


473 


481 


488 


496 


504 


549 


557 


565 


572 


580 


5 

6 


3.5 
4.2 


626 


633 


641 


648 


656 


7 


4.9 


702 


709 


717 


724 


732 


8 
9 


5.6 
6.3 


778 


785 


793 


800 


808 


853 


861 


868 


876 


884 






929 


937 


944 


952 


959 







*005 *012 *020 *027 *035 
080 087 095 103 110 
155 163 170 178 185 
230 238 245 253 260 
305 313 320 328 335 

380 388 395 403 410 



1 


0.8 


2 


1.6 


3 


2.4 


4 


3.2 


5 


4.0 


6 


4.8 


7 


5.6 


8 


6.4 


9 


7.2 



P. p. 



94 



Logarithms of Numbers. 





Num 


. 580 to 619. 


Log 


. 763 to 792. 






N 


L 


1 


2 


3 


& 


5 


.6 


.7 


8 9 


P. P. 


580 


76 343 


350 


358 


365 


373 


380 


388 


395 


403 410 


8 


581 


418 


425 


433 


440 


448 


455 


462 


470 


477 485 




582 


492 


500 


507 


515 


522 


530 


537 


545 


552 559 


1 
2 


1.5 


583 


567 


574 


582 


589 


597 


604 


612 


619 


626 634 


3 


2.4 


584 


641 


649 


656 


664 


671 


678 


686 


693 


701 708 


4 
5 


3.2 
4.0 


585 


716 


723 


730 


738 


745 


753 


760 


768- 


775 782 


6 

7 
8 


4.8 
5.6 
6.4 


586 


790 


797 


805 


812 


819 


827 


834 


842 


849 856 


587 


864 


871 


879 


886 


893 


901 


908 


916 


923 930 


9 


7.2 


588 


938 


945 


953 


960 


967 


975 


982 


989 


997 *004 




589 


77 012 


019 


026 


034 


041 


048 


056 


063 


070 078 




590 


085 


093 


100 


107 


115 


122 


129 


137 


144 151 




591 


159 


166 


173 


181 


188 


195 


203 


210 


217 225 




592 


232 


240 


247 


254 


262 


269 


276 


283 


291 298 




593 


305 


313 


320 


327 


335 


342 


349 


357 


364 371 




594 


379 


386 


393 


401 


408 


415 


422 


430 


437 444 




595 


452 


459 


466 


474 


481 


488 


495 


503 


510 517 




596 


525 


532 


539 


546 


554 


561 


568 


576 


583 590 




597 


597 


605 


612 


619 


627 


634 


641 


648 


656 663 


7 


598 


670 


677 


685 


692 


699 


706 


714 


721 


728 735 




599 


743 


750 


757 


764 


772 


779 


786 


793 


801 808 


1 
2 


0.7 
1.4 


600 


815 


822 


830 


837 


844 


851 


859 


866 


873 880 


3 
4 
5 


2.1 

2.8 
3.5 


601 


887 


895 


902 


909 


916 


924 


931 


938 


945 952 


602 


960 


967 


974 


981 


988 


996 *003 *010 *017 *025 


6 


4.2 


603 


78 032 


039 


046 


053 


061 


068 


075 


082 


089 097 


7 
8 
9 


4.9 
5.6 
6.3 


604 


104 


111 


118 


125 


132 


140 


147 


154 


161 168 


605 


176 


183 


190 


197 


204 


211 


219 


226 


233 240 




606 


247 


254 


262 


269 


276 


283 


290 


297 


305 312 




607 


319 


326 


333 


340 


347 


355 


362 


369 


376 383 




608 


390 


398 


405 


412 


419 


426 


433 


440 


447 455 




609 


462 


469 


476 


483 


490 


497 


504 


512 


519 526 




610 


533 


540 


547 


554 


561 


569 


576 


583 


590 597 




611 


604 


611 


618 


625 


633 


640 


647 


654 


661 668 




612 


675 


682 


689 


696 


704 


711 


718 


725 


732 739 




613 


746 


753 


760 


767 


774 


781 


789 


796 


802 810 




614 


817 


824 


831 


838 


845 


852 


859 


866 


873 880 




615 


888 


895 


902 


909 


916 


923 


930 


937 


944 951 




616 


958 


965 


972 


979 


986 


993 *000 *007 *014 *021 


i 


617 


79 029 


036 


043 


050 


057 


064 


071 


078 


085 092 




618 


099 


106 


113 


120 


127 


134 


141 


148 


155 162 




619 


169 


176 


183 


190 


197 


204 


211 


218 


225 232 


i 


620 


239 


246 


253 


260 


267 


274 


281 


288 


295 302 


) 


N 


L 


■ 


2 


3 


4 


5 


6 


7 


8 9 


p. p. 



Logarithms of Numbers. 



95 



Num. 620 to 659. Log. 792 to 819. 



N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


620 


79 239 


246 


253 


260 


267 


274 


281 


288 


295 


302 




621 


309 


316 


323 


330 


337 


344 


351 


358 


365 


372 




622 


379 


386 


393 


400 


407 


414 


421 


428 


435 


442 




623 


449 


456 


463 


470 


477 


484 


491 


498 


505 


511 




624 


518 


525 


532 


539 


546 


553 


560 


567 


574 


581 




625 


588 


595 


602 


609 


616 


623 


630 


637 


644 


650 




626 


657 


664 


671 


678 


685 


692 


699 


706 


713 


720 




627 


727 


734 


741 


748 


754 


761 


768 


775 


782 


789 




628 


796 


803 


810 


817 


824 


831 


837 


844 


851 


858 




629 


865 


872 


879 


886 


893 


900 


906 


913 


920 


927 




630 


934 


941 


948 


955 


962 


969 


975 


982 


989 


996 




631 


80 003 


010 


017 


024 


030 


037 


044 


051 


058 


065 




632 


072 


079 


085 


092 


099 


106 


113 


120 


127 


134 




633 


140 


147 


154 


161 


168 


175 


182 


188 


195 


202 




634 


209 


216 


223 


229 


236 


243 


250 


257 


264 


271 




635 


277 


284 


291 


298 


305 


312 


318 


325 


332 


339 




636 


346 


353 


359 


366 


373 


380 


387 


393 


400 


407 


7 


637 


414 


421 


428 


434 


441 


448 


455 


462 


468 


475 


638 


482 


489 


496 


502 


509 


516 


523 


530 


536 


543 


1 


0.7 


639 


550 


557 


564 


570 


577 


584 


591 


598 


604 


611 


2 
3 


1.4 

2.1 


640 


618 


625 


632 


638 


645 


652 


659 


665 


672 


679 


4 
5 
6 


2.8 
3.5 
4.2 


641 


686 


693 


699 


706 


713 


720 


726 


733 


740 


747 


642 


754 


760 


767 


774 


781 


787 


794 


801 


808 


814 


7 


4.9 


643 


821 


828 


835 


841 


848 


855 


862 


868 


875 


882 


8 
9 


5.6 

6.3 


644 


889 


895 


902 


909 


916 


922 


929 


936 


943 


949 


645 


956 


963 


969 


976 


983 


990 


996 *003 *010 *017 




646 


81 023 


030 


037 


043 


050 


057 


064 


070 


077 


084 




647 


090 


097 


104 


111 


117 


124 


131 


137 


144 


151 




648 


158 


164 


171 


178 


184 


191 


198 


204 


211 


218 




649 


224 


231 


238 


245 


251 


258 


265 


271 


278 


285 




650 


291 


298 


305 


311 


318 


325 


331 


338 


345 


351 




651 


358 


365 


371 


378 


385 


391 


398 


405 


411 


418 




652 


425 


431 


438 


445 


451 


458 


465 


471 


478 


485 




653 


491 


498 


505 


511 


518 


525 


531 


538 


544 


551 




654 


558 


564 


571 


578 


584 


591 


598 


604 


611 


617 




655 


624 


631 


637 


644 


651 


657 


664 


671 


677 


684 




656 


690 


697 


704 


710 


717 


723 


730 


737 


743 


750 




657 


757 


763 


770 


776 


783 


790 


796 


803 


809 


816 




658 


823 


829 


836 


842 


849 


856 


862 


869 


875 


882 




659 


889 


895 


902 


908 


915 


921 


928 


935 


941 


948 




660 


954 


961 


968 


974 


981 


987 


994 *000 *007 *014 




N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



96 



LOGARITHMS OF NUMBERS. 





Num 


. 660 to 699. 


Log 


819 to 845. 








N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 

■am 


660 


81 954 


961 


968 


974 


981 


987 


994 *000 *007 *014 


7 


661 


82 020 


027 


033 


040 


046 


053 


060 


066 


073 


079 


1 

2 


0.7 
1.4 


662 


086 


092 


099 


105 


112 


119 


125 


132 


138 


145 


663 


151 


158 


164 


171 


178 


184 


191 


197 


204 


210 


3 


2.1 


664 


217 


223 


230 


236 


243 


249 


256 


263 


269 


276 


4 
5 


2.8 
3.5 


665 


282 


289 


295 


302 


308 


315 


321 


328 


334 


341 


6 

7 
8 


4.2 
4.9 

5.6 


666 


347 


354 


360 


367 


373 


380 


387 


393 


400 


406 


667 


413 


419 


426 


432 


439 


445 


452 


458 


465 


471 


9 


6.3 


668 


478 


484 


491 


497 


504 


510 


517 


523 


530 


536 




669 


543 


549 


556 


562 


569 


575 


582 


588 


595 


601 




670 


607 


614 


620 


627 


633 


640 


646 


653 


659 


666 




671 


672 


679 


685 


692 


698 


705 


711 


718 


724 


730 




672 


737 


743 


750 


756 


763 


769 


776 


782 


789 


795 




673 


802 


808 


814 


821 


827 


834 


840 


847 


853 


860 




674 


866 


872 


879 


885 


892 


898 


905 


911 


918 


924 




675 


930 


937 


943 


950 


956 


963 


969 


975 


982 


988 




676 


995 *001 *008 *014 *020 


*027 *033 *040 *046 *052 




677 


83 059 


065 


072 


078 


085 


091 


097 


104 


110 


117 


6 


678 


123 


129 


136 


142 


149 


155 


161 


168 


174 


181 




679 


187 


193 


200 


206 


213 


219 


225 


232 


238 


245 


1 

2 


0.6 
1.2 


680 


251 


257 


264 


270 


276 


283 


289 


296 


302 


308 


3 
4 
5 


1.8 
2.4 
3.0 


681 


315 


321 


327 


334 


340 


347 


353 


359 


366 


372 


682 


378 


385 


391 


398 


404 


410 


417 


423 


429 


436 


6 


3.6 


683 


442 


448 


455 


461 


467 


474 


480 


487 


493 


499 


7 
8 
9 


4.2 
4.8 
5.4 


684 


506 


512 


518 


525 


531 


537 


544 


550 


556 


563 


685 


569 


575 


582 


588 


594 


601 


607 


613 


620 


626 




686 


632 


639 


645 


651 


658 


664 


670 


677 


683 


689 




687 


696 


702 


708 


715 


721 


727 


734 


740 


746 


753 




688 


759 


765 


771 


778 


784 


790 


797 


803 


809 


816 




689 


822 


828 


835 


841 


847 


853 


860 


866 


872 


879 




690 


885 


891 


897 


904 


910 


916 


923 


929 


935 


942 




691 


948 


954 


960 


967 


973 


979 


985 


992 


998 *004 




692 


84 Oil 


017 


023 


029 


036 


042 


048 


055 


061 


067 




693 


073 


080 


086 


092 


098 


105 


111 


117 


123 


130 




694 


136 


142 


148 


155 


161 


167 


173 


180 


186 


192 




695 


198 


205 


211 


217 


223 


230 


236 


242 


248 


255 




696 


261 


267 


273 


280 


286 


292 


298 


305 


311 


317 




697 


323 


330 


336 


342 


348 


354 


361 


367 


373 


379 




698 


386 


392 


398 


404 


410 


417 


423 


429 


435 


442 




699 


448 


454 


460 


466 


473 


479 


485 


491 


497 


504 


4 


700 


510 


516 


522 


528 


535 


541 


547 


553 


559 


566 




N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



Logarithms of Numbers. 



97 





Num. 


700 to 739. 


Log 


. 845 to 869. 








N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


700 


84 510 


516 


522 


528 


535 


541 


547 


553 


559 


566 




701 


572 


578 


584 


590 


597 


603 


609 


615 


621 


628 




702 


634 


640 


646 


652 


658 


665 


671 


677 


683 


689 




703 


696 


702 


708 


714 


720 


726 


733 


739 


745 


751 




704 


757 


763 


770 


776 


782 


788 


794 


800 


807 


813 




705 


819 


825 


831 


837 


844 


850 


856 


862 


868 


874 




706 


880 


887 


893 


899 


905 


911 


917 


924 


930 


936 




707 


942 


948 


954 


960 


967 


973 


979 


985 


991 


997 




708 


85 003 


009 


016 


022 


028 


034 


040 


046 


052 


058 




709 


065 


071 


077 


083 


089 


095 


101 


107 


114 


120 




710 


126 


132 


138 


144 


150 


156 


163 


169 


175 


181 




711 


187 


193 


199 


205 


211 


217 


224 


230 


236 


242 




712 


248 


254 


260 


266 


272 


278 


285 


291 


297 


303 




713 


309 


315 


321 


327 


333 


339 


345 


352 


358 


364 




714 


370 


376 


382 


388 


394 


400 


406 


412 


418 


425 




715 


431 


437 


443 


449 


455 


461 


467 


473 


479 


485 




716 


491 


497 


503 


509 


516 


522 


528 


534 


540 


546 




717 


552 


558 


564 


570 


576 


582 


588 


594 


600 


606 


6 


718 


612 


618 


625 


631 


637 


643 


649 


655 


661 


667 




719 


673 


679 


685 


691 


697 


703 


709 


715 


721 


727 


1 

2 


1.2 


720 


733 


739 


745 


751 


757 


763 


769 


775 


781 


788 


3 

4 


1.8 
2.4 


721 


794 


800 


806 


812 


818 


824 


830 


836 


842 


848 


5 


3.0 


722 


854 


860 


866 


872 


878 


884 


890 


896 


902 


908 


6 

• 7 
8 


3.6 
4.2 
4.8 


723 


914 


920 


926 


932 


938 


944 


950 


956 


962 


968 


724 


974 


980 


986 


992 


998 


*004 *010 *016 *022 *028 


9 


5.4 


725 


86 034 


040 


046 


052 


058 


064 


070 


076 


082 


088 




726 


094 


100 


106 


112 


118 


124 


130 


136 


141 


147 




727 


153 


159 


165 


171 


177 


183 


189 


195 


201 


207 




728 


213 


219 


225 


231 


237 


243 


249 


255 


261 


267 




729 


273 


279 


285 


291 


297 


303 


308 


314 


320 


326 




730 


332 


338 


344 


350 


356 


362 


368 


374 


380 


386 




731 


392 


398 


404 


410 


415 


421 


427 


433 


439 


445 




732 


451 


457 


463 


469 


475 


481 


487 


493 


499 


504 




733 


510 


516 


522 


528 


534 


540 


546 


552 


558 


564 




734 


570 


576 


581 


587 


593 


599 


605 


611 


617 


623 




735 


629 


635 


641 


646 


652 


658 


664 


670 


676 


682 




736 


688 


694 


700 


705 


711 


717 


723 


729 


735 


741 




737 


747 


753 


759 


764 


770 


776 


782 


788 


794 


800 




738 


806 


812 


817 


823 


829 


835 


841 


847 


853 


859 




739 


864 


870 


876 


882 


888 


894 


900 


906 


911 


917 




740 


923 


929 


935 


941 


947 


953 


958 


964 


970 


976 




N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



98 



LOGAKITHMS OF NUMBERS. 



Num. 740 to 779. Log. 869 to 892. 



N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


740 


86 923 


929 


935 


941 


947 


953 


958 


964 


970 


976 




741 


982 


988 


994 


999 *005 


*011 *017 *023 *029 *035 




742 


87 040 


046 


052 


058 


064 


070 


075 


081 


087 


093 




743 


099 


105 


111 


116 


122 


128 


134 


140 


146 


151 




744 


157 


163 


169 


175 


181 


186 


192 


198 


204 


210 




745 


216 


221 


227 


233 


239 


245 


251 


256 


262 


268 




746 


274 


280 


286 


291 


297 


303 


309 


315 


320 


326 




747 


332 


338 


344 


349 


355 


361 


367 


373 


379 


384 




748 


390 


396 


402 


408 


413 


419 


425 


431 


437 


442 




749 


448 


454 


460 


466 


471 


477 


483 


489 


495 


500 




750 


506 


512 


518 


523 


529 


535 


541 


547 


552 


558 




751 


564 


570 


576 


581 


587 


593 


599 


604 


610 


616 




752 


622 


628 


633 


639 


645 


651 


656 


662 


668 


674 




753 


679 


685 


691 


697 


703 


708 


714 


720 


726 


731 




754 


737 


743 


749 


754 


760 


766 


772 


777 


783 


789 




755 


795 


800 


806 


812 


818 


823 


829 


835 


841 


846 




756 


852 


858 


864 


869 


875 


881 


887 


892 


898 


904 




757 


910 


915 


921 


927 


933 


938 


944 


950 


955 


961 


6 


758 


967 


973 


978 


984 


990 


996 *001 *007 *013 *018 




759 


88 024 


030 


036 


041 


047 


053 


058 


064 


070 


076 


1 

2 


1.2 


760 


081 


087 


093 


098 


104 


110 


116 


121 


127 


133 


3 
4 


1.8 

2.4 


761 


138 


144 


150 


156 


161 


167 


173 


178 


184 


190 


5 


3.0 


762 


195 


201 


207 


213 


218 


224 


230 


235 


241 


247 


6 
7 
8 


3.6 
4.2 
4.8 


763 


252 


258 


264 


270 


275 


281 


287 


292 


298 


304 


764 


309 


315 


321 


326 


332 


338 


343 


349 


355 


360 


9 


5.4 


765 


366 


372 


377 


383 


389 


395 


400 


406 


412 


417 




766 


423 


429 


434 


440 


446 


451 


457 


463 


468 


474 




767 


480 


485 


491 


497 


502 


508 


513 


519 


525 


530 




768 


536 


542 


547 


553 


559 


564 


570 


576 


581 


587 




769 


593 


598 


604 


610 


615 


621 


627 


632 


638 


643 




770 


649 


655 


660 


666 


672 


677 


683 


689 


694 


700 




771 


705 


711 


717 


722 


728 


734 


739 


745 


750 


756 




772 


762 


767 


773 


779 


784 


790 


795 


801 


807 


812 




773 


818 


824 


829 


835 


840 


846 


852 


857 


863 


868 




774 


874 


880 


885 


891 


897 


902 


908 


913 


919 


925 




775 


930 


936 


941 


947 


953 


958 


964 


969 


975 


981 




776 


986 


992 


997 *003 *009 


*014 *020 *025 *031 *037 




777 


89 042 


048 


053 


059 


064 


070 


076 


081 


087 


092 




778 


098 


104 


109 


115 


120 


126 


131 


137 


143 


148 




779 


154 


159 


165 


170 


176 


182 


187 


193 


198 


204 




780 


209 


215 


221 


226 


232 


237 


243 


248 


254 


260 




N 


L 


i 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



Logarithms of Numbers. 



99 



Num. 780 to 819. Log. 892 to 913. 



N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


780 


89 209 


215 


221 


226 


232 


237 


243 


248 


254 


260 




781 


265 


271 


276 


282 


287 


293 


298 


304 


310 


315 




782 


321 


326 


332 


337 


343 


348 


354 


360 


365 


371 




783 


376 


382 


387 


393 


398 


404 


409 


415 


421 


426 




784 


432 


437 


443 


448 


454 


459 


465 


470 


476 


481 




785 


487 


492 


498 


504 


509 


515 


520 


526 


531 


537 




786 


542 


548 


553 


559 


564 


570 


575 


581 


586 


592 




787 


597 


603 


609 


614 


620 


625 


631 


636 


642 


647 




788 


653 


658 


664 


669 


675 


680 


686 


691 


697 


702 


r 


789 


708 


713 


719 


724 


730 


735 


741 


746 


752 


757 




790 


763 


768 


774 


779 


785 


790 


796 


801 


807 


812 




791 


818 


823 


829 


834 


840 


845 


851 


856 


862 


867 




792 


873 


878 


883 


889 


894 


900 


905 


911 


916 


922 




793 


927 


933 


938 


944 


949 


955 


960 


966 


971 


977 




794 


982 


988 


993 


998 *004 


*009 *015 *020 *026 *031 




795 


90 037 


042 


048 


053 


059 


064 


069 


075 


080 


086 




796 


091 


097 


102 


108 


113 


119 


124 


129 


135 


140 




797 


146 


151 


157 


162 


168 


173 


179 


184 


189 


195 


5 


798 


200 


206 


211 


217 


222 


227 


233 


238 


244 


249 


-1 A C 


799 


255 


260 


266 


271 


276 


282 


287 


293 


298 


304 


1 

2 


1.0 


800 


309 


314 


320 


325 


331 


336 


342 


347 


352 


358 


3 
4 


1.5 
2.0 


801 


363 


369 


374 


380 


385 


390 


396 


401 


407 


412 


5 


2.5 


802 


417 


423 


428 


434 


439 


445 


450 


455 


461 


466 


6 

7 
8 


3.0 
3.5 
4.0 


803 


472 


477 


482 


488 


493 


499 


504 


509 


515 


520 


804 


526 


531 


536 


542 


547 


553 


558 


563 


569 


574 


9 


4.5 


805 


580 


585 


590 


596 


601 


607 


612 


617 


623 


628 




806 


634 


639 


644 


650 


655 


660 


666 


671 


677 


682 




807 


687 


693 


698 


703 


709 


714 


720 


725 


730 


736 




808 


741 


747 


752 


757 


763 


768 


773 


779 


784 


789 




809 


795 


800 


806 


811 


816 


822 


827 


832 


838 


843 




810 


849 


854 


859 


865 


870 


875 


881 


886 


891 


897 




811 


902 


907 


913 


918 


924 


929 


934 


940 


945 


950 




812 


956 


961 


966 


972 


977 


982 


988 


993 


998 *004 




813 


91 009 


014 


020 


025 


030 


036 


041 


046 


052 


057 




814 


062 


068 


073 


078 


084 


089 


094 


100 


105 


110 




815 


116 


121 


126 


132 


137 


142 


148 


153 


158 


164 




816 


169 


174 


180 


185 


190 


196 


201 


206 


212 


217 




817 


222 


228 


233 


238 


243 


249 


254 


259 


265 


270 




818 


275 


281 


286 


291 


297 


302 


307 


312 


318 


323 




819 


328 


334 


339 


344 


350 


355 


360 


365 


371 


376 




820 


381 


387 


392 


397 


403 


408 


413 


418 


424 


429 




N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



100 



Logarithms of Numbers. 



Num. 820 to 859. Log. 913 to 934. 



N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


820 


91 381 


387 


392 


397 


403 


408 


413 


418 


424 


429 




821 


434 


440 


445 


450 


,455 


461 


466 


471 


477 


482 




822 


487 


492 


498 


503 


508 


514 


519 


524 


529 


535 




823 


540 


545 


551 


556 


561 


566 


572 


577 


582 


587 




824 


593 


598 


603 


609 


614 


619 


624 


630 


635 


640 




825 


645 


651 


656 


661 


666 


672 


677 


682 


687 


693 




826 


698 


703 


709 


714 


719 


724 


730 


735 


740 


745 




827 


751 


756 


761 


766 


772 


777 


782 


787 


793 


798 




828 


803 


808 


814 


819 


824 


829 


834 


840 


845 


850 




829 


855 


861 


866 


871 


876 


882 


887 


892 


897 


903 




830 


908 


913 


918 


924 


929 


934 


939 


944 


950 


955 




831 


960 


965 


971 


976 


981 


986 


991 


997 *002 *007 




832 


92 012 


018 


023 


028 


033 


038 


044 


049 


054 


059 




833 


065 


070 


075 


080 


085 


091 


096 


101 


106 


111 




834 


117 


122 


127 


132 


137 


143 


148 


153 


158 


163 




835 


169 


174 


179 


184 


189 


195 


200 


205 


210 


215 




836 


221 


226 


231 


236 


241 


247 


252 


257 


262 


267 




837 


273 


278 


283 


288 


293 


298 


304 


309 


314 


319 


5 


838 


324 


330 


335 


340 


345 


350 


355 


361 


366 


371 


1 
2 


0.5 
1.0 


839 


376 


381 


387 


392 


397 


402 


407 


412 


418 


423 


840 


428 


433 


438 


443 


449 


454 


459 


464 


469 


474 


3 
4 


1.5 

2.0 


841 


480 


485 


490 


495 


500 


505 


511 


516 


521 


526 


5 


2.5 


842 


531 


536 


542 


547 


552 


557 


562 


567 


572 


578 


6 
7 
8 


3.0 
3.5 
4.0 


843 


583 


588 


593 


598 


603 


609 


614 


619 


624 


629 


844 


634 


639 


645 


650 


655 


660 


665 


670 


675 


681 


9 


4.5 


845 


686 


691 


696 


701 


706 


711 


716 


722 


727 


732 




846 


737 


742 


747 


752 


758 


763 


768 


773 


778 


783 




847 


788 


793 


799 


804 


809 


814 


819 


824 


829 


834 




848 


840 


845 


850 


855 


860 


865 


870 


875 


881 


886 




849 


891 


896 


901 


906 


911 


916 


921 


927 


932 


937 




850 


942 


947 


952 


957 


962 


967 


973 


978 


983 


988 




851 


993 


998 *003 *008 *013 


*018 *024 *029 *034 *039 




852 


93 044 


049 


054 


059 


064 


069 


075 


080 


085 


090 




853 


095 


100 


105 


110 


115 


120 


125 


131 


136 


141 




854 


146 


151 


156 


161 


166 


171 


176 


181 


186 


192 




855 


197 


202 


207 


212 


217 


222 


227 


232 


237 


242 




856 


247 


252 


258 


263 


268 


273 


278 


283 


288 


293 




857 


298 


303 


308 


313 


318 


323 


328 


334 


339 


344 




858 


349 


354 


359 


364 


369 


374 


379 


384 


389 


394 




859 


399 


404 


409 


414 


420 


425 


430 


435 


440 


445 




860 


450 


455 


460 


465 


470 


475 


480 


485 


490 


495 




N 


L 


* 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



Logarithms of Numbers. 



xoi 



Num. 860 to 899. Log. 934 to 954. 



N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


860 


93 450 


455 


460 


465 


470 


475 


480 


485 


490 


495 




861 


500 


505 


510 


515 


520 


526 


531 


536 


541 


546 




862 


551 


556 


561 


566 


571 


576 


581 


586 


591 


596 




863 


601 


606 


611 


616 


621 


626 


631 


636 


641 


646 




864 


651 


656 


661 


666 


671 


676 


682 


687 


692 


697 




865 


702 


707 


712 


717 


722 


727 


732 


737 


742 


747 




866 


752 


757 


762 


767 


772 


777 


782 


787 


792 


797 




867 


802 


807 


812 


817 


822 


827 


832 


837 


842 


847 




868 


852 


857 


862 


867 


872 


877 


882 


887 


892 


897 




869 


902 


907 


912 


917 


922 


927 


932 


937 


942 


947 




870 


952 


957 


962 


967 


972 


977 


982 


987 


992 


997 




871 


94 002 


007 


012 


017 


022 


027 


032 


037 


042 


047 




872 


052 


057 


062 


067 


072 


077 


082 


086 


091 


096 




873 


101 


106 


111 


116 


121 


126 


131 


136 


141 


146 




874 


151 


156 


161 


166 


171 


176 


181 


186 


191 


196 




875 


201 


206 


211 


216 


221 


226 


231 


236 


240 


245 




876 


250 


255 


260 


265 


270 


275 


280 


285 


290 


295 




877 


300 


305 


310 


315 


320 


325 


330 


335 


340 


345 


5 


878 


349 


354 


359 


364 


369 


374 


379 


384 


389 


394 


1 

2 


0.5 
1.0 


879 


399 


404 


409 


414 


419 


424 


429 


433 


438 


443 


880 


448 


453 


458 


463 


468 


473 


478 


483 


488 


493 


3 
4 


1.5 

2.0 


881 


498 


503 


507 


512 


517 


522 


527 


532 


537 


542 


5 


2.5 


882 


547 


552 


557 


562 


567 


571 


576 


581 


586 


591 


6 

7 
8 


3.0 
3.5 
4.0 


883 


596 


601 


606 


611 


616 


621 


626 


630 


635 


640 


884 


645 


650 


655 


660 


665 


670 


675 


680 


685 


689 


9 


4.5 


885 


694 


699 


704 


709 


714 


719 


724 


729 


734 


738 




886 


743 


748 


753 


758 


763 


768 


773 


778 


783 


787 




887 


792 


797 


802 


807 


812 


817 


822 


827 


832 


836 




888 


841 


846 


851 


856 


861 


866 


871 


876 


880 


885 




889 


890 


895 


900 


905 


910 


915 


919 


924 


929 


934 




890 


939 


944 


949 


954 


959 


963 


968 


973 


978 


983 




891 


988 


993 


998 *002 *007 


*012 *017 *022 *027 *032 




892 


95 036 


041 


046 


051 


056 


061 


066 


071 


075 


080 




893 


085 


090 


095 


100 


105 


109 


114 


119 


124 


129 




894 


134 


139 


143 


148 


153 


158 


163 


168 


173 


177 




895 


182 


187 


192 


197 


202 


207 


211 


216 


221 


226 




896 


231 


236 


240 


245 


250 


255 


260 


265 


270 


274 




897 


279 


284 


289 


294 


299 


303 


308 


313 


318 


323 




898 


328 


332 


337 


342 


347 


352 


357 


361 


366 


371 




899 


376 


381 


386 


390 


395 


400 


405 


410 


415 


419 




900 


424 


429 


434 


439 


444 


448 


453 


458 


463 


468 




N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



102 




Logarithms of Numbers. 






Num. 900 to 939. Log. 954 to 973. 


N 


L 


01234 5678 


9 


P. P. 



900 


96 424 


429 


434 


439 


444 


901 


472 


477 


482 


asi 


4; J 2 


902 


521 


525 


530 


535 


540 


903 


569 


574 


578 


583 


588 


904 


617 


622 


626 


631 


r, . 


905 


665 


670 


674 


679 


6S4 


906 


713 


71S 


722 


-_" 


732 


907 


761 




770 


775 


7S0 


908 


809 


S13 


818 


S23 


S2S 


909 


856 


861 


866 


.-71 


875 


910 


904 


909 


914 


91S 


923 


911 


952 


9-^7 


961 


966 


971 


912 


999 


•004 *009 *014 *019 


913 


96 047 


052 


057 


061 


066 


91-4 


095 


099 


104 


109 


114 


915 


142 


147 


152 


156 


161 


916 


190 


194 


199 


204 


209 


917 


237 


242 


246 


251 


256 


91S 


2&4 


28 


2-4 


298 


303 


919 


332 


336 


341 


346 


350 


920 


379 


&S4 


3SS 


393 


39S 


921 


426 


431 


435 


440 


445 


922 


473 


47S 


4 S3 


183 


492 


923 


520 


525 


530 


534 


539 


924 


567 


572 


577 


581 


5 So 


925 


614 


619 


624 


628 


633 


926 


661 


666 


670 




680 


927 




713 


717 


722 


727 


928 


755 


759 


764 


769 


774 


929 


802 


806 


Sll 


S16 


S20 


930 




853 


858 


8 _ 


S67 


931 


895 


900 


904 


909 


914 


932 


942 


946 


951 


966 


960 


933 


988 


993 


997 


*002 


*007 


934 


97 035 


039 


044 


049 


053 


938 


081 


086 


090 


095 


100 


936 


128 


132 


137 


142 


146 


937 


174 


179 


183 


188 


192 


938 


220 


225 


230 


234 


239 


939 




271 


276 


280 


2S5 


940 


313 


317 


322 


327 


331 



44S 453 45S 463 468 

497 501 506 511 516 

545 550 564 559 564 

593 59S 602 607 612 

641 646 6-50 6-55 660 

6S9 694 69S 703 70S 

7:^7 742 746 751 756 

786 7S9 794 799 804 

332 B37 842 847 B5S 

880 886 890 S95 S99 

928 933 93S 942 £47 

976 9S0 9S5 990 995 

•023 *028 *033 *088 *042 

071 076 080 0S5 090 

US 123 12S 133 137 

166 171 175 ISO 185 

213 21S 223 227 232 

261 265 270 275 280 

30S 313 317 322 327 

355 360 365 369 374 

402 407 412 417 421 

450 4.54 459 464 468 

497 501 506 511 515 

544 54S 553 568 562 

591 595 600 605 609 

638 642 647 652 656 

685 6S9 694 699 703 

731 736 741 745 7-50 

7"- 783 788 792 797 

S25 S30 ,-34 839 844 

872 .-- £ 

91S 923 928 932 937 

965 970 974 99 

*011 *016 *021 *025 *O90 

058 063 067 072 077 

104 109 114 US 123 

151 155 160 165 169 

197 202 206 211 216 

243 24- 263 257 262 

290 294 299 304 308 

336 340 345 350 354 



8 



P. P. 



Logarithms of Numbers. 



103 





Num 


940 to 979. 


Log. 


973 to 


£>91. 








N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


940 


97 313 


317 


322 


327 


331 


336 


340 


345 


350 


354 




941 


359 


364 


368 


373 


377 


382 


387 


391 


396 


400 




942 


405 


410 


414 


419 


424 


428 


433 


437 


442 


447 




943 


451 


456 


460 


465 


470 


474 


479 


483 


488 


493 




944 


497 


502 


506 


511 


516 


520 


525 


529 


534 


539 




945 


543 


548 


552 


557 


562 


566 


571 


575 


580 


585 




946 


589 


594 


598 


603 


607 


612 


617 


621 


626 


630 




947 


635 


640 


644 


649 


653 


658 


663 


667 


672 


676 




948 


681 


685 


690 


695 


699 


704 


708 


713 


717 


722 




949 


727 


731 


736 


740 


745 


749 


754 


759 


763 


768 


5 


950 


772 


777 


782 


786 


791 


795 


800 


804 


809 


813 


1 


0.5 


951 


818 


823 


827 


832 


836 


841 


845 


850 


855 


859 


2 
3 
4 


1.0 
1.5 

2.0 


952 


864 


868 


873 


877 


882 


886 


891 


896 


900 


905 


953 


909 


914 


918 


923 


928 


932 


937 


941 


946 


950 


5 


2.5 


954 


955 


959 


964 


968 


973 


978 


982 


987 


991 


996 


6 

7 


3.0 
3.5 


955 


98 000 


005 


009 


014 


019 


023 


028 


032 


037 


041 


8 
9 


4.0 
4.5 


956 


046 


050 


055 


059 


064 


068 


073 


078 


082 


087 


957 


091 


096 


100 


105 


109 


114 


118 


123 


127 


132 




958 


137 


141 


146 


150 


155 


159 


164 


168 


173 


177 




959 


182 


186 


191 


195 


200 


204 


209 


214 


218 


223 




960 


227 


232 


236 


241 


245 


250 


254 


259 


263 


268 




961 


272 


277 


281 


286 


290 


295 


299 


304 


308 


313 




962 


318 


322 


327 


331 


336 


340 


345 


349 


354 


358 




963 


363 


367 


372 


376 


381 


385 


390 


394 


399 


403 




964 


408 


412 


417 


421 


426 


430 


435 


439 


444 


448 




965 


453 


457 


462 


466 


471 


475 


480 


484 


489 


493 


4 


966 


498 


502 


507 


511 


516 


520 


525 


529 


534 


538 


1 


0.4 


967 


543 


547 


552 


556 


561, 


565 


570 


574 


579 


583 


2 


0.8 


968 


588 


592 


597 


601 


605 


610 


614 


619 


623 


628 


3 

4 


1.2 
1.6 


969 


632 


637 


641 


646 


650 


655 


659 


664 


668 


673 


5 


io 


970 


677 


682 


686 


691 


695 


700 


704 


709 


713 


717 


6 

7 


2.4 

2.8 


971 


722 


726 


731 


735 


740 


744 


749 


753 


758 


762 


8 


3.2 


972 


767 


771 


776 


780 


784 


789 


793 


798 


802 


807 


9 


3.6 


973 


811 


816 


820 


825 


829 


834 


838 


843 


847 


851 




974 


856 


860 


865 


869 


874 


878 


883 


887 


892 


896 




975 


900 


905 


909 


914 


918 


923 


927 


932 


936 


941 




976 


945 


949 


954 


958 


963 


967 


972 


976 


981 


985 




977 


989 


994 


998 *003 *007 


*012 *016 *021 *025 *029 




978 


99 034 


038 


043 


047 


052 


056 


061 


065 


069 


074 




979 


078 


083 


087 


092 


096 


100 


105 


109 


114 


118 




980 


123 


127 


131 


136 


140 


145 


149 


154 


158 


162 




N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



104 



Logarithms of Numbers. 



Num. 980 to 1000. Log. 991 to 999. 



N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


980 


99 123 


127 


131 


136 


140 


145 


149 


154 


158 


162 




981 


167 


171 


176 


180 


185 


189 


193 


198 


202 


207 




982 


211 


216 


220 


224 


229 


233 


238 


242 


247 


251 




983 


255 


260 


264 


269 


273 


277 


282 


286 


291 


295 




984 


300 


304 


308 


313 


317 


322 


326 


330 


335 


339 




985 


344 


348 


352 


357 


361 


366 


370 


374 


379 


383 




986 


388 


392 


396 


401 


405 


410 


414 


419 


423 


427 




987 


432 


436 


441 


445 


449 


454 


458 


463 


467 


471 




988 


476 


480 


484 


489 


493 


498 


502 


506 


511 


515 




989 


520 


524 


528 


533 


537 


542 


546 


550 


555 


559 


4 


990 


564 


568 


572 


577 


581 


585 


590 


594 


599 


603 


1 


0.4 


991 


607 


612 


616 


621 


625 


629 


634 


638 


642 


647 


2 
3 


0.8 
12 


992 


651 


656 


660 


664 


669 


673 


677 


682 


686 


691 


4 


1.6 


993 


695 


699 


704 


708 


712 


717 


721 


726 


730 


734 


5 


2.0 


994 


739 


743 


747 


752 


756 


760 


765 


769 


774 


778 


6 
7 


2.4 

2.8 


995 


782 


787 


791 


795 


800 


804 


808 


813 


817 


822 


8 
9 


3.2 
3.6 


996 


826 


830 


835 


839 


843 


848 


852 


856 


861 


865 






997 


870 


874 


878 


883 


887 


891 


896 


900 


904 


909 




998 


913 


917 


922 


926 


930 


935 


939 


944 


948 


952 




999 


957 


961 


965 


970 


974 


978 


983 


987 


991 


996 




1000 


000 000 


043 


087 


130 


174 


217 


260 


304 


347 


391 




N 


L 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. 


P. 



Logarithms of Important Numbers. 



Number. 


Logarithm. 


7T 


= 


3.141 593 


0.497 150 


I* 


= 


4.188 790 


0.622 089 


i* 


= 


0.523 599 


1.718 999 


l 


= 


0.318 310 


1.502 850 


Tr2 


= 


9.869 604 


0.994 300 


1 


= 


0.101 321 


1.005 700 


vz 


= 


1.772 454 


0.248 575 


1 


= 


0,564 190 


1.751 425 


fn 


= 


1.464 592 


0.165 717 


1 


— 


0.682 784 


1.834 283 


Vs 


= 


1.240 701 


0.093 667 



Geometry. 



105 



GEOMETRY. 

No attempt will be made to give the successive propositions of geometry, 
as these can be found in the standard text-books. Instead will be given 
such constructions as will be found useful to the engineer, followed by the 
mensuration of bodies in one, two, and three dimensions. 

A straight line is usually best obtained by the use of a straight edge, 
such as a T square, or one of the sides of a draughtsman's triangle. In 
some cases, however, when very long centre lines are required, it is not ad- 
visable to place too much reliance upon any long straight edge. A fine 
thread tightly stretched between points may be used to advantage, a com- 
paratively short straight edge being used to connect points marked off 
upon the line of the thread. 

Right angles are best obtained by use of a draughtsman's triangle or set 
square, but where this is not available, or where the angle is to be laid out 
upon a large scale, as upon the ground or on a floor, the following con- 
structions may be found useful : 

To divide a given line, AB, into two equal 
parts, and to erect a perpendicular through the 
middle : 

With the ends, A and B, as centres, draw the 
dotted circle arcs with a radius greater than half A— 
the line. Through the crossings of the arcs draw 
the perpendicular, CD, which divides the line 
into two equal parts. 



)C 



From a given point, C, on the line, AB, to erect 
a perpendicular, CD : 

With C as a centre, find two points, A and B, 
on the line at equal distances from C. With A 
and B as centres, draw the dotted circle arcs at 
D. From the crossing, D, draw the required per- 
pendicular, DC. 



From a given point, C, at a distance from the 
line, AB, to draw a perpendicular to the line : 

With C as a centre, draw the dotted circle arc 
so that it cuts the line at A and B. With 'A and 
B as centres, draw the dotted cross arcs at D with 
equal radii. Draw the required perpendicular 
through C and D. 



C 

:'d 



\ 



X D 



\ 



At the end, A, of a given line, AB, to erect a 
perpendicular, AC: 

With any point, D, as a centre at a distance 
from the line, and with AD as radius, draw the 
dotted circle arc so that it cuts the line at E; 
through E and D draw the diameter, EC; then 
join C and A, which will give the required per- 
pendicular. 



The division of a line into any required 
number of parts may best be done by con- 
tinual bisection as far as possible. When 
the limit has been reached in this manner 
the portions thus obtained may be divided 
by trial and error, using fine dividers with 
screw adjustment, or the following con- 
struction may be used : 

Let it be required to divide A C into three 
equal parts. From A draw any convenient 
hue, AB, and on it step off with the dividers any equal spaces, 1, 2, 3. 
Then join 3 with C, and draw from 1 and 2 lines parallel to 3C; these will 
divide AC into three equal parts at I, II, and III. 




106 



Geometry. 




Constructions with Angles. 



On a given line, AB, and at the point, B, to 
construct an angle equal to the angle, CDE : 

With D as a centre, draw the dotted arc, 
CE; and with the same radius and B as a 
centre, draw the arc, GF; then make OF equal 
to CE; then join BF, which will form the 
required angle, FBG = CDE. 



To divide the angle, ACB y into two equal 
parts: 

With C as a centre, draw the dotted arc, DE ; 
with D and E as centres, draw the cross arcs at 
2^ with equal radii. Join CF, which divides the 
angle into the required parts. 

Angles: ACF = FCB - %{ACB). 





To divide an angle into two equal parts 
when the lines do not extend to a meeting 
point: 

Draw the lines CD and CE parallel and at 
equal distances from the lines AB and FG. 
With C as a centre, draw the dotted arc, BG; 
and with H and G as centres, draw the cross 
arcs, H. Join CH, which divides the angle 
into the required equal parts. 




To trisect a right angle, ACB : 

With any convenient radius, CB, strike a 
circular arc with C as a centre. With the 
dividers open to the same radius sweep short 
arcs from A and B as centres. These will 
intersect the first arc at D and E. Through E 
and D draw lines passing through C. These 
lines will trisect the right angle, A CB. 



Angles may he laid off by means of a protractor graduated in degrees 
and subdivisions, but unless the instrument is accurately made and care- 
fully used the results are not very reliable. A more accurate method is to 
use a table of chords. With the dividers open to any convenient distance 
sweep an arc. Multiply the radius used by the tabular value in the table 
on page 107 for the required angle, and sweep the distance as a chord upon 
the arc. The remaining side of the angle may then be drawn. 

A convenient radius is 10 inches. Con- 
sidering this as units, one inch will be 0.1, 
one-tenth of an inch will be 0.01, and one- 
hundredth of an inch will be 0.001, and the 
chord may be taken directly from the table : 
Thus, for an angle of 17° we have from the 
table a value for the chord of 0.2956, or for a 
radius of 10 the chord is 2.956, and the angle, 
AOB = 17°, laid out far more accurately than 
would be possible with any but the most 
elaborate protractor of large size. When the chord can be taken on a 
vernier scale the length can readily be taken to as high a degree of pre- 
cision as can be laid out on a drawing-board. Intermediate values may 
be taken by direct proportion. 

The chord for any angle is equal to twice the sine of half of the angle. 




Geometry. 



107 



Table of Chords. 



Deg. 


Chord. 


Deg. 


Chord. 


Deg. 


Chord. 


Deg. 


Chord. 


Deg. 


Chord. 


1 


0.0175 


19 


0.3301 


37 


0.6346 


55 


0.9235 


73 


1.1896 


2 


0.0349 


20 


0.3473 


38 


0.6511 


56 


0.9389 


74 


1.2036 


3 


0.0524 


21 


0.3645 


39 


0.6676 


57 


0.9543 


75 


1.2175 


4 


0.0698 


22 


0.3816 


40 


0.6840 


58 


0.9696 


76 


1.2313 


5 


0.0872 


23 


0.3987 


41 


0.7004 


59 


0.9848 


77 


1.2450 


6 


0.1047 


24 


0.4158 


42 


0.7167 


60 


1.0000 


78 


1.2586 


7 


0.1221 


25 


0.4329 


43 


0.7330 


61 


1.0151 


79 


1.2722 


8 


0.1395 


26 


0.4499 


44 


0.7492 


62 


1.0301 


80 


1.2856 


9 


0.1569 


27 


0.4669 


45 


0.7654 


63 


1.0450 


81 


1.2989 


10 


0.1743 


28 


0.4838 


46 


0.7815 


64 


1.0598 


82 


1.3121 


11 


0.1917 


29 


0.5008 


47 


0.7975 


65 


1.0746 


83 


1.3252 


12 


0.2091 


30 


0.5176 


48 


0.8135 


66 


1.0893 


84 


1.3383 


13 


0.2264 


31 


0.5345 


49 


0.8294 


67 


1.1039 


85 


1.3512 


14 


0.2437 


32 


0.5513 


50 


0.8452 


68 


1.1184 


86 


1.3640 


15 


0.2611 


33 


0.5680 


51 


0.8610 


69 


1.1328 


87 


1.3767 


16 


0.2783 


34 


0.5847 


52 


0.8767 


70 


1.1472 


88 


1.3893 


17 


0.2956 


35 


0.6014 


53 


0.8924 


71 


1.1614 


89 


1.4018 


18 


0.3129 


36 


0.6180 


54 


0.9080 


72 


1.1756 


90 


1.4142 



Construction of Polygons. 



To inscribe a square in a given circle : 

Draw the diameter, AB, and through the centre 

erect the perpendicular, CD, _ and complete the 

square as shown in the illustration. 





To describe a square about a given circle : 
Draw the diameters, AB and CD, at right angles to 
one another ; with the radius of the circle, and A, B, 
C, and D as centres, draw the four dotted half circles 
which cross one another in the corners of the square, 
and thus solve the problem. 



To inscribe a pentagon in a given circle : 
Draw the diameter, AB, and from the centre, C, 
erect the perpendicular, CD. Bisect the radius, A C, 
at E; with E as centre, and DE as radius, draw the 
arc, DF, and the straight line, DF, is the length of 
the side of the pentagon. 





To construct a pentagon on a given side, AB : 
From B erect BC perpendicular to and half the 
length of AB; draw a line from A through C and 
beyond ; with C as a centre and CB as radius, draw 
the arc, BD, cutting this last line at D; then the 
chord, BD, is the radius of the circle circumscribing 
the pentagon. With A and B as centres, and BD as 
radius, draw the cross in the centre. 



108 



Geometry. 




To construct a pentagon on a given side, AB, 
without resort to its centre : 

From B erect Bo perpendicular and equal to AB; 
with C as centre and Co as radius, draw the arc, Do; 
then AD is the diagonal of the pentagon. With AD 
as radius and A as centre, draw the arc, DE ; and 
with B as centre and AB as radius, finish the cross, E, 
and thus complete the pentagon. 



To construct a hexagon in a given circle : 
The radius of the circle is equal to the side of the f 
hexagon. \ 



To construct an octagon on 
the given line, AB : 

Prolong AB through B. 
With B as centre and AB as radius, draw semi- 
circle, AFDEC; from B, draw BI at right angles 
to AB; divide the angles, ABD and DBC, each into 
two equal parts ; then BE is one side of the octa- 
gon. With A and E as centres, and radius AE, 
draw the arcs, HKE and AKI, which determine 
the points H and J, and thus complete the octagon 
as shown in the illustration. 





To cut off the corners of a square so as to make a 
regular octagon : 

With the corners as centres, draw circle arcs through 
the centre of the square to the sides, which determines 
the sides of the octagon. 




K J, 




To construct any regular polygon on a given 
line, AB, without resort to its centre : 

Extend AB through B, and, with B as a 
centre, draw the half circle, ADC. Divide the 
half circle into as many parts as the number of 
sides in the polygon, and complete the construc- 
tion as shown in the illustration. 



Table of Polygons. 





Side. 




Area. 




Apothem. 




120° 

90° 

72° 

60° 
51° 2V 42" 

45° 

40° 

36° 
32° 43' 38" 

30° 



60° 
90° 
108° 
120° 
128° 
135° 
140° 
144° 
147° 
150° 



17' 



47' 



1.73205 


0.4330 


1.41421 


1.0000 


1.17557 


1.7205 


1.00000 


2.5980 


0.86776 • 


3.6339 


0.76536 


4.8284 


0.68404 


6.1820 


0.61801 


7.6942 


0.56346 


9.3656 


0.51763 


11.1961 



0.5000 
0.7071 
0.8090 
0.8660 
0.9009 
0.9238 
0.9396 
0.9510 
0.9595 
0.9659 



The Circle. 



109 



To find the length of side of any polygon, multiply the radius of the 
circumscribing circle by the tabular number. To find the arc. multiply 
the square of the side'by the tabular number. To find the apothem. 
. multiply radius of circumscribing circle by tabular number. 



The Circle. 

yotation. 



d = diameter of the circle. 

r = radius of the circle. 

p = periphery or circumference. 



c = chord of a segment, length of. 
h = height of a segment. 

side of a regular polygon. 



a = area of a circle or part thereof. v — centre angle. 



b = length of a circle-arc. 



if = polygon angle. 



All measures must be expressed in terms of the same unit. 



Formulas for the Circle. 



Periphery jpr Circumfir- 


Diameter and Radius. 


Area of the Circle. 


Vncf. 

P = Trd = 3.14o\ 


- 3.14 


a = —— = 0.7854^. 


p = 2rrr = (L28 


2s S.28 


a = wr* — 3.14r-. 


p = 2 J -a = 3.54^ a. 


d = Z% - = L1281 a. 


4- 12.oo 


2 1 4 a 


r = -W — = 0.cv64* a. 


m _ j£_ _ j± 



n = 3.141 59-2 653 589 793 238 462 643 353 279 502 N>4 197 169 399 I 



2-- 5.283 18E 


T 4- 


= 0.785 386 


1 


= 0.315 310 


360 

- 


114.5915 


3rr = 9.424 77<- 


K» 


= 1.047 197 


- 


= 0.636 619 


IT- = 


9.569 650 


i- = 12L566 ■" 


-,t 


= 1.570 796 


3 

TT 








5?r = 15.707 963 


K* 


= 0.392 699 


= 0.954 929 


) TT = 


1.772 453 


6rr = 15.549 5-56 


1 «?" 


= 0.523 599 


4 

77 


= 1.273-239 


7 


0.564 189 


7- = 21.991 14S 
Stt = 25.132 741 




= 0.261 799 
= 2.094 394 


6 

TT 

B 

- 


= 1.909 859 
= 2.546 475 


V*- 


1.253 314 


9ir = 2-S.274 391 


*h* 


= 0.006 726 


12 

TT 


= 3.819 716 


4- 


0.797 SS4 






Log. 7T = 0.4 ( 


>7 149 872 69413 







110 


Circumference and Area of Circles. 






Circum. 


Area. 




Circum. 


Area. 




Circum. 


Area. 


Diam- 


S~\ 


/^H^v 


Diam- 


x^\ 


/^^v 


Diam- 


^—\ 


^^. 


eter. 


O 


IP 


eter. 


O 


IP 


eter. 


O 


© 


1 


3.1416 


0.7854 


51 


160.22 


2042.8 


101 


317.30 


8011.9 


2 


6.2832 


3.1416 


52 


163.36 


2123.7 


102 


320.44 


8171.3 


3 


9.4248 


7.0686 


53 


166.50 


2206.2 


103 


323.58 


8332.3 


4 


12.566 


12.5664 


54 


169.65 


2290.2 


104 


326.73 


8494.9 


5 


15.708 


19.6350 


55 


172.79 


2375.8 


105 


329.87 


8659.0 


6 


18.850 


28.2743 


56 


175.93 


2463.0 


106 


333.01 


8824.7 


7 


21.991 


38.4845 


57 


179.07 


2551.8 


107 


336.15 


8992.0 


8 


25.133 


50.2655 


58 


182.21 


2642.1 


108 


339.29 


9160.9 


9 


28.274 


63.6173 


59 


185.35 


2734.0 


109 


342.43 


9331.3 


10 


31.416 


78.54 


60 


188.50 


2827.4 


110 


345.58 


9503.3 


11 


34.558 


95.03 


61 


191.64 


2922.5 


111 


348.72 


9676.9 


12 


37.699 


113.10 


62 


194.78 


3019.1 


112 


351.86 


9852.0 


13 


40.841 


132.73 


63 


197.92 


3117.2 


113 


355.00 


10028.8 


14 


43.982 


153.94 


64 


201.06 


3217.0 


114 


358.14 


10207.0 


15 


47.124 


176.71 


65 


204.20 


3318.3 


115 


361.28 


10386.9 


16 


50.265 


201.06 


66 


207.35 


3421.2 


116 


364.42 


10568.3 


17 


53.407 


226.98 


67 


210.49 


3525.7 


117 


367.57 


10751.3 


18 


56.549 


254.47 


68 


213.63 


3631.7 


118 


370.71 


10935.9 


19 


59.690 


283.53 


69 


216.77 


3739.3 


119 


373.85 


11122.0 


20 


62.832 


314.16 


70 


219.91 


3818.5 


120 


376.99 


11310 


21 


65.973 


346.36 


71 


223.05 


3959.2 


121 


380.13 


11499 


22 


69.115 


380.13 


72 


226.19 


4071.5 


122 


383.27 


11690 


23 


72.257 


415.48 


73 


229.34 


4185.4 


123 


386.42 


11882 


24 


75.398 


452.39 


74 


232.48 


4300.8 


124 


389.56 


12076 


25 


78.540 


490.87 


75 


235.62 


4417.9 


125 


392.70 


12272 


26 


81.681 


530.93 


76 


238.76 


4536.5 


126 


395.84 


12469 


27 


84.823 


572.56 


77 


241.90 


4656.6 


127 


398.98 


12668 


28 


87.965 


615.75 


78 


245.04 


4778.4 


128 


402.12 


12868 


29 


91.106 


660.52 


79 


248.19 


4901.7 


129 


405.27 


13070 


30 


94.248 


706.86 


80 


251.33 


5026.6 


130 


408.41 


13273 


31 


97.389 


754.77 


81 


254.47 


5153.0 


131 


411.55 


13478 


32 


100.53 


804.25 


82 


257.61 


5281.0 


132 


414.69 


13685 


33 


103.67 


855.30 


83 


260.75 


5410.6 


133 


417.83 


13893 


34 


106.81 


907.92 


84 


263.89 


5541.8 


134 


420.97 


14103 


35 


109.96 


962.11 


85 


267.04 


5674.5 


135 


424.12 


14314 


36 


113.10 


1017.88 


86 


270.18 


5808.8 


136 


427.26 


14527 


37 


116.24 


1075.21 


87 


273.32 


5944.7 


137 


430.40 


14741 


38 


119.38 


1134.11 


88 


276.46 


6082.1 


138 


433.54 


14957 


39 


122.52 


1194.59 


89 


279.60 


6221.1 


139 


436.68 


15175 


40 


125.66 


1256.63 


90 


282.74 


6361.7 


140 


439.82 


15394 


41 


128.81 


1320.25 


91 


285.88 


6503.9 


141 


442.96 


15615 


42 


131.95 


1385.44 


92 


289.03 


6647.6 


142 


446.11 


15837 


43 


135.09 


1452.20 


93 


292.17 


6792.9 


143 


449.25 


16061 


44 


138.23 


1520.52 


94 


295.31 


6939.8 


144 


452.39 


16286 


45 


141.37 


1590.43 


95 


298.45 


7088.2 


145 


455.53 


16513 


46 


144.51 


1661.90 


96 


301.59 


7238.2 


146 


458.67 


16742 


47 


147.65 


1734.94 


97 


304.73 


7389.8 


147 


461.81 


16972 


48 


150.80 


1809.55 


98 


307.88 


7543.0 


148 


464.96 


17203 


49 


153.94 


1885.74 


99 


311.02 


7697.7 


149 


468.10 


17437 


50 


157.08 


1963.50 


100 


314.16 


7854.0 


150 


471.24 


17671 







CIRCUMFERENCE 


and Area of Circles. 


111 




Circum. 


Area. 




Circum. 


Area. 


Diam- 


Circum. 


Area. 


Diam- 


/->, 


/^m^ 


Diam- 


X^N 


i^H^ 


S~\ 


itf^t 


eter. 


O 


IP 


eter. 


O 


IP 


1 eter. 


o 


fP 


151 


474.38 


17908 


201 


631.46 


31731 


251 


788.54 


49481 


152 


477.52 


18146 


202 


634.60 


32047 


252 


791.68 


49876 


153 


480.66 


18385 


203 


637.74 


32365 


253 


794.82 


50273 


154 


483.81 


18627 


204 


640.89 


32685 


254 


797.96 


50671 


155 


486.95 


18869 


205 


644.03 


33006 


255 


801.11 


51071 


156 


490.09 


19113 


206 


647.17 


33329 


256 


804.25 


51472 


157 


493.23 


19359 


207 


650.31 


33654 


257 


807.39 


51875 


158 


496.37 


19607 


208 


653.45 


33979 


258 


810.53 


52279 


159 


499.51 


19856 


209 


656.59 


34307 


259 


813.67 


52685 


160 


502.65 


20106 


210 


659.73 


34636 


260 


816.81 


53093 


161 


505.80 


20358 


211 


662.88 


34967 


261 


819.96 


53502 


162 


508.94 


20612 


212 


666.02 


35299 


262 


823.10 


53913 


163 


512.08 


20867 


213 


669.16 


35633 


263 


826.24 


54325 


164 


515.22 


21124 


214 


672.30 


35968 


264 


829.38 


54739 


165 


518.36 


21382 


215 


675.44 


36305 


265 


832.52 


55155 


166 


521.50 


21642 


216 


678.58 


36644 


266 


835.66 


55572 


167 


521.65 


21904 


217 


681.73 


36984 


267 


838.81 


55990 


168 


527.79 


22167 


218 


684.87 


37325 


268 


841.95 


56410 


169 


530.93 


22432 


219 


688.01 


37668 


269 


845.09 


56832 


170 


534.07 


22698 


220 


691.15 


38013 


270 


848.23 


57256 


171 


537.21 


22966 


221 


694.29 


38360 


271 


851.37 


57680 


172 


540.35 


23235 


222 


697.43 


38708 


272 


854.51 


58107 


173 


543.50 


23506 


: 223 


700.58 


39057 


273 


857.66 


58535 


174 


546.64 


23779 


224 


703.72 


39408 


274 


860.80 


58965 


175 


549.78 


24053 


225 


706.86 


39761 


275 


863.94 


59396 


176 , 


552.92 


24328 


226 


710.00 


40115 


276 


867.08 


59828 


177 ' 


556.06 


24606 


227 


713.14 


40471 


277 


870.22 


60263 


178 


559.20 


24885 


228 


716.28 


40828 


278 


873.36 


60699 


179 


562.35 


25165 


229 


719.42 


41187 


279 


876.50 


61136 


180 


565.49 


25447 


230 


722.57 


41548 


280 


879.65 


61575 


181 


568.63 


25730 


231 


725.71 


41910 


281 


882.79 


62016 


182 


571.77 


26016 


i 232 


728.85 


42273 


282 


885.93 


62458 


183 


574.91 


26302 


233 


731.99 


42638 


283 


889.07 


62902 


184 


578.05 


26590 


234 


735.13 


43005 


284 


892.21 


63347 


185 


581.19 


26880 


235 


738.27 


43374 


285 


895.35 


63794 


186 


584.34 


27172 


236 


741.42 


43744 


286 


898.50 


64242 


187 


587.48 


27465 


237 


744.56 


44115 


287 


901.64 


64692 


188 


590.62 


27759 


238 


747.70 


44488 


288 


904.78 


65144 


189 


593.76 


28055 


239 


750.84 


44863 


289 


907.92 


65597 


190 


596.90 


28353 


240 


753.98 


45239 


290 


911.06 


66052 


191 


600.04 


28652 


241 


757.12 


45617 


291 


914.20 


66508 


192 


603.19 


28953 


242 


760.27 


45996 


292 


917.35 


66966 


193 


606.33 


29255 


243 


763.41 


46377 


293 


920.49 


67426 


194 


609.47 


29559 


244 


766.55 


46759 


294 


923.63 


67887 


195 


612.61 


29865 


245 


769.69 


47144 


295 


926.77 


68349 


196 


615.75 


30172 


246 


772.83 


47529 


296 


929.91 


68813 


197 


618.89 


30481 


247 


775.97 


47916 


297 


933.05 


69279 


198 


622.04 


30791 


248 


779.12 


48305 


298 


936.19 


69747 


199 


625.18 


31103 


249 


782.26 


48695 


299 


939.34 


70215 


200 


628.32 


31416 


250 


785.40 


49087 


300 


942.48 


70686 



112 



Circumference and Area of Circles. 





Circum. 


Area. 




Circum. 


Area. 




Circum 


Area. 


Diam- 


S~\ 


/^H^v 


Diam- 


/^\ 


^^. 


Diam- 


/*~V 


^||5w 


eter. 


o 


IP 


eter. 


o 


© 


eter. 


O 


IP 


301 


945.62 


71158 


351 


1102.70 


96 762 


401 


1259.78 


126 293 


302 


948.76 


71631 


352 


1105.84 


97 314 


402 


1262.92 


126 923 


303 


951.90 


72107 


353 


1108.98 


97 868 


403 


1266.06 


127 556 


304 


955.04 


72583 


354 


1112.12 


98 423 


404 


1269.20 


128190 


305 


958.19 


73062 


355 


1115.27 


98 980 


405 


1272.35 


128 825 


306 


961.33 


73542 


356 


1118.41 


99 538 


406 


1275.49 


129 462 


307 


964.47 


74023 


357 


1121.55 


100 098 


407 


1278.63 


130100 


308 


967.61 


74506 


358 


1124.69 


100 660 


408 


1281.77 


130 741 


309 


970.75 


74991 


359 


1127.83 


101 223 


409 


1284.91 


131382 


310 


973.89 


75477 


360 


1130.97 


101 788 


410 


1288.05 


132 025 


311 


977.04 


75964 


361 


1134.11 


102 354 


411 


1291.19 


132 670 


312 


980.18 


76454 


362 


1137.26 


102 922 


412 


1294.34 


133 317 


313 


983.32 


76945 


363 


1140.40 


103 491 


413 


1297.48 


133 965 


314 


986.46 


77437 


364 


1143.54 


104 062 


414 


1300.62 


134 614 


315 


989.60 


77931 


365 


1146.68 


104 635 


415 


1303.76 


135 265 


316 


992.74 


78427 


366 


1149.82 


105 209 


416 


1306.90 


135 918 


317 


995.88 


78924 


367 


1152.96 


105 785 


417 


1310.04 


136 572 


318 


999.03 


79423 


368 


1156.11 


106 362 


418 


1313.19 


137 228 


319 


1002.17 


79923 


369 


1159.25 


106 941 


419 


1316.33 


137 885 


320 


1005.31 


80425 


370 


1162.39 


107 521 


420 


1319.47 


138 544 


321 


1008.45 


80928 


371 


1165.53 


108 103 


421 


1322.61 


139 205 


322 


1011.59 


81433 


372 


1168.67 


108 687 


422 


1325.75 


139 867 


323 


1014.73 


81940 


373 


1171.81 


109 272 


423 


1328.89 


140 531 


324 


1017.88 


82448 


374 


1174.96 


109 858 


424 


1332.04 


141 196 


325 


1021.02 


82958 


375 


1178.10 


110 447 


425 


1335.18 


141 863 


326 


1024.16 


83469 


376 


1181.24 


111 036 


426 


1338.32 


142 531 


327 


1027.30 


83982 


377 


1184.38 


111628 


427 


1341.46 


143 201 


328 


1030.44 


84496 


378 


1187.52 


112 221 


428 


1344.60 


143 872 


329 


1033.58 


85012 


379 


1190.66 


112 815 


429 


1347.74 


144 545 


330 


1036.73 


85530 


380 


1193.81 


113 411 


430 


1350.88 


145 220 


331 


1039.87 


86049 


381 


1196.95 


114 009 


431 


1354.03 


145 896 


332 


1043.01 


86570 


382 


1200.09 


114 608 


432 


1357.17 


146 574 


333 


1046.15 


87092 


383 


1203.23 


115 209 


433 


1360.31 


147 254 


334 


1049.29 


87616 


384 


1206.37 


115 812 


434 


1363.45 


147 934 


335 


1052.43 


88141 


385 


1209.51 


116 416 


435 


1366.59 


148 617 


336 


1055.58 


88668 


386 


1212.65 


117 021 


436 


1369.73 


149 301 


337 


1058.72 


89197 


387 


1215.80 


117 628 


437 


1372.88 


149 987 


338 


1061.86 


89727 


388 


1218.94 


118 237 


438 


1376.02 


150 674 


339 


1065.00 


90259 


389 


1222.08 


118 847 


439 


1379.16 


151 363 


340 


1068.14 


90792 


390 


1225.22 


119 459 


440 


1382.30 


152 053 


341 


1071.28 


91327 


391 


1228.36 


120 072 


441 


1385.44 


152 745 


342 


1074.42 


91863 


392 


1231.50 


120 687 


442 


1388.58 


153 439 


343 


1077.57 


92401 


393 


1234.65 


121304 


443 


1391.73 


154134 


344 


1080.71 


92941 


394 


1237.79 


121 922 


444 


1394.87 


154 830 


345 


1083.85 


93482 


395 


1240.93 


122 542 


445 


1398.01 


155 528 


346 


1086.99 


94025 


396 


1244.07 


123 163 


446 


1401.15 


156 228 


347 


1090.13 


94569 


397 


1247.21 


123 786 


44T 


1404.29 


156 930 


348 


1093.27 


95115 


398 


1250.35 


124 410 


448 


1407.43 


157 633 


349 


1096.42 


95662 


399 


1253.50 


125 036 


449 


1410.58 


158 337 


350 


1099.56 


96211 


400 


1256.64 


125 664 


450 


1413.72 


159 043 



Circumference and Area of Circles. 



113 





Circum. 


Area. 




Circum. 


Area. 




Circum. Area. 


Diam- 


s~~*\ 


4^g^t 


Diam- 


/^\ 


i^H^ 


Diam- 


S~\ 


i^H^ 


eter. 


o 


IP 


eter. 


O 


IP 


eter. 


o 


w 


451 


1416.86 


159 751 


501 


1573.94 


197 136 


551 


1731.02 


238 448 


452 


1420.00 


160 460 


502 


1577.08 


197 923 


552 


1734.16 


239 314 


453 


1423.14 


161 171 


503 


1580.22 


198 713 


; 553 


1737.40 


240 182 


454 


1426.28 


161 8S3 


504 


1583.36 


199 504 


| 554 


1740.44 


241 051 


455 


1429.42 


162 597 


505 


1586.50 


200 296 


; 555 


1743.58 


241922 


456 


1432.57 


163 313 


506 


1589.65 


201 090 


! 556 


1746.73 


242 795 


457 


1435.71 


164 030 


507 


1592.79 


201 8S6 


j -557 


1749.87 


243 669 


458 


1438.85 


164 748 


508 


1595.93 


202 6S3 


558 


1753.01 


244 545 


459 


1441.99 


165 468 


509 


1599.07 


203 482 


; 559 


1756.15 


245 422 


460 


1445.13 


166 190 


510 


1602.21 


204 2S2 


560 


1759.29 


246 301 


461 


1448.27 


166 914 


511 


1605.35 


205 084 


561 


1762.43 


247 181 


462 


1451.42 


167 639 


512 


1608.50 


205 887 


562 


1765.58 


248 063 


463 


1454.56 


168 365 


513 


1611.64 


206 692 


563 


1768.72 


248 947 


464 


1457.70 


169 093 


514 


1614.78 


207 499 


564 


1771.86 


249 832 


465 


1460.84 


169 823 


515 


1617.92 


208 307 


565 


1775.00 


250 719 


466 


1463.98 


170 554 


516 


1621.06 


209 117 


566 


1778.14 


251 607 


467 


1467.12 


171287 


517 


1624.20 


209 928 


567 


1781.28 


252 497 


468 


1470.27 


172 021 


518 


1627.35 


210 741 


568 


1784.42 


253 388 


469 


1473.41 


172 757 


519 


1630.49 


211 556 


569 


1787.57 


254 281 


470 


1476.55 


173 494 


520 


1633.63 


212 372 


570 


1790.71 


255176 


471 


1479.69 


174 234 


521 


1636.77 


213 189 


571 


1793.85 


256 072 


472 


1482.83 


174 974 


522 


1639.91 


214 008 


572 


1796.99 


256 970 


473 


1485.97 


175 716 


523 


1643.05 


214 829 


573 


1S00.13 


257 869 


474 


1489.11 


176 460 


524 


1646.20 


215 651 


574 


1803.27 


258 770 


1 475 


1492.26 


177 205 


525 


1649.34 


216 475 


575 


1806.42 


259 672 


476 


1495.40 


177 952 


526 


1652.48 


217 301 


576 


1809.56 


260 576 


477 


1498.54 


178 701 


527 


1655.62 


218128 


577 


1812.70 


261 482 


478 


1501.68 


179 451 


528 


1658.76 


218 956 


578 


1815.84 


262 389 


479 


1504.82 


180 203 


529 


1661.90 


219 787 


579 


1818.98 


263 298 


480 


1507.96 


180 956 


530 


1665.04 


220 618 


5S0 


1822.12 


264 208 


481 


1511.11 


181 711 


531 


1668.19 


221452 


581 


1825.27 


265 120 


482 


1514.25 


182 467 


532 


1671.33 


222 287 


582 


1828.41 


266 033 


483 


1517.39 


183 225 


533 


1674.47 


223 123 


583 


1831.55 


266 948 


484 


1520.53 


183 9S4 


534 


1677.61 


223 961 


584 


1834.69 


267 865 


485 


1523.67 


184 745 


535 


1680.75 


224 801 


585 


1837.83 


268 7S3 


| 486 


1526.81 


185 508 


536 


1683.89 


225 642 


586 


1840.97 


269 702 


487 


1529.96 


186 272 


537 


1687.04 


226 484 


587 


1844.11 


270 624 


488 


1533.10 


187 038 


538 


1690.18 


227 329 


588 


1847.26 


271547 


489 


1536.24 


187 805 


539 


1693.32 


228175 


589 


1850.40 


272 471 


' 490 


1539.38 


188 574 


540 


1696.46 


229 022 


590 


1853.54 


273 397 


491 


1542.52 


189 345 


541 


1699.60 


229 871 


, 591 


1856.68 


274 325 


i 492 


1545.66 


190117 


542 


1702.74 


230 722 


592 


1859.82 


275 254 


i 493 


1548.81 


190 890 


543 


1705.88 


231 574 


593 


1862.96 


276 184 


494 


1551.95 


191665 


544 


1709.03 


232 428 


594 


1S66.11 


277 117 


1 495 


1555.09 


192 442 


545 


1712.17 


233 283 


595 


1869.25 


278 051 


496 


1558.23 


193 221 


546 


1715.31 


234 140 


596 


1872.39 


278 986 


497 


1561.37 


194 000 


547 


1718.45 


234 998 


597 


1875.53 


279 923 


i 498 


1564.51 


194 782 


548 


1721.59 


235 858 


598 


1878.67 


280 862 


| 499 


1567.65 


195 565 


549 


1724.73 


236 720 


599 


1881.81 


281802 


500 


1570.80 


196 350 


550 


1727.88 


237 583 1 


600 


1884.96 


282 743 



114 




Circumference 


and Area of Circles. 






Circum. 


Area. 


| 


Circum. 


Area. 




Circum. 


Area. 


Diam- 


/^\ 


/^H^v 


Diam- 


/"-*\ 


^1^ 


Diam- 


y^\ 


,*ass^ 


eter. 


o 


w 


eter. 


O 


IP 


eter. 


O 


w 


601 


1888.10 


283 687 


651 


2045.18 


332 853 


701 


2202.26 


385 945 


602 


1891.24 


284 631 


652 


2048.32 


333 876 


702 


2205.40 


387 047 


603 


1894.38 


285 578 


653 


2051.46 


334 901 


703 


2208.54 


388 151 


604 


1897.52 


286 526 


654 


2054.60 


335 927 


704 


2211.68 


389 256 


605 


1900.66 


287 475 


655 


2057.74 


336 955 


705 


2214.82 


390 363 


606 


1903.81 


288 426 


656 


2060.88 


337 985 


706 


2217.96 


391 471 


607 


1906.95 


289 379 


657 


2064.03 


339 016 


707 


2221.11 


392 580 


608 


1910.09 


290 333 


658 


2067.17 


340 049 


708 


2224.25 


393 692 


609 


1913.23 


291 289 


659 


2070.31 


341 083 


709 


2227.39 


394 805 


610 


1916.37 


292 247 


660 


2073.45 


342 119 


710 


2230.53 


395 919 


611 


1919.51 


293 206 


661 


2076.59 


343 157 


711 


2233.67 


397 035 


612 


1922.65 


294 166 


662 


2079.73 


344 196 


712 


2236.81 


398 153 


613 


1925.80 


295 128 


663 


2082.88 


345 237 


713 


2239.96 


399 272 


614 


1928.94 


296 092 


664 


2086.02 


346 279 


714 


2243.10 


400 393 


615 


1932.08 


297 057 


665 


2089.16 


347 323 


715 


2246.24 


401 515 


616 


1935.22 


298 024 


666 


2092.30 


348 368 


716 


2249.38 


402 639 


617 


1938.36 


298 992 


667 


2095.44 


349 415 


717 


2252.52 


403 765 


618 


1941.50 


299 962 


668 


2098.58 


350 464 


718 


2255.66 


404 892 


619 


1944.65 


300 934 


669 


2101.73 


351 514 


719 


2258.81 


406 020 


620 


1947.79 


301 907 


670 


2104.87 


352 565 


720 


2261.95 


407 150 


621 


1950.93 


302 882 


671 


2108.01 


353 618 


721 


2265.09 


408 282 


622 


1954.07 


303 858 


672 


2111.15 


354 673 


722 


2268.23 


409 416 


623 


1957.21 


304 836 


673 


2114.29 


355 730 


723 


2271.37 


410 550 


624 


1960.35 


305 815 


674 


2117.43 


356 788 


724 


2274.51 


411 687 


625 


1963.50 


306 796 


675 


2120.58 


357 847 


725 


2277.65 


412 825 


626 


1966.64 


307 779 


676 


2123.72 


358 908 


726 


2280..80 


413 965 


627 


1969.78 


308 763 


677 


2126.86 


359 971 


727 


2283.94 


415106 


628 


1972.92 


309 748 


678 


2130.00 


361 035 


728 


2287.08 


416 248 


629 


1976.06 


310 736 


679 


2133.14 


362 101 


729 


2290.22 


417 393 


630 


1979.20 


311725 


680 


2136.28 


363 168 


730 


2293.36 


418 539 


631 


1982.35 


312 715 


681 


2139.42 


364 237 


731 


2296.50 


419 686 


632 


1985.49 


313 707 


682 


2142.57 


365 308 


732 


2299.65 


420 835 


633 


1988.63 


314 700 


683 


2145.71 


366 380 


733 


2302.79 


421 986 


634 


1991.77 


315 696 


684 


2148.85 


367 453 


734 


2305.93 


423 139 


635 


1994.91 


316 692 


685 


2151.99 


368 528 


735 


2309.07 


424 292 


636 


1998.05 


317 690 


686 


2155.13 


369 605 


736 


2312.21 


425 447 


637 


2001.19 


318 690 


687 


2158.27 


370 684 


737 


2315.35 


426 604 


638 


2004.34 


319 692 


688 


2161.42 


371764 


738 


2318.50 


427 762 


639 


2007.48 


320 695 


689 


2164.56 


372 845 


739 


2321.64 


428 922 


640 


2010.62 


321 699 


690 


2107.70 


373 928 


740 


2324.78 


430 084 


641 


2013.67 


322 705 


691 


2170.84 


375 013 


741 


2327.92 


431 247 


642 


2016.90 


323 713 


692 


2173.98 


376 099 


742 


2331.06 


432 412 


643 


2020.04 


324 722 


693 


2177.12 


377 187 


743 


2334.30 


433 578 


644 


2023.19 


325 733 


694 


2180.27 


378 276 


744 


2337.34 


434 746 


645 


2026.33 


326 745 


695 


2183.41 


379 367 


715 


2340.49 


435 916 


646 


2029.47 


327 759 


696 


2186.55 


380 459 


746 


2343.63 


437 087 


647 


2032.61 


328 775 


697 


2189.69 


381554 


747 


2346.77 


438 259 


648 


3035.75 


329 792 


698 


2192.83 


382 649 


7 is 


2349.91 


4:59 433 


649 


2038.89 


330 810 


699 


2195.97 


383 746 


749 


2353.05 


440 609 


650 


2042.04 


331 831 


700 


2199.11 


384 845 


7.50 


2356.19 


441 786 







Circumference 


and Area of Circles. 


115 




Circum. 


Area. 




Circum. 


Area. 




Circum. 


Area. 


Diam- 


x— V 


/^^v 


Diam- 


/^\ 


>^H^ 


Diam- 


S~*\ 


^m^ 


eter. 


O 


IP 


eter. 


O 


IP 


eter. 


O 


IP 


751 


2359.34 


442 965 


801 


2516.42 


503 912 


851 


2673.50 


568 786 


752 


2362.48 


444 146 


802 


2519.56 


505 171 


852 


2676.64 


570 124 


753 


2365.62 


445 328 


803 


2522.70 


506 432 


853 


2679.78 


571 463 


754 


2368.76 


446 511 


804 


2525.84 


507 694 


854 


2682.92 


572 803 


755 


2371.90 


447 697 


805 


2528.98 


508 958 


855 


2686.06 


574 146 


756 


2375.04 


448 883 


806 


2532.12 


510 223 


856 


2689.20 


575 490 


757 


2378.19 


450 072 


807 


2535.27 


511 490 


857 


2692.34 


576 835 


758 


2381.33 


451 262 


808 


2538.41 


512 758 


858 


2695.49 


578 182 


759 


2384.47 


452 453 


809 


2541.55 


514 028 


859 


2698.63 


579 530 


760 


2387.61 


453 646 


810 


2544.69 


515 300 


860 


2701.77 


580 880 


761 


2390.75 


454 841 


811 


2547.83 


516 573 


861 


2704.91 


582 232 


762 


2393.89 


456 037 


812 


2550.97 


517 848 


862 


2708.05 


583 585 


763 


2397.04 


457 234 


813 


2554.11 


519 124 


863 


2711.19 


584 940 


764 


2400.18 


458 434 


814 


2557.26 


520 402 


864 


2714.34 


586 297 


765 


2403.32 


459 635 


815 


2560.40 


521 681 


865 


2717.48 


587 655 


766 


2406.46 


460 837 


816 


2563.54 


522 962 


866 


2720.62 


589 014 


767 


2409.60 


462 041 


817 


2566.68 


524 245 


867 


2723.76 


590 375 


768 


2412.74 


463 247 


818 


2569.82 


525 529 


868 


2726.90 


591738 


769 


2415.88 


464 454 


819 


2572.96 


526 814 


869 


2730.04 


593 102 


770 


2419.03 


465 663 


820 


2576.11 


528 102 


870 


2733.19 


594 468 


771 


2422.17 


466 873 


821 


2579.25 


529 391 


871 


2736.33 


595 835 


772 


2425.31 


468 085 


822 


2582.39 


530 681 


872 


2739.47 


597 204 


773 


2428.45 


469 298 


823 


2585.53 


531 973 


873 


2742.61 


598 575 


774 


2431.59 


470 513 


824 


2588.67 


533 267 


874 


2745.75 


599 947 


775 


2434.73 


471 730 


825 


2591.81 


534 562 


875 


2748.89 


601 320 


776 


2437.88 


472 948 


826 


2594.96 


535 858 


876 


2752.04 


602 696 


777 


2441.02 


474 168 


827 


2598.10 


537 157 


877 


2755.18 


604 073 


778 


2444.16 


475 389 


828 


2601.24 


538 456 


878 


2758.32 


605 451 


779 


2447.30 


476 612 


829 


2604.38 


539 758 


879 


2761.46 


606 831 


780 


2450.44 


477 836 


830 


2607.52 


541061 


880 


2764.60 


608 212 


781 


2453.58 


479 062 


831 


2610.66 


542 365 


881 


2767.74 


609 595 


782 


2456.73 


480 290 


832 


2613.81 


543 671 


882 


2770.88 


610 980 


783 


2459.87 


481 519 


833 


2616.95 


544 979 


883 


2774.03 


612 366 


784 


2463.01 


482 750 


834 


2620.09 


546 288 


8S4 


2777.17 


613 754 


785 


2466.15 


483 982 


835 


2623.23 


547 599 


885 


2780.31 


615 143 


7S6 


2469.29 


485 216 


836 


2626.37 


548 912 


886 


2783.45 


616 534 


787 


2472.43 


486 451 


837 


2629.51 


550 226 


887 


2786.59 


617 927 


788 


2475.58 


487 688 


838 


2632.65 


551541 


888 


2789.73 


619 321 


789 


2478.72 


488 927 


839 


2635.80 


552 858 


889 


2792.88 


620 717 


790 


2481.86 


490 167 


840 


2638.94 


554 177 


890 


2796.02 


622 114 


791 


2485.00 


491 409 


841 


2642.08 


555 497 


891 


2799.16 


623 513 


792 


2488.14 


492 652 


842 


2645.22 


556 819 


892 


2S02.30 


624 913 


793 


2491.28 


493 897 


843 


2648.36 


558 142 


893 


2805.44 


626 315 


794 


2494.42 


495 143 


844 


2651.50 


559 467 


894 


2S0S.5S 


627 71S 


795 


2497.57 


496 391 


845 


2654.65 


560 794 


895 


2811.73 


629 124 


796 


2500.71 


497 641 


846 


2657.79 


562 122 


896 


2814.87 


630 530 


797 


2503.85 


498 892 


847 


2660.93 


563 452 


897 


2818.01 


631 938 


798 


2506.99 


500 145 


848 


2664.07 


564 783 


898 


2821.15 


633 348 


799 


2510.13 


501 399 


849 


2667.21 


566 116 


899 


2824.29 


634 760 


800 


2513.27 


502 655 


850 


2670.35 


567 450 


900 


2827.43 


636 173 



116 




Circumference 


and Area op Circles. 






Circum. 


Area. 




Circum. 


Area. 




Circum. 


Area. 


Diam- 


S~\ 


^H^w 


Diam- 


S~\ 


/^H^v 


Diam- 


j^^\ 


^s^. 


eter. 


O 


w 


eter. 


o 


w 


eter. 


O 


w^ 


901 


2830.58 


637 587 


934 


2934.25 


685 147 


967 


3037.92 


734 417 


902 


2833.72 


639 003 


935 


2937.39 


686 615 


968 


3041.06 


735 937 


903 


2836.86 


640 421 


936 


2940.53 


688 084 


969 


3044.20 


737 458 


904 


2840.00 


641 840 


937 


2943.67 


689 555 


970 


3047.34 


738 981 


905 


2843.14 


643 261 


938 


2946.81 


691028 


971 


3050.49 


740 506 


906 


2846.28 


644 683 


939 


2949.96 


692 502 


972 


3053.63 


742 032 


907 


2849.42 


646 107 


940 


2953.10 


693 978 


973 


3056.77 


743 559 


908 


2852.57 


647 533 


941 


2956.24 


695 455 


974 


3059.91 


745 088 


909 


2855.71 


648 960 


942 


2959.38 


696 934 


975 


3063.05 


746 619 


910 


2858.85 


650 388 


943 


2962.52 


698 415 


976 


3066.19 


748 151 


911 


2861.99 


651 818 


944 


2965.66 


699 897 


977 


3069.34 


749 685 


912 


2865.13 


653 250 


945 


2968.81 


701 380 


978 


3072.48 


751 221 


913 


2868.27 


654 684 


946 


2971.95 


702 865 


979 


3075.62 


752 758 


914 


2871.42 


656 118 


947 


2975.09 


704 352 


980 


3078.76 


754 296 


915 


2874.56 


657 555 


948 


2978.23 


705 840 


981 


3081.90 


755 837 


916 


2877.70 


658 993 


949 


2981.37 


707 330 


982 


3085.04 


757 378 


917 


2880.84 


660 433 


950 


2984.51 


708 822 


983 


3088.19 


758 922 


918 


2883.98 


661 874 


951 


2987.65 


710 315 


984 


3091.33 


760 466 


919 


2887.12 


663 317 


952 


2990.80 


711 809 


985 


3094.47 


762 013 


920 


2890.27 


664 761 


953 


2993.94 


713 307 


986 


3097.61 


763 561 


921 


2893.41 


666 207 


954 


2997.08 


714 803 


987 


3100.75 


765 111 


922 


2896.55 


667 654 


955 


3000.22 


716 303 


988 


3103.89 


766 662 


923 


2899.69 


669 103 


956 


3003.36 


717 804 


989 


3107.04 


768 215 


924 


2902.83 


670 554 


957 


3006.50 


719 306 


990 


3110.18 


769 769 


925 


2905.97 


672 006 


958 


3009.65 


720 810 


991 


3113.32 


771 325 


926 


2909.11 


673 460 


959 


3012.79 


722 316 


992 


3116.46. 


772 882 


927 


2912.26 


674 915 


960 


3015.93 


723 823 


993 


3119.60 


774 441 


928 


2915.40 


676 372 


961 


3019.07 


725 332 


994 


3122.74 


776 002 


929 


2918.54 


677 831 


962 


3022.21 


726 842 


995 


3125.88 


777 564 


930 


2921.68 


679 291 


963 


3025.35 


728 354 


996 


3129.03 


779 128 


931 


2924.82 


680 752 


964 


3028.50 


729 867 


997 


3132.17 


780 693 


932 


2927.96 


682 216 


965 


3031.64 


731 382 


998 


3135.31 


782 260 


933 


2931.11 


683 680 


966 


3034.78 


732 899 


999 


3138.45 


783 828 


Note.— Wh 
any diameter 
ence as have t 
places in this « 


en it is des 
riot in the t 
een pointec 
\rea as hav 


ired to find th 
able, point ofT 
1 off in the diai 
b been pointed 


e circumfe 
as many pi 
neter, and 
off in the c 


rence correspoi 
aces in the cir 
loint off twice i 
iameter. Thus 


iding to 
cumfer- 
19 many 


Diami 


'Mrs. 


Circumferen 


zes. 


Areas. 




9 
91 

916 
9160 


16 
6 


28.777 
287.77 
2877.7 
28777. 




65.8992 
6 589.93 
658 993. 
65 899 321. 




Wh 
consis 
differ* 
the gi 
addin 
diame 

Re 


en it is 
ting of a 
mce bet 
ven dian 
g the re 
ter. Th 
quired tl 


desired to 
whole nur 
ween the t 
leter lies ar 
suit to the 
us : 
le circumfe 


find tl 
nber ar 
abular 
id mul 
tabuls 

rence i 


le circu] 
id a deci 
figures 
tiplying 
u* value 

or the d 


mference oi 
mal, it maj 
for the dia 
this differe 
correspond 

iameter 916 


• area i 
be do 
meters 
ice by 
ing to 

.27? 


or any c 
ne by tal 

betweei 
the deci 

the nea 


iameter 
ling the ^ 
i which 
mal and 
:t lower 



Arcs and Segments of Circles. 



117 



We have Circumference 917 = 2880.84 

Circumference 916 = 2877.70 
Difference, 3.14 

3.14 X 27 = 0.8478 
Circumference 916.27 = 2877.70 + 0.85 = 2878.55 

For the area corresponding to the same diameter we have 

Area 917 = 660433 

Area 916 = 658993 

Difference, 1440 

1440 X 0.27 = 388.8 

Area 916.27 = 658993 + 388.8 = 659381.8 



Arcs and Segments of Circles. 

The table starting below enables the following values to be determined : 
angle at centre = v, radius = r, length of arc = b, area of segment = a, 
surface of spherical segment = a, volume of spherical segment = c, and 
length of chord = c. 

The quantities given are the height or versedsine of the arc = h, and 
the length of the chord = c. 

To use the table, divide the length of the chord by the height. Look 
for the nearest value to this quotient in the first, or extreme left-hand 
column, and opposite this value will be found the corresponding values 
for the various coefficients, k, for a chord of unit length. These values, 
multiplied by the length of the given chord, will give the required 
lengths ; by the square of the chord, will give the required surfaces ; and 
by the cube of the chord, will give the required volume. 

Thus, for a chord, c = 25, and height, h — 5, we have 

!-• 

The nearest value to this in the table is 5.0134. 
We then have 

Angle at centre, v = 87° ; 

Radius, r-25 X 0.72637 = 18.159 ; 

Length of arc, 5 = 25 X 1.1027 = 27.567 ; 

Area of segment, a = 25 2 X 0.13704 = 85.65 ; 

Surface of spherical segment, a = 25 2 X 0.91036 = 568.97 ; 
Volume of spherical segment, c = 25 3 X 0.08340 = 1303.12, 



Chord diY. 


Centre 


Radius 


Cir. arc 


Area seg. 


Surface 


Solidity 


Chord 


by height. 


angle v. 


r — kc. 


b = kc. 


a = kc 2 . 


a = kc 2 . 


c = kc 3 . 


c = kr. 


^Ul^ 


K> 


<3> 


<3> 


f^gm^ 


dfi^^ 




/ N. 


\ c s 


\ ,* 


Xx' 


458.08 


i 


57.296 


1.0000 


.00109 


.78539 


.00085 


.01744 


229.18 


2 


28.649 


1.0000 


.00218 


.78549 


.00172 


.03490 


152.77 


3 


19.101 


1.0000 


.00327 


.78562 


.00255 


.05234 


< 114.57 


4 


14.327 


1.0000 


.00436 


.78574 


.00310 


.06978 


84.747 


5 


11.462 


1.0001 


.00647 


.78586 


.00401 


.08722 


76.375 


6 


9.5530 


1.0003 


.00741 


.78599 


.00514 


.10466 


65.943 


7 


8.1902 


1.0004 


.00910 


.78621 


.00592 


.12208 


57.273 


8 


7.1678 


1.0006 


.01089 


.78630 


.00686 


.13950 


! 50.902 


9 


6.3728 


1.0008 


.01254 


.78665 


.00772 


.15690 


i 45.807 


10 


5.7368 


1.0011 


.01407 


.78695 


.00857 


.17430 



118 




Arcs and Segments of 


Circles 






Chord div. 
by height. 


Centre 
angle v. 


Radius 
r = kc. 


Cir. arc 

b = kc. 


Area seg. 
a = kc 2 . 


Surface 
a = kc 2 . 


Solidity 
C = kc z . 


Chord 
c = kr. 


/"HTT^ 




<P 


\ s ~* 


<^m^ 


t^Htat 


<^f!k 


f N. 


\y 


N x ,' 


\ w X 


41.203 


11 


5.2167 


1.0013 


.01552 


.78730 


.00964 


.19168 


38.133 


12 


4.7834 


1.0016 


.01695 


.78762 


.01031 


.20904 


35.221 


13 


4.4168 


1.0019 


.01841 


.78794 


.01114 


.22640 


32.742 


14 


4.1027 


1.0023 


.02000 


.78832 


.01199 


.24372 


30.514 


15 


3.8307 


1.0027 


.02157 


.78889 


.01288 


.26104 


28.601 


16 


3.5927 


1.0029 


.02269 


.78909 


.01375 


.27834 


26.915 


17 


3.3827 


1.0034 


.02434 


.78969 


.01462 


.29560 


25.412 


18 


3.1962 


1.0039 


.02592 


.79028 


.01542 


.31286 


24.068 


19 


3.0293 


1.0044 


.02744 


.79084 


.01635 


.33008 


22.860 


20 


2.8793 


1.0048 


.02878 


.79140 


.01722 


.34728 


21.760 


21 


2.7440 


1.0054 


.03040 


.79234 


.01802 


.36446 


20.777 


22 


2.6222 


1.0059 


.03178 


.79300 


.01897 


.38160 


19.862 


23 


2.5080 


1.0066 


.03343 


.79340 


.01984 


.39872 


19.028 


24 


2.4050 


1.0072 


.03493 


.79416 


.02072 


.41582 


18.261 


25 


2.3101 


1.0078 


.03639 


.79486 


.02159 


.43286 


17.553 


26 


2.2233 


1.0084 


.03784 


.79530 


.02248 


.44990 


16.970 


27 


2.1418 


1.0091 


.03970 


.79639 


.02315 


.46688 


16.288 


28 


2.0673 


1.0101 


.04115 


.79748 


.02424 


.48384 


15.721 


29 


1.9969 


1.0105 


.04230 


.79811 


.02511 


.50076 


15.191 


30 


1.9319 


1.0113 


.04385 


.79907 


.02600 


.51762 


14.970 


31 


1.8710 


1.0121 


.04476 


.80002 


.02692 


.53446 


14.230 


32 


1.8140 


1.0129 


.04710 


.80098 


.02778 


.55126 


13.796 


33 


1.7605 


1.0138 


.04842 


.80181 


.02866 


.56802 


13.382 


34 


1.7102 


1.0146 


.04989 


.80300 


.02956 


.58479 


12.994 


35 


1.6628 


1.0155 


.05137 


.80405 


.03046 


.60140 


12.733 


36 


1.6184 


1.0167 


.05311 


.80531 


.03137 


.61802 


12.473 


37 


1.5758 


1.0174 


.05401 


.80622 


.03226 


.63460 


11.931 


38 


1.5358 


1.0184 


.05628 


.80713 


.03328 


.65112 


11.621 


39 


1.4979 


1.0194 


.05755 


.80850 


.03418 


.66760 


11.342 


40 


1.4619 


1.0204 


.05899 


.80987 


.03506 


.68404 


11.060 


41 


1.4266 


1.0207 


.06001 


.81046 


.03589 


.70040 


10.791 


42 


1.3952 


1.0226 


.06196 


.81240 


.03680 


.71672 


10.534 


43 


1.3643 


1.0237 


.06359 


.81377 


.03773 


.73300 


10.289 


44 


1.3347 


1.0248 


.06574 


.81505 


.03864 


.74920 


10.043 


45 


1.3066 


1.0260 


.06628 


.81756 


.03890 


.76536 


9.8303 


46 


1.2797 


1.0272 


.06826 


.81795 


.04050 


.78146 


9.6153 


47 


1.2539 


1.0290 


.06998 


.81939 


.04143 


.79748 


9.4092 


48 


1.2289 


1.0297 


.07138 


.82064 


.04247 


.81346 


9.2113 


49 


1.2057 


1.0309 


.07290 


.82244 


.04330 


.82938 


9.0214 


50 


1.1831 


1.0323 


.07453 


.82384 


.04424 


.84522 


8.8387 


51 


1.1614 


1.0336 


.07611 


.82562 


.04519 


.86102 


8.6629 


52 


1.1406 


1.0349 


.07758 


.82729 


.04614 


.87674 


8.4462 


53 


1.1206 


1.0364 


.07959 


.82896 


.04685 


.89238 


8.3306 


54 


1.1014 


1.0378 


.08083 


.83072 


.04805 


.90798 


8.1733 


55 


1.0828 


1.0393 


.08246 


.83249 


.04901 


.92348 







Arcs and Segments of 


Circles 




119 


Chord div. 


Centre 


Radius 


Cir. arc 


Area seg. 


Surface 


Solidity 


Chord . 


by height. 


angle v. 


r = kc. 


b = kc. 


a = he 2 . 


a = kc 2 . 


C = kc 3 . 


c = ku 


SHJi^. 


x> 


<I> 


\> 


t^KK^\ 






^ v. 


\ C ' 

V 


XvX' 


\^ 


8.0215 


56 


1.0650 


1.0407 


.08400 


.83422 


.05002 


.93894 


7.8750 


57 


1.0478 


1.0422 


.08579 


.83602 


.05098 


.95430 


7.7334 


58 


1.0313 


1.0431 


.08680 


.83796 


.05191 


.96960 


7.5895 


59 


1.0154 


1.0454 


.08891 


.84064 


.05299 


.98484 


7.4565 


60 


1.0000 


1.0470 


.09106 


.84266 


.05400 


1.0000 


7.3358 


61 


.98515 


1.0486 


.09209 


.84380 


.05466 


1.0150 


7.2118 


62 


.97080 


1.0503 


.09375 


.84581 


.05583 


1.0300 


7.0914 


63 


.95694 


1.0520 


.09540 


.84791 


.05684 


1.0450 


6.9748 


64 


.94352 


1.0537 


.09697 


.84996 


.05784 


1.0598 


6.8616 


65 


.93058 


1.0555 


.09865 


.85215 


.05885 


1.0746 


6.7512 


66 


.91804 


1.0573 


.10036 


.85441 


.05987 


1.0892 


6.6453 


67 


.90590 


1.0591 


.10201 


.85640 


.06088 


1.1038 


6.5469 


68 


.89415 


1.0610 


.10367 


.85815 


.06181 


1.1184 


6.4902 


69 


.88276 


1.0629 


.10520 


.86082 


.06201 


1.1328 


6.3431 


70 


.87172 


1.0648 


.10710 


.86350 


.06396 


1.1471 


6.2400 


71 


.86102 


1.0668 


.10887 


.86699 


.06515 


1.1614 


6.1553 


72 


.85065 


1.0687 


.11046 


.86834 


.06604 


1.1755 


6.0652 


73 


.84058 


1.0708 


.11225 


.87081 


.06709 


1.1896 


5.9773 


74 


.83082 


1.0728 


.11385 


.87344 


.06815 


1.2036 


5.8918 


75 


.82134 


1.0749 


.11563 


.87590 


.06921 


1.2175 


5.8084 


76 

77 


.81213 
.80319 


1.0770 


.11736 


.87853 
.88120 


.07037 
.07136 


1.2313 


5.7271 


1.0792 


.11910 


1.2450 


5.6478 


78 


.79449 


1.0814 


.12072 


.88389 


.07244 


1.2586 


5.5704 


79 


.78606 


1.0836 


.12281 


.88677 


.07352 


1.2721 


5.4949 


80 


.77786 


1.0859 


.12441 


.88949 


.07462 


1.2855 


5.4254 


81 


.76988 


1.0882 


.12660 


.89161 


.07512 


1.2989 


5.3492 


82 


.76212 


1.0905 


.12793 


.89520 


.07683 


1.3121 


5.2705 


83 


.75458 


1.0920 


.12958 


.89958 


.07819 


1.3252 


5.2101 


84 


.74724 


1.0953 


.13157 


.90095 


.07907 


1.3383 


5.1429 


85 


.74009 


1.0977 


.13330 


.90420 


.07960 


1.3512 


5.0772 


86 


.73314 


1.1012 


.13546 


.90734 


.08102 


1.3639 


5.0134 


87 


.72637 


1.1027 


.13704 


.91036 


.08340 


1.3767 


4.9501 


88 


.71978 


1.1054 


.13893 


.91363 


.08436 


1.3893 


4.8886 


89 


.71336 


1.1079 


.14078 


.91696 


.08530 


1.4018 


4.8216 


90 


.70710 


1.1105 


.14279 


.92210 


.08621 


1.4142 


4.7694 


91 


.70101 


1.1132 


.14449 


.92352 


.08716 


1.4265 


4.7117 


92 


.69508 


1.1159 


.14643 


.92476 


.08798 


1.4387 


4.6615 


93 


.68930 


1.1186 


.14817 


.92914 


.08932 


1.4507 


4.5999 


94 


.68366 


1.1211 


.15009 


.93385 


.09076 


1.4627 


4.5453 


95 


.67817 


1.1242 


.15211 


.93746 


.09197 


1.4745 


4.4845 


96 


.67282 


1.1271 


.15375 


.94272 


.09348 


1.4863 


4.4398 


97 


.66760 


1.1300 


.15600 


.94470 


.09442 


1.4979 


4.3859 


98 


.66250 


1.1329 


.15801 


.94852 


.09567 


1.5094 


4.3383 


99 


.65754 


1.1359 


.15995 


.95236 


.09693 


1.5208 


4.2862 


100 


.65270 


1.1382 


.16180 


.95682 


.09831 


1.5321 



120 



Arcs and Segments of Circles. 



Chord div. 


Centre 


Radius 


dr. arc 


Area seg. 


Surface 


Solidity 


Chord 


by height. 


angle v. 


r = kc. 


b = kc. 


a = kc*. 


a = kc*. 


c feo». 


c = kr. 


s^m^ 


K> 


<i> 


x> 


d^HI^ 


^fiSlStok 


<d9fe 


/-— ^ 


v \/'' 


X.'' 


\ M *' 


4.2406 


101 


.64798 


1.1420 


.16393 


.96011 


.09956 


1.5432 


4.1930 


102 


.64338 


1.1451 


.16610 


.96112 


.10076 


1.5543 


4.1570 


103 


.63889 


1.1483 


.16925 


. 965(58 


.10215 


1.5652 


4.1006 


104 


.63450 


1.1515 


.17001 


.97216 


.10313 


1.5760 


4.0555 


105 


.63023 


1.1547 


.17204 


.97613 


.10471 


1.5867 


4.0113 


106 


.62607 


1.1580 


.17414 


.98067 


.10601 


1.5973 


3.9679 


107 


.62200 


1.1614 


.17619 


.98195 


.10735 


1.6077 


3.9252 


108 


.61803 


1.1648 


.17832 


.98931 


.10870 


1.6180 


3.8832 


109 


.61416 


1.1682 


.18011 


.99:57(5 


.11007 


1.6282 


3.8419 


110 


.61039 


1.1716 


.18257 


.99827 


.11149 


1.6383 


3.8013 


111 


.60670 


1.1752 


.18472 


1.0028 


.11284 


1.6482 


3.7612 


112 


.60325 


1.1790 


.18696 


1.0077 


.11426 


1.6581 


3.7221 


118 


.59960 


1.1823 


.18900 


1.0122 


.11566 


1.6677 


3.6837 


114 


.59618 


1.1859 


.19117 


1.0169 


.11709 


1.6773 


3.6451 


115 


.59284 


L.1897 


.19339 


1.0218 


.11853 


1.6867 


3.6086 


116 


.58959 


1.1934 


.19559 


1.0266 


.11995 


1.6961 


3.5712 


117 


.58641 


1.1972 


.19787 


1.0317 


.12145 


1.7053 


3.5349 


118 


.58331 


1.2011 


.20009 


1.0368 


.12294 


1.7143 


3.4992 


119 


.58030 


1.2050 


.20227 


1.0417 


.12444 


1.7282 


3.4641 


120 


.57735 


1.2089 


.20153 


1.0472 


.12596 


1.7320 


3.4296 


121 


.57450 


1.2130 


.20678 


1.0525 


.12718 


1.7107 


3.3953 


122 


.57168 


1.2177 


.20945 


1.0578 


.12903 


1.7492 


3.3616 


123 


.56895 


1.2213 


.21175 


1.0634 


.13060 


1.7576 


3.3285 


124 


.56628 


1.2253 


.21399 


1.0690 


.18218 


1.7659 


3.2940 


125 


.56370 


1.2295 


.21538 


1.075:5 


.13391 


1.7740 


3.2637 


126 


.56116 


1.2338 


.21859 


1.0803 


.18558 


1.7820 


3.2319 


127 


.55870 


L.2383 


.22121 


1.0862 


.13701 


1.7898 


3.2006 


128 


.55630 


L.2425 


.22370 


1.0921 


.13866 


1.7976 


3.1716 


129 


.55396 


1.2470 


.22617 


1.0974 


.14028 


1.8051 


3.1393 


130 


.55169 


1.2515 


.22866 


1.10-10 


.14202 


1.81 26 


3.1093 


131 


.54947 


1.2561 


.23113 


1.1101 


.14371 


1.8199 


3.0805 


182 


.54732 


1.2607 


.28372 


1.1164 


.14537 


1.8271 


3.0555 


138 


.54522 


1.2654 


.23608 


1.1212 


.14676 


1.8341 


3.0216 


134 


.54318 


1.2701 


.28892 


1.1295 


.11801 


1.8410 


2.9777 


185 


.54120 


1.2749 


.'J I IDS 


1.1363 


.15209 


1.8477 


2.9651 


188 


.5:5927 


1.2798 


.24364 


1.1428 


.15252 


1.8543 


2.9374 


137 


.53740 


1.2847 


.24676 


1.1495 


.15422 


1.8608 


2.9115 


138 


.58557 


1.2897 


.219:18 


L.1558 


.15605 


1.8673 


2.8829 


139 


.5:5380 


1.2948 


.25222 


1.1684 


.15807 


1 .87:53 


2.8662 


140 


.53209 


L.2999 


.25485 


1.1705 


.15996 


1.8794 


2.8299 


141 


.58042 


1.3063 


.25759 


1.1777 


.1(5-01 


1 .8853 


2.8038 


142 


.52881 


1.3065 


.259:56 


1.1853 


.1688] 


1.8910 4 


2. 7781 


L48 


.52724 


1.3157 


.26820 


1.1925 


.16577 


1 .8966 


2.7527 


144 


.52578 


L.82M 


.26604 


1.2000 


.16776 


1.9021 


2.7276 


145 


.52426 


L.8265 


.26889 


1.2077 


.16966 


1.9074 







Arcs and Segments of 


Circles 




121 


Chord div. 


Centre 


Radius 


Cir. aro 


Area Beg. 


Surface 


Solidity 


Chord 


by height. 


angle v. 


r Ac. 


b ho. 


a = kc 2 . 


a fee*. 


C = kc 3 . 


c — kr. 


^ITTx, 


x> 


r • 


S s7 


^ 


rf^Illlt 


^rlltob 


f — s. 


V 


N w '' 


2.7002 


146 


.52284 


1.3320 


.27196 


1.2166 


.17209 


1.9126 


2.6816 


117 


.52147 


1.3,377 


.27119 


1.2219 


.17405 


1.9176 


2.6588 


MS 


.52015 


1.3433 


.27772 


1.2318 


.17005 


1.9225 


2<6301 


149 


.51887 


1.3191 


.28108 


1.23,90 


.17809 


1.9272 


2.6064 


150 


.51764 


1.3,519 


.28369 


1.2170 


.18023 


1.9318 


2.6830 


151 


.51645 


1.3G0S 


.28074 


1.2503, 


.18006 


1.9363 


2.6598 


L52 


.51530 


1.3GGS 


.28983 


1.3018 


.18751 


1.9406 


2.5239 


153 


.51120 


1.3729 


.29397 


1.2801 


.18815 


1.9447 


2.5148 


154 


.51315 


1.3790 


.29007 


1.2824 


.18913 


1.9487 


2.4919 


155 


.51214 


1.3,852 


.29928 


1.2914 


.19147 


1.9526 


2.4099 


15G 


.51117 


1.3919 


.30259 


1.3004 


.19374 


1.9563 


•j, His 


157 


.51014 


1.3973 


.30560 


1.3094 


.19007 


1.9598 


2.4262 


158 


.50936 


1.4043 


.30905 


1.3191 


.19851 


1.9632 


2.4047 


159 


.50851. 


1.4109 


.31239 


1.32S7 


.20095 


1.90G3 


2.3835 


1G0 


.50771 


1.4175 


.31575 


1.3308 


.20312 


1.9096 


2.3613 


161 


.50695 


1.4243 


.31931 


1.3490 


.20009 


1.9725 


2.3417 


162 


.50623 


1.4311 


.82263 


1.3583 


.20S47 


1.9753 


2.32U 


163 


.50555 


1,13,80 


.32618 


1.3,082 


.21105 


1.9780 


2.3004 


164 


.50191 


1.4450 


.32969 


1.3791 


.21371 


1.9805 


2.2805 


165 


.50131 


1,1520 


.33327 


1.3895 


.21634 


1.9829 


2.2605 


166 


.50374 


1,1592 


.33684 


1,4021 


.21904 


1.9851 


2.2408 


167 


.50823 


1.4665 


.31048 


1.4111 


.32177 


1.9871 


2.2212 


168 


.50275 


1,1739 


.34122 


1.4222 


.22450 


1.9890 


2.2013 


169 


.50231 


1.4813 


.34802 


1,13,14 


.22706 


1.9908 


2.1826 


170 


.50191 


1,1 8S9 


.35230 


1.4476 


.23028 


1.9924 


2.1636 


171 


.50151 


1.4966 


.35563 


1.4505 


.23266 


1.9938 


2.1447 


172 


.50122 


1.5044 


::^x>;\ 


1.4684 


.23650 


1.9951 


2.1271 


173 


.50093 


1.5123, 


.36337 


1.4797 


.23900 


1.9962 


2.1075 


174 


.50068 


1.5202 


.36747 


1.4927 


.24225 


1.9972 ( 


2.0892 


175 


.50017 


1.5283 


.3,7152 


1.5052 


.24537 


1.9981 


2.0710 


176 


.50030 


1.5365 


.37562 


1.5179 


.24850 


1.9988 


2.0530 


177 


.50017 


1.5448 


.37974 


1.5308 


.25179 


1.9993 


2.0352 


178 


.50007 


1.5533 


.3S401 


1.5439 


.25531 


1.9996 


2.0175 


179 


.50002 


1.5018 


.38838 


1.5573 


.25810 


1.9999 


2.0000 


ISO 


.50000 


1.5708 


.39309 


1.5708 


.20179 


2.0000 



122 



The Ellipse. 



The Ellipse. 




Xotation. 

a = semi-major axis. 

b = semi-minor axis. 
f,/' = foci. 

x = abscissa = hori- 
zontal distance 
from centre to 
base of vertical 
u nd e r a n y 
point, p, on 
perimeter. 

y = ordinate = verti- 
cal distance 
from horizontal 
axis to point, p, 
on perimeter. 

Equation of ellipses, referred to axes through centre : 

a 2yS _|_ ^2 B a 2fc2. 

Construction : given the semi-axes, a and b. 

Find the foci,/,/', by taking the semi-major axis, a, in the dividers and 
sweeping arcs from B, intersecting the major axis at/ and/'. By attaching 
a string to pins at/ and/', and making the length of the string equal to 
2a, the curve can be drawn by 
moving a pencil around in the 
bight of the string. 

Points on the perimeter of 
an ellipse may be found as 
follows : 

Mark off on a straight- 
edged piece of paper the dis- 
tances rt = a, rs = 6; then, 
when t is on the minor axis 
and s on the major axis, r will 
be on a point in the curve, 
and so any number of points 
may be found. 

To draw a normal or a 
tangent at any point, p, on 
the perimeter of an ellipse, 
draw lines, fp,f'p, from the loint, p, to the two foci. A line, rs, bisecting 

the angle, A)/', will be the 
v m normal, and a line, mn, at 

right angles to the normal 
will be the tangent. The 
construction of the nor- 
mal indicates the proper 
angles for joints in ellipti- 
cal arches. 

The perimeter of an 
ellipse can be accurately 
computed only by the 
summation of a series. 

A good, approximate 
formula is that of Bous- 
sinesq, which is very close, 
when a is not more than 
three times greater than b. 





Perimeter 



_.-*(». •+»_*•*). 



The quantity in the parenthesis is the radius of a circle of equivalent 
perimeter to an ellipse whose major and minor semi-axes are a and b. 



The Ellipse. 



123 



Example. Let a = 5, b — 2. 



; 3.6689 — 23.052. 



The true perimeter of any ellipse may be computed from the following 
series : 

t, • « xs T. 1/a — b\ 2 1 /a — fr\ 4 , 1 /a — fr\ 6 ~| 

Penmeter = S = .(a + 5)[l + T (^ 1; ) + ^-^) + -(_).._). 

Calling the quantity within the brackets k, we have 

S= tt(o, + 6)fc. 
In the following table are given values of k for successive values of 



a — b 

Example. Let a = 7, 5 



, this rendering the application of the formula simple. 
1. 
We have 



^1 = 4 = 0.75. 

a + fr 8 



In the table, for 0.75 we have k = 1.1466 ; hence, 
S = ttX8X 1-1166 = 28.817. 
By the Boussinesq formula we get 

S = 29.388. 



Perimeter of Ellipse. 

Values of k for successive Values of 



a — 5 



a — b 
aT+b 


k 


a — 6 

a + 6 


k 


a — b 

a + b 


h 


a — b 
a + & 


k 


a — b 
a + b 


k 


0.01 


1.0000 


0.21 


1.0110 


0.41 


1.0431 


0.61 


1.0954 


0.81 


1.1721 


0.02 


1.0001 


0.22 


1.0122 


0.42 


1.0450 


0.62 


1.0986 


0.82 


1.1768 


0.03 


1.0002 


0.23 


1.0133 


0.43 


1.0472 


0.63 


1.1016 


0.83 


1.1813 


0.01 


1.0004 


0.24 


1.0145 


0.44 


1.0494 


0.64 


3.1048 


0.84 


1.1859 


0.05 


1.0006 


0.25 


1.0158 


0.45 


1.0516 


0.65 


1.1083 


0.85 


1.1903 


0.06 


1.0009 


0.26 


1.0173 


0.46 


1.0538 


0.66 


1.1115 


0.86 


1.1950 


0.07 


1.0012 


0.27 


1.0186 


0.47 


1.0561 


0.67 


1.1157 


0.87 


1.2000 


0.08 


1.0016 


0.28 


1.0200 


0.48 


1.0585 


0.68 


1.1193 


0.88 


1.2049 


0.09 


1.0020 


0.29 


1.0215 


0.49 


1.0608 


0.69 


1.1229 


0.89 


1.2100 


0.10 


1.0025 


0.30 


1.0226 


0.50 


1.0635 


0.70 


1.1267 


0.90 


1.2154 


0.11 


1.0029 


0.31 


1.0245 


0.51 


1.0661 


0.71 


1.1306 


0.91 


1.2207 


0.12 


1.0031 


0.32 


1.0261 


0.52 


1.0686 


0.72 


1.1345 


0.92 


1.2263 


0.13 


1.0040 


0.33 


1.0276 


0.53 


1.0713 


0.73 


1.1383 


0.93 


1.2315 


0.14 


1.0047 


0.34 


1.0291 


0.54 


1.0740 


0.74 


1.1423 


0.94 


1.2374 


0.15 


1.0054 


0.35 


1.0311 


0.55 


1.0768 


0.75 


1.1466 


0.95 


1.2430 


0.16 


1.0062 


0.36 


1.0331 


0.56 


1.0798 


0.76 


1.1509 


0.96 


1.2486 


0.17 


1.0070 


0.37 


1.0349 


0.57 


1.0827 


0.77 


1.1550 


0.97 


1.2546 


0.18 


1.0080 


0.38 


1.0369 


0.58 


1.0857 


0.78 


1.1593 


0.98 


1.2605 


0.19 


1.0090 


0.39 


1.0389 


0.59 


1.0889 


0.79 


1.1637 


0.99 


1.2665 


0.20 


1.0100 


0.40 


1.0404 


0.60 


1.0922 


0.80 


1.1677 


1.00 


1.2732 



The area of an ellipse is readily found by the formula 

Area = A = nab. 

This is simply obtained by taking the product, ab, as the diameter of a 
circle and looking up the corresponding area in the table of circles, pages 
110-116. 



1LM 



Thk Parabola. 



The Parabola. 




Notation, 

x = abscissa, for any point on the curve. 
y = ordinate. 

/ focus. 
■■ vertex. 

/> semi parameter ■■ double ordinate through focus. 
en directrix. 

in '.,/> distance of focus from vertex — distance of directrix from 
vertex. 
Equation : 

1/a «■ 2px; 



Construction of Parabola. 

Given j>osition of vertex, 0, and focus,/; 




Take the distance. in fO, and lay It ofl from 0to a; ,i will then be 
the point where the directrix cuts the horizontal axis, at any point, a', 
erect a verities 1, and with the distance, i"'. In the dividers, iweep an arc 
with /as a centre ; the Intersection of this arc with the vertical will be a 
point In the curve, in Like manner the points b, c, or any others maybe 

found 

(liven the rise and span of the cur \ 

Lav off the span, A B, and height, C: divide i Sand /-' Pinto any 
number of equal pint , i • , and EO and OF into the same number of 
equal parts. Join L-2-8 with 0, and the Intersection of these lines with 



- 



The Pakamola. 



125 




verticals through a. b, C, etc., will be Taints "> the Curve. The accuracy of 
ghe curve willaepend upon the number of division*, 

Length of Parabolic Curve. 




Let A— height, * - Bpan, / length <>f curve. 

»-[»+*(4)-S(i)l 

This isa close approximation when the rise is small In proportion to 
the apan. 

The exact formula for the length of an are oi a parabola from the 
yertez to a point whose co ordinates are x and y is 

«-?[v?(i>-?j.^^(Vf+^?)} 

Area of Parabola. 




L<-t « span, /< height. 



Area %*&• 



, 



126 



Cycloidal Curves. 



The Hyperbola. 




Notation. 

x = abscissa for any point on the curve. 

y = ordinate. 
/,/' = foci. 
A, B = vertices, A-B = transverse axis. 

Equation of the hyperbola : 

a iy2 _ D 2 X 1 = — a W. 

Construction of the Curve. 

Given the transverse axis, A-B, and the foci, /,/' : 

Take any points, 1, '2, 3, 4, etc., on the axis, OX, and make fa = Bl, 
fa = Al, fb = J32, fb = A2, etc., thus obtaining as many points on the 
curves as may be required. 



Cycloidal Curves. 

Cycloidal curves are those generated by the path of a point on a circle 
which rolls upon a given line. They are principally used for tooth pro- 
files in wheel gearing. 

We may consider the usual forms of cycloidal curves as generated by 
one circle rolling upon another, the rolling circle being called the 
generating circle, and the stationary one the base circle. 

When the base circle is of infinitely great diameter it may be considered 
as a straight line, and the curve is the orthocycloid, usually called the 
common cycloid. When the generating circle rolls on the outside of the 
base circle, the curve is called the epicycloid; when it rolls inside of the 
base circle it is called the hypocycloid. 

When the rolling circle is of infinitely great diameter it may be con- 
sidered as a straight line, and the curve is called an involute, or more 
correctly an evolute. 

We shall here give only the geometrical methods of construction of the 
four curves, taking up their applications in connection with the practical 
constructions. 

Common Cycloid. 

Let D be the generating circle : 

Lay off C, equal to one-half the circumference of the circle, D. Divide 
C and the half circumference of D into the same number of equal parts, 



Areas of Plane Figures. 



127 



1, 2, 3, 4, 5, 6, and V, 2', 3', 4', 
5', 6'. Erect ordinates from 1, 

2, 3, etc., and draw horizontal? 
from r, 2', 3', etc. Then make 
aa' = VI, bb f = 2'to, cc f = 3'n, 
d'd = 4'o, ee' = 5'p. Then a 7 , 
&', c 7 , etc., will "be points on 
the curve. 

Epicycloid and Hypo= 
cycloid. 

The construction of both 
epicycloid and hypo cycloid is 
similar to that of the common 
cycloid, modified only by the 

fact that the base is circular instead of straight. The following construc- 
tion applies to both curves, the only change being that due to the rolling 
being external and internal. 

In each case the arc, 1-4, on the base circle is made equal in length to 
the semi-circumference of the generating circle. Radial lines are drawn 






from the centre of the base circle through 1, 2, 3, 4, and arcs struck from 
through 1', 2', 3', 4'. Then aa f = Vn, W = 2'm, c& = d'l, and the curve is 
drawn through a', b', c' , d. 



Areas of Plane Figures. 

a = area ; other dimensions as in the figures. 
Square. 




Rectangle. 

r<- a > 




/ \ 


A / 

1 / 

b l 

1 


a == ab, 




a = bVd? — 62. 



128 



Areas of Plane Figures. 



Triangle. 



Triangle. 




Circle. 




a = nr* = 0.7854d 2 , 
a = ^ = 0.0796P2. 




i = yj)h, 



b L /c2-a2-52N2 
8 TV" I 25 ) 



Trapezium. 




»*-6 

i = %(a[ft + ft'] + &ft' + eft). 



Circle Ring. 




a = tt(£2 — r 2 ) = ir(.R + r)(E — r), 
a = 0.7854(X>2 — rf2). 

Or take the difference between 
the areas of the inner and outer 
circles, as found in the tables of 
areas of circles. 



Sector. 





114.5 * 



a== y 2 [br — c(r — ft)], 

7rr 2 V _ c . M 

a = -360 + T (r - 7i) ' 



Surfaces of Solids. 



129 



Quadrant. 



Spandrel. 





a = 0.785r3 = 0.3927 c 2 . 



i = 0.215r2 = 0.1075c 2 . 



The area of any irregular figure is best found by Simpson's Rule, as 
follows : 



c 


"o 




h 2 


h 3 


h 


h 5 


h 6 


h 


D 

h s 


/ 




+—d — > 












B 



Divide the base, AB, into any even number of parts, d (in the illustra- 
tion, 8 parts), and erect the ordinates, ho, hi, h<>, etc. Then the area, a, of 
the figure, ABCD, will be 

a = j(h + 4.hi + 2h 2 + 4/i 3 + 2ft 4 + i.h b + 2h Q + 4/* 7 + h). 

It will be observed that the coefficients of the ordinates are alternately 
4 and 2, with the exception of the first and last. 

When the figure is drawn to scale, the area is best measured by a plani- 
meter, but if this Is not available, Simpson's Rule is practically as correct 
as any. The degree of accuracy will naturally depend upon the number 
of divisions taken. 



Surfaces of Solids. 



Sphere. 




a = 47rr2 = 12.5664r2 = nd 2 . 



The surface of any sphere may 
readily be found by multiplying 
the area of a circle of the same 
diameter by 4, using the Table of 
Areas of Circles. 



, 



Torus, or Ring of Circular 
Cross Section. 




. = 47^^ = 39.4784ifr, 
, = 9.8696£>d 



130 



Volumes of Solids. 



Sector of a Sphere. 




Circle Zone. 




a== 2irrh^— (C2 + 4A2). 



Cone. 




I = ttEs, 



I = TTRVR2 + #2. 



Frustum of a 


Cone. 






^\ 


A""~> 






s j 

'■;■'{■ U 


(A | 

4ttnl\V 








-0— ** 




D — cT 


j? = 


ds 


d 


f^+d), 






180D 


180(D- 


-d) 




J2 


s 


* 





Volumes of Solids. 



c = content of the various bodies in terms of the dimensions given in the 
figures. 



Sphere. 




4irr* 

c = -5- = 4.18879?-\ 

c = ~- = 0.5236^. 
o 

For Table of Volumes of 
Spheres, see page 132. 



Torus. 




c = 2n 2 Rr 2 = 19.74.Rr2 
C = 2.463Dd 2 . 



Volumes of Solids. 



131 



Sphere Sector. 




c = %nr*h = 2.0944r2A, 



c = %irr8(r =f ^r 2 — %c 2 ). 



Segment of a Sphere. 




C = 7^2(7.-1^), 
f C 2 + 4A 2 



C = 7l7i 2 



Sh 



yj). 



Cone. 




c = —5- = lM7r-h, 

u 

c = 0.2618M. 



Conic Frustum. 




c = y s nh(R 2 + 2fr + r 2 ), 



Cylinder. 




c = 7rr 2 /i = 0.785dPft, 
c = ^- = 0.0796p2^. 

47T 



Ellipsoid. 




c = 0.4247r2jfr2 = 4.1847i2r2, 
c = 0.0537r 2 2)d 2 = 0.5231-Dd 2 . 



Paraboloid. 




t* r - - *\ 

c = %7rr2/i = 1.5707r2ft. 



Pyramid. 




C = %*h 



-X^-T 



132 



Trigonometry. 



Pyramidic Frustum. 




Wedge Frustum. 

si 



h *» 



c = j{A + a + VAa). 




c = ^(a + 6). 



Volumes of Spheres. 

D = diameter. 



D 


Volume. 


D 


Volume. 


D 


Volume. 


D 


Volume. 


D 


Volume. 


1 


0.523599 


21 


4849.048 


41 


36086.95 


61 


118846.9 


81 


278261.8 


2 


4.188790 


22 


5575.280 


42 


38792.38 


62 


124788.2 


82 


288695.6 


3 


14.13717 


23 


6370.626 


43 


41629.77 


63 


130924.3 


83 


299387.0 


4 


33.51032 


24 


7238.228 


44 


44602.24 


64 


137258.3 


84 


310339.1 


5 


65.44984 


25 


8181.230 


45 


47712.93 


65 


143793.3 


85 


321555.1 


6 


113.0974 


26 


9202.770 


46 


50965.00 


66 


150532.5 


86 


333038.2 


7 


179.5943 


27 


10306.00 


47 


54361.60 


67 


157479.1 


87 


344791.4 


8 


268.0826 


28 


11494.04 


48 


57905.83 


68 


164636.2 


88 


356818.0 


9 


381.7035 


29 


12770.05 


49 


61600.86 


69 


172006.9 


89 


369120.9 


10 


523.5988 


30 


14137.17 


50 


65449.84 


70 


179594.3 


90 


381703.5 


11 


696.9100 


31 


15598.53 


51 


69455.90 


71 


187401.7 


91 


394568.8 


12 


904.7785 


32 


17157.25 


52 


73622.17 


72 


195432.2 


92 


407720.0 


13 


1150.347 


33 


18816.56 


53 


77951.80 


73 


203688.8 


93 


421160.4 


14 


1436.755 


34 


20579.52 


54 


82447.94 


74 


212174.8 


94 


434892.8 


15 


1767.146 


35 


22449.29 


55 


87113.74 


75 


220893.3 


95 


448920.4 


16 


2144.660 


36 


24429.02 


56 


91952.32 


76 


229847.3 


96 


463246.7 


17 


2572.441 


37 


26521.84 


57 


96966.82 


77 


239040.1 


97 


477874.4 


18 


3053.628 


38 


28730.91 


58 


102160.4 


78 


248474.8 


98 


492807.0 


19 


3591.364 


39 


31059.35 


59 


107536.2 


79 


258154.6 


99 


508047.3 


20 


4188.790 


40 


33510.32 


60 


113097.4 


80 


268082.6 


100 


523598.8 



TRIGONOMETRY. 
Angular Functions. 

In order to obtain a clear idea of the various functions by which angu- 
lar values may be expressed in terms of straight lines, let it be supposed 
that we have a straight line, X'X, and that from a point, 0, on this line 
we have an arm, OR, which may be moved like a crank about as a 
centre. The arm, OR, will then make various angles with the line, X'X, 
according to the position which is given to it. 

If we take a radius, Oc = unity on any convenient scale, and describe 
a circle about 0, we find that there are a number of ways in which we 
can measure the angle, a, which the arm, OR, makes with the line, X'X. 

Thus, we may erect a perpendicular from c until it reaches OR at d, and 
the distance, cd, will be the tangent of the angle, a (written tan a). Or, we, 
may drop a perpendicular from b to OX at a, and we have ab, the sine of' 
the angle, a. Again, we may measure the distance, Oa, the cosine of a ; or 
ac, the versed-sine of a ; or/V, the cotangent of a. If we had given any one 
of the distances, measured on the same scale as the radius, Oc f we can con- 
struct the angle, a. 



Trigonometry. 



133 



By supposing the arm, OR, to be gradually moved about 0, so that the 
angle, a, steadily increases, we may observe the manner in which these 
functions vary. At first, when the angle is equal to zero, the sine, tangent, 
**and versed-sine are also equal to zero, while the cosine and secant are both 
equal to the radius, or equal to unity. As the angle increases the sine, 
tangent, and versed-sine increase, while the cosine diminishes. At 45° the 



cotangent «/fi 




sine and cosine are equal to each other and equal to y$ 2 = 0.7071, while 
the tangent and cotangent are also equal to each other and also equal to 
the radius or unity. At 90° the cosine and cotangent become equal to zero 
and the sine equals the radius. For angles between 90° and 180° the cosine 
and cotangent become negative ; between 180° and 270° the sine and co- 
sine, tangent and cotangent, are all negative ; and between 270° and 360° 
the sine and tangent are negative, the cosine and cotangent positive. 
Distances measured above X'X and to the right of YY' are positive; 
those measured to the left of YY' and below X'X are negative. 
Referring again to the diagram, the functions are : 

ah — sine, ac — versed sine, cd == tangent, Od = secant, 

aO = cosine, fg = co versed sine, fe = cotangent, Oe — cosecant. 



Trigonometric Tables. 

In the following tables the values of the various angular functions are 
given for every degree and minute of the quadrant for a radius of unity. 
If any other radius is used, the tabular values are to be multiplied by the 
actual length of the radius. These tables of so-called Natural Functions 
are followed by tables of the Logarithmic Angular Functions, these being 
the logarithms of the natural functions. If the computations are made 
by the ordinary processes of multiplication and division, the natural func- 
tions are used, and if logarithms of numbers are used, the logarithms of 
the angular functions are to be used with them. 

In the logarithmic functions the characteristics have been increased by 
10, in order to avoid negative characteristics ; hence, the corresponding 
number of tens are to be subtracted from the final result. 



134 



Natukal Functions. 



0° 




Natural Trigonometrical Functions. 


179° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Co tang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.00000 


1.0000 


Infinite. 


.00000 


Infinite. 


1.0000 


.00000 


1.0000 


60' 


1 


. 0029 


.99971 


3437.7 


. 0029 


3437.7 


.0000 


. 0000 


.0000 


59 


2 


. 0058 


. 9942 


1718.9 


. 0058 


1718.9 


.0000 


. 0000 


.0000 


58 


3 


. 0087 


. 9913 


1145.9 


. 0087 


1145.9 


.0000 


. 0000 


.0000 


57 


4 


. 0116 


. 9884 


859.44 


. 0116 


859.44 


.0000 


. 0000 


.0000 


56 


5 


.00145 


.99854 


687.55 


.00145 


687.55 


1.0000 


.00000 


1.0000 


55 


6 


. 0174 


. 9825 


572.96 


. 0174 


572.96 


.0000 


. 0000 


.0000 


54 


7 


. 0204 


. 9796 


491.11 


. 0204 


491.11 


.0000 


. 0000 


.0000 


53 


8 


. 0233 


. 9767 


429.72 


. 0233 


429.72 


.0000 


. 0000 


.0000 


52 


9 


. 0262 


. 9738 


381.97 


. 0262 


381.97 


.0000 


. 0000 


.0000 


51 


10 


.00291 


.99709 


343.77 


.00291 


343.77 


1.0000 


.00000 


.99999 


50 


11 


. 0320 


. 9680 


312.52 


. 0320 


312.52 


.0000 


. 0000 


. 9999 


49 


12 


. 0349 


. 9651 


286.48 


. 0349 


286.48 


.0000 


. 0001 


. 9999 


48 


13 


. 0378 


. 9622 


64.44 


. 0378 


64.44 


.0000 


. 0001 


. 9999 


47 


14 


. 0407 


. 9593 


45.55 


. 0407 


45.55 


.0000 


. 0001 


. 9999 


46 


15 


.00436 


.99564 


229.18 


.00436 


229.18 


1.0000 


.00001 


.99999 


45 


16 


. 0465 


. 9534 


14.86 


. 0465 


14.86 


.0000 


. 0001 


. 9999 


44 


17 


. 0494 


. 9505 


02.22 


. 0494 


02.22 


.0000 


. 0001 


. 9999 


43 


18 


. 0524 


. 9476 


190.99 


. 0524 


190.98 


.0000 


. 0001 


. 9999 


42 


19 


. 0553 


. 9447 


80.93 


. 0553 


80.93 


.0000 


. 0001 


. 9998 


41 


20 


.00582 


.99418 


171.89 


.00582 


171.88 


1.0000 


.00002 


.99998 


40 


21 


. 0611 


. 9389 


63.70 


. 0611 


63.70 


.0000 


. 0002 


. 9998 


39 


22 


. 0640 


. 9360 


56.26 


. 0640 


56.26 


.0000 


. 0002 


. 9998 


38 


23 


. 0669 


. 9331 


49.47 


. 0669 


49.46 


.0000 


. 0002 


. 9998 


37 


24 


. 0698 


. 9302 


43.24 


. 0698 


43.24 


.0000 


. 0002 


. 9997 


36 


25 


.00727 


.99273 


137.51 


.00727 


137.51 


1.0000 


.00003 


.99997 


35 


26 


. 0756 


. 9244 


32.22 


. 0756 


32.22 


.0000 


. 0003 


. 9997 


34 


27 


. 0785 


. 9215 


27.32 


. 0785 


27.32 


.0000 


. 0003 


. 9997 


33 


28 


. 0814 


. 9185 


22.78 


. 0814 


22.77 


.0000 


. 0003 


. 9997 


32 


29 


. 0843 


. 9156 


18.54 


. 0844 


18.54 


.0000 


. 0003 


. 9996 


31 


30 


.00873 


.99127 


114.59 


.00873 


114.59 


1.0000 


.00004 


.99996 


30 


31 


. 0902 


. 9098 


10.90 


. 0902 


10.89 


.0000 


. 0004 


. 9996 


29 


32 


. 0931 


. 9069 


07.43 


. 0931 


07.43 


.0000 


. 0004 


. 9996 


28 


33 


. 0960 


. 9040 


04.17 


. 0960 


04.17 


.0000 


. 0005 


. 9995 


27 


34 


. 0989 


. 9011 


01.11 


. 0989 


01.11 


.0000 


. 0005 


. 9995 


26 


35 


.01018 


.98982 


98.223 


.01018 


98.218 


1.0000 


.00005 


.99995 


25 


36 


. 1047 


. 8953 


5.495 


. 1047 


5.489 


.0000 


. 0005 


. 9994 


24 


37 


. 1076 


. 8924 


2.914 


. 1076 


2.908 


.0000 


. 0006 


. 9994 


23 


38 


. 1105 


. 8895 


0.469 


. 1105 


0.463 


.0001 


. 0006 


. 9994 


22 


39 


. 1134 


. 8865 


88.149 


. 1134 


88.143 


.0001 


. 0006 


. 9993 


21 


40 


.01163 


.98836 


85.946 


.01164 


85.940 


1.0001 


.00007 


.99993 


20 


41 


. 1193 


. 8807 


3.849 


. 1193 


3.843 


.0001 


. 0007 


. 9993 


19 


42 


. 1222 


. 8778 


1.853 


. 1222 


1.847 


.0001 


. 0007 


. 9992 


18 


43 


. 1251 


. 8749 


79.950 


. 1251 


79.943 


.0001 


. 0008 


. 9992 


17 


44 


. 1280 


. 8720 


8.133 


. 1280 


8.126 


.0001 


. 0008 


. 9992 


16 


45 


.01309 


.98691 


76.396 


.01309 


76.390 


1.0001 


.00008 


.99991 


15 


46 


. 1338 


. 8662 


4.736 


. 1338 


4.729 


.0001 


. 0009 


. 9991 


14 


47 


. 1367 


. 8633 


3.146 


. 1367 


3.139 


.0001 


. 0009 


. 9991 


13 


48 


. 1396 


. 8604 


1.622 


. 1396 


1.615 


.0001 


. 0010 


. 9990 


12 


49 


. 1425 


. 8575 


0.160 


. 1425 


0.153 


.0001 


. 0010 


. 9990 


11 


50 


.01454 


.98546 


68.757 


.01454 


68.750 


1.0001 


.00010 


.99989 


10 


51 


. 1483 


. 8516 


7.409 


. 1484 


7.402 


.0001 


. 0011 


. 9989 


9 


52 


. 1512 


. 8487 


6.113 


. 1513 


6.105 


.0001 


. 0011 


. 9988 


8 


53 


. 1.542 


. 8458 


4.866 


. 1542 


4.858 


.0001 


. 0012 


. 9988 


7 


51 


. 1571 


. 8429 


3.664 


. 1571 


3.657 


.0001 


. 0012 


. 9988 


6 


55 


.01600 


.98400 


62.507 


.01600 


62.499 


1.0001 


.00013 


.99987 


5 


56 


. 1629 


. 8371 


1.391 


. 1629 


1.383 


.0001 


. 0013 


. 9987 


4 


57 


. 1658 


. 8342 


0.314 


. 1658 


0.306 


.0001 


. 0014 


. 9987 


3 


58 


. 1687 


. 8313 


59.274 


. 1687 


59.2«6 


.0001 


. 0014 


. 9986 


2 


59 


. 171(5 


. 8284 


8.270 


. 1716 


8.261 


.0001 


. 0015 


. 9985 


1 


60 


. 1745 


. 8255 


7.299 


. 1745 


7.290 


.0001 


. 0015 


. 9985 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



90° 



89° 



Natural Functions. 



135 



1° 




Natural Trigonometrical Functions. 


178° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Yrs. sin. 


Cosine. 


M. 





.01745 


.98255 


57.299 


.01745 


57.290 


1.0001 


.00015 


.99985 


60 


1 


. 1774 


. 8226 


6.359 


. 1775 


6.350 


.0001 


. 0016 


. 9984 


59 


2 


. 1803 


. 8196 


5.450 


. 1804 


5.441 


.0001 


. 0016 


. 9984 


58 


3 


. 1832 


. 8167 


4.570 


. 1833 


4.561 


.0002 


. 0017 


. 9983 


57 


4 


. 1861 


. 8138 


3.718 


. 1862 


3.708 


.0002 


. 0017 


. 9983 


56 


5 


.01891 


.98109 


52.891 


.01891 


52.882 


1.0002 


.00018 


.99982 


55 


6 


. 1920 


. 8080 


2.090 


. 1920 


2.081 


.0002 


. 0018 


. 9981 


54 


7 


. 1949 


. 8051 


1.313 


. 1949 


1.303 


.0002 


. 0019 


. 9981 


53 


8 


. 1978 


. 8022 


0.558 


. 1978 


0.548 


.0002 


. 0019 


. 9980 


52 


9 


. 2007 


. 7993 


49.826 


. 2007 


49.816 


.0002 


. 0020 


. 9980 


51 


10 


.02036 


.97964 


49.114 


.02036 


49.104 


1.0002 


.00021 


.99979 


50 


11 


. 2065 


. 7935 


8.422 


. 2066 


8.412 


.0002 


. 0021 


. 9979 


49 


12 


. 2094 


. 7906 


7.750 


. 2095 


7.739 


.0002 


. 0022 


. 9978 


48 


13 


. 2123 


. 7877 


7.096 


. 2124 


7.085 


.0002 


. 0022 


. 9977 


47 


14 


. 2152 


. 7847 


6.460 


. 2153 


6.449 


.0002 


. 0023 


. 9977 


46 


15 


.02181 


.97818 


45.840 


.02182 


45.829 


1.0002 


.00024 


.99976 


45 


16 


. 2210 


. 7789 


5.237 


. 2211 


5.226 


.0002 


. 0024 


. 9975 


44 


17 


. 2240 


. 7760 


4.650 


. 2240 


4.638 


.0002 


. 0025 


. 9975 


43 


18 


. 2269 


. 7731 


4.077 


. 2269 


4.066 


.0002 


. 0026 


. 9974 


42 


19 


. 2298 


. 7702 


3.520 


. 2298 


3.508 


.0003 


. 0026 


. 9974 


41 


20 


.02327 


.97673 


42.976 


.02327 


42.964 


1.0003 


.00027 


.99973 


40 


21 


. 2356 


. 7644 


2.445 


. 2357 


2.433 


.0003 


. 0028 


. 9972 


39 


22 


. 2385 


. 7615 


1.928 


. 2386 


1.916 


.0003 


. 0028 


. 9971 


38 


23 


. 2414 


. 7586 


1.423 


. 2415 


1.410 


.0003 


. 0029 


. 9971 


37 


24 


. 2443 


. 7557 


0.930 


. 2444 


0.917 


.0003 


. 0030 


. 9970 


36 


25 


.02472 


.97528 


40 448 


.02473 


40.436 


1.0003 


.00030 


.99969 


35 


26 


. 2501 


. 7499 


39.978 


. 2502 


39.965 


.0003 


. 0031 


. 9969 


34 


27 


. 2530 


. 7469 


9.518 


. 2531 


9.506 


.0003 


. 0032 


. 9968 


33 


28 


. 2559 


. 7440 


9.069 


. 2560 


9.057 


.0003 


. 0033 


. 9967 


32 


29 


. 2589 


. 7411 


8.631 


. 2589 


8.618 


.0003 


. 0033 


. 9966 


31 


30 


.02618 


.97382 


38.201 


.02618 


38.188 


1.0003 


.00034 


.99966 


30 


31 


. 2647 


. 7353 


7.782 


. 2648 


7.769 


.0003 


. 0035 


. 9965 


29 


32 


. 2676 


. 7324 


7.371 


. 2677 


7.358 


.0003 


. 0036 


. 9964 


28 


33 


. 2705 


. 7295 


6.969 


. 2706 


6.956 


.0004 


. 0036 


. 9963 


27 


34 


. 2734 


. 7266 


6.576 


. 2735 


6.563 


.0004 


. 0037 


. 9963 


26 


35 


.02763 


.97237 


36.191 


.02764 


36.177 


1.0004 


.00038 


.99962 


25 


36 


. 2792 


. 7208 


5.814 


. 2793 


5.800 


.0004 


. 0039 


. 9961 


24 


37 


. 2821 


. 7179 


5.445 


. 2822 


5.431 


.0004 


. 0040 


. 9960 


23 


38 


. 2850 


. 7150 


5.084 


. 2851 


5.069 


.0004 


. 0041 


. 9959 


22 


39 


. 2879 


. 7121 


4.729 


. 2880 


4.715 


.0004 


. 0041 


. 9958 


21 


40 


.02908 


.97091 


34.382 


.02910 


34.368 


1.0004 


.00042 


.99958 


20 


41 


. 2937 


. 7062 


4.042 


. 2939 


4.027 


.0004 


. 0043 


. 9957 


19 


42 


. 2967 


. 7033 


3.708 


. 2968 


3.693 


.0004 


. 0044 


. 9956 


18 


43 


. 2996 


. 7004 


3.381 


. 2997 


3.366 


.0004 


. 0045 


. 9955 


17 


44 


. 3025 


. 6975 


3.060 


. 3026 


3.045 


.0004 


. 0046 


. 9954 


16 


45 


.03054 


.96946 


32.745 


.03055 


32.730 


1.0005 


.00046 


.99953 


15 


46 


. 3083 


. 6917 


2.437 


. 3084 


2.421 


.0005 


. 0047 


. 9952 


14 


47 


. 3112 


. 6888 


2.134 


. 3113 


2.118 


.0005 


. 0048 


. 9951 


13 


48 


. 3141 


. 6859 


1.836 


. 3143 


1.820 


.0005 


. 0049 


. 9951 


12 


49 


. 3170 


. 6830 


1.544 


. 3172 


1.528 


.0005 


. 0050 


. 9950 


11 


50 


.03199 


.96801 


31.257 


.03201 


31.241 


1.0005 


.00051 


.99949 


10 


51 


. 3228 


. 6772 


0.976 


. 3230 


0.960 


.0005 


. 0052 


. 9948 


9 


52 


. 3257 


. 6743 


0.699 


. 3259 


0.683 


.0005 


. 0053 


. 9947 


8 


53 


. 3286 


. 6713 


0.428 


. 3288 


0.411 


.0005 


. 0054 


. 9946 


7 


54 


. 3315 


. 6684 


0.161 


. 3317 


0.145 


.0005 


. 0055 


. 9945 


6 


55 


.03344 


.96655 


29.899 


.03346 


29.882 


1.0005 


.00056 


.99944 


5 


56 


. 3374 


. 6626 


9.641 


. 3375 


9.624 


.0006 


. 0057 


. 9943 


4 


57 


. 3403 


. 6597 


9.388 


. 3405 


9.371 


.0006 


. 0058 


. 9942 


3 


58 


. 3432 


. 6568 


9.139 


. 3434 


9.122 


.0006 


. 0059 


. 9941 


2 


59 


. 3461 


. 6539 


8.894 


. 3463 


8.877 


.0006 


. 0060 


. 9940 


1 


60 


. 3490 


. 6510 


8.654 


. 3492 


8.636 


.0006 


. 0061 


. 9939 





M. 


Cosine. 


Yrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



91° 



88° 



136 



Natural Functions. 



2° 




Natural Trigonometrical Functions. 


177° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


! Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.03490 


.96510 


28.654 


.03492 


28.636 


1.0006 


.00061 


.99939 


60 


1 


. 3519 


. 6481 


8.417 


. 3521 


8.399 


.0006 


. 0062 


. 9938 


59 


2 


. 3548 


. 6452 


8.184 


. 3550 


8.166 


.0006 


. 0063 


. 9937 


58 


3 


. 3577 


. 6423 


7.955 


. 3579 


7.937 


.0006 


. 0064 


. 9936 


57 


4 


. 3606 


. 6394 


7.730 


. 3608 


7.712 


.0006 


. 0065 


. 9935 


56 


5 


.03635 


.96365 


27.508 


.03638 


27.490 


1.0007 


.00066 


.99934 


55 


6 


. 3664 


. 6336 


7.290 


. 3667 


7.271 


.0007 


. 0067 


. 9933 


54 


7 


. 3693 


. 6306 


7.075 


. 3696 


7.056 


.0007 


. 0068 


. 9932 


53 


8 


. 3722 


. 6277 


6.864 


. 3725 


6.845 


.0007 


. 0069 


. 9931 


52 


9 


. 3751 


. 6248 


6.655 


. 3754 


6.637 


.0007 


. 0070 


. 9930 


51 


10 


.03781 


.96219 


26.450 


.03783 


26.432 


1.0007 


.00071 


.99928 


50 


11 


. 3810 


. 6190 


6.249 


. 3812 


6.230 


.0007 


. 0073 


. 9927 


49 


12 


. 3839 


. 6161 


6.050 


. 3842 


6.031 


.0007 


. 0074 


. 9926 


48 


13 


. 3868 


. 6132 


5.854 


. 3871 


5.835 


.0007 


. 0075 


. 9925 


47 


14 


. 3897 


. 6103 


5.661 


. 3900 


5.642 


.0008 


. 0076 


. 9924 


46 


15 


.03926 


.96074 


25.471 


.03929 


25.452 


1.0008 


.00077 


.99923 


45 


16 


. 3955 


. 6045 


5.284 


. 3958 


5.264 


.0008 


. 0078 


. 9922 


44 


17 


. 3984 


. 6016 


5.100 


. 3987 


5.080 


.0008 


. 0079 


. 9921 


43 


18 


. 4013 


. 5987 


4.918 


. 4016 


4.898 


.0008 


. 0080 


. 9919 


42 


19 


. 4042 


. 5958 


4.739 


. 4045 


4.718 


.0008 


. 0082 


. 9918 


41 


20 


.04071 


.95929 


24.562 


.04075 


24.542 


1.0008 


.00083 


.99917 


40 


21 


. 4100 


. 5900 


4.388 


. 4104 


4.367 


.0008 


. 0084 


. 9916 


39 


22 


. 4129 


. 5870 


4.216 


. 4133 


4.196 


.0008 


. 0085 


. 9915 


38 


23 


. 4158 


. 5841 


4.047 


. 4162 


4.026 


.0009 


. 0086 


. 9913 


37 


24 


. 4187 


. 5812 


3.880 


. 4191 


3.859 


.0009 


. 0088 


. 9912 


36 


25 


.04217 


.95783 


23.716 


.04220 


23.694 


1.0009 


.00089 


.99911 


35 


26 


. 4246 


. 5754 


, 3.553 


. 4249 


3.532 


.0009 


. 0090 


. 9910 


34 


27 


. 4275 


. 5725 


3.393 


. 4279 


3.372 


.0009 


. 0091 


. 9908 


33 


28 


. 4304 


. 5696 


3.235 


. 4308 


3.214 


.0009 


. 0093 


. 9907 


32 


29 


. 4333 


. 5667 


3.079 


. 4337 


3.058 


.0009 


. 0094 


. 9906 


31 


30 


.04362 


.95638 


22.925 


.04366 


22.904 


1.0009 


.00095 


.99905 


30 


81 


. 4391 


. 5609 


2.774 


. 4395 


2.752 


.0010 


. 0096 


. 9903 


29 


32 


. 4420 


. 5580 


2.624 


. 4424 


2.602 


.0010 


. 0098 


. 9902 


28 


33 


. 4449 


. 5551 


2.476 


. 4453 


2.454 


.0010 


. 0099 


. 9901 


27 


34 


. 4478 


. 5522 


2.330 


. 4483 


2.308 


.0010 


. 0100 


. 9900 


26 


35 


.04507 


.95493 


22.186 


.04512 


22.164 


1.0010 


.00102 


.99898 


25 


36 


. 4536 


. 5464 


2.044 


. 4541 


2.022 


.0010 


. 0103 


. 9897 


24 


37 


. 4565 


. 5435 


1.904 


. 4570 


1.881 


.0010 


. 0104 


. 9896 


23 


38 


. 4594 


. 5405 


1.765 


. 4599 


1.742 


.0010 


. 0106 


. 9894 


22 


39 


. 4623 


. 5376 


1.629 


. 4628 


1.606 


.0011 


. 0107 


. 9893 


21 


40 


.04652 


.95347 


21.494 


.04657 


21.470 


1.0011 


.00108 


.99892 


20 


41 


. 4681 


. 5318 


1.360 


. 4687 


1.337 


.0011 


. 0110 


. 9890 


19 


42 


. 4711 


. 5289 


1.228 


. 4716 


1.205 


.0011 


. 0111 


. 9889 


18 


43 


. 4740 


. 5260 


1.098 


. 4745 


1.075 


.0011 


. 0112 


. 9888 


17 


44 


. 4769 


. 5231 


0.970 


. 4774 


0.946 


.0011 


. 0114 


. 9886 


16 


45 


.04798 


.95202 


20.843 


.04803 


20.819 


1.0011 


.00115 


.99885 


15 


46 


. 4827 


. 5173 


0.717 


. 4832 


0.693 


.0012 


. 0116 


. 9883 


14 


47 


. 4856 


. 5144 


0.593 


. 4862 


0.569 


.0012 


. 0118 


. 9882 


13 


48 


. 4885 


. 5115 


0.471 


. 4891 


0.446 


.0012 


. 0119 


. 9881 


12 


49 


. 4914 


. 5086 


0.350 


. 4920 


0.325 


.0012 


. 0121 


. 9879 


11 


50 


.04943 


.950-)/ 


20.230 


.04949 


20.205 


1.0012 


.00122 


.99878 


10 


51 


. 4972 


. 5028 


0.112 


. 4978 


0.087 


.0012 


. 0124 


. 9876 


9 


52 


. 5001 


. 4999 


19.995 


. 5007 


19.970 


.0012 


. 0125 


. 9875 


8 


53 


. 5030 


. 4970 


9.880 


. 5037 


9.854 


.0013 


. 0127 


. 9873 


7 


54 


. 5059 


. 4941 


9.766 


. 5066 


9.740 


.0013 


. 0128 


. 9872 


6 


55 


.05088 


.94912 


19.653 


.05095 


19.627 


1.0013 


.00129 


.99870 


5 


56 


. 5117 


. 4883 


9.541 


. 5124 


9.515 


.0013 


. 0131 


. 9869 


4 


57 


. 5146 


. 4853 


9.431 


. 5153 


9.405 


.0013 


. 0132 


. 9867 


3 


58 


. 5175 


. 4824 


9.322 


. 5182 


9.296 


.0013 


. 0134 


. 9866 


2 


59 


. 5204 


. 4795 


9.214 


. 5212 


9.188 


.0013 


. 0135 


. 9864 


1 


60 


. 5234 


. 4766 


9.107 


. 5241 


9.081 


.0014 


. 0137 


. 9863 





M. 


Cosine. 


Vrs. Bin. 


Secant, j 


Co tang. 


Tang. 1 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



92° 



87° 



Natural Functions. 



137 



3° 




Natural Trigonometrical Functions. 


176° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Co tang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.05234 


.94766 


19.107 


.05241 


19.081 


1.0014 


.00137 


.99863 


60 


1 


. 5263 


. 4737 


9.002 


. 5270 


8.975 


.0014 


. 0138 


. 9861 


59 


2 


. 5292 


. 4708 


8.897 


. 5299 


8.871 


.0014 


. 0140 


. 9860 


58 


3 


. 5321 


. 4679 


8.794 


. 5328 


8.768 


.0014 


. 0142 


. 9858 


57 


4 


. 5350 


. 4650 


8.692 


. 5357- 


8.665 


.0014 


. 0143 


. 9857 


56 


5 


.05379 


.94621 


18.591 


.05387 


18.564 


1.0014 


.00145 


.99855 


55 


6 


. 5408 


, 4592 


8.491 


. 5416 


8.464 


.0015 . 


. 0146 


. 9854 


54 


7 


. 5437 


. 4563 


8.393 


. 5445 


8.365 


.0015 


. 0148 


. 9852 


53 


8 


. 5466 


. 4534 


8.295 


. 5474 


8.268 


.0015 


. 0149 


. 9850 


52 


9 


. 5495 


. 4505 


8.198 


. 5503 


8.171 


.0015 


. 0151 


. 9849 


51 


10 


.05524 


.94476 


18.103 


.05532 


18.075 


1.0015 


.00153 


.99847 


50 


11 


. 5553 


. 4447 


8.008 


. 5562 


7.980 


.0015 


. 0154 


. 9846 


49 


12 


. 5582 


. 4418 


7.914 


. 5591 


7.886 


.0016 


. 0156 


. 9844 


48 


13 


. 5611 


. 4389 


7.821 


. 5620 


7.793 


.0016 


. 0157 


. 9842 


47 


14 


. 5640 


. 4360 


7.730 


. 5649 


7.701 


.0016 


. 0159 


. 9841 


46 


15 


.05669 


.94331 


17.639 


.05678 


17.610 


1.0016 


.00161 


.99839 


45 


16 


. 5698 


. 4302 


7.549 


. 5707 


7.520 


.0016 


. 0162 


. 9837 


44 


17 


. 5727 


. 4273 


7.460 


. 5737 


7.431 


.0016 


. 0164 


. 9836 


43 


18 


. 5756 


. 4244 


7.372 


. 5766 


7.343 


.0017 


. 0166 


. 9834 


42 


19 


. 5785 


. 4214 


7.285 


. 5795 


7.256 


.0017 


. 0167 


. 9832 


41 


20 


.05814 


.94185 


17.198 


.05824 


17.169 


1.0017 


.00169 


.99831 


40 


21 


. 5843 


. 4156 


7.113 


. 5853 


7.084 


.0017 


. 0171 


. 9829 


39 


22 


. 5872 


. 4127 


7.028 


. 5883 


6.999 


.0017 


. 0172 


. 9827 


38 


23 


. 5902 


. 4098 


6.944 


. 5912 


6.915 


.0017 


. 0174 


. 9826 


37 


24 


. 5931 


. 4069 


6.861 


. 5941 


6.832 


.0018 


. 0176 


. 9824 


36 


25 


.05960 


.94040 


16.779 


.05970 


16.750 


1.0018 


.00178 


.99822 


35 


26 


. 5989 


. 4011 


6.698 


. 5999 


6.668 


.0018 


. 0179 


. 9820 


34 


27 


. 6018 


. 3982 


6.617 


. 6029 


6.587 


.0018 


. 0181 


. 9819 


33 


28 


. 6047 


. 3953 


6.538 


. 6058 


6.507 


.0018 


. 0183 


. 9817 


32 


29 


. 6076 


. 3924 


6.459 


. 6087 


6.428 


.0018 


. 0185 


. 9815 


31 


30 


.06105 


.93895 


16.380 


.06116 


16.350 


1.0019 


.00186 


.99813 


30 


31 


. 6134 


. 3866 


6.303 


. 6145 


6.272 


.0019 


. 0188 


. 9812 


29 


32 


. 6163 


. 3837 


6.226 


. 6175 


6.195 


.0019 


. 0190 


. 9810 


28 


33 


. 6192 


. 3808 


6.150 


. 6204 


6.119 


.0019 


. 0192 


. 9808 


27 


34 


. 6221 


. 3777 


6.075 


. 6233 


6.043 


.0019 


. 0194 


. 9806 


26 


35 


.06250 


.93750 


16.000 


.06262 


15.969 


1.0019 


.00195 


.99804 


25 


36 


. 6279 


. 3721 


5.926 


. 6291 


5.894 


.0020 


. 0197 


. 9803 


24 


37 


. 6308 


. 3692 


5.853 


. 6321 


5.821 


.0020 


. 0199 


. 9801 


23 


38 


. 6337 


. 3663 


5.780 


. 6350 


5.748 


.0020 


. 0201 


. 9799 


22 


39 


. 6366 


. 3634 


5.708 


. 6379 


5.676 


.0020 


. 0203 


. 9797 


21 


40 


.06395 


.93605 


15.637 


.06408 


15.605 


1.0020 


.00205 


.99795 


20 


41 


. 6424 


. 3576 


5.566 


. 6437 


5.534 


.0021 


. 0206 


. 9793 


19 


42 


. 6453 


. 3547 


5.496 


. 6467 


5.464 


.0021 


. 0208 


. 9791 


18 


43 


. 6482 


. 3518 


5.427 


. 6496 


5.394 


.0021 


. 0210 


. 9790 


17 


44 


. 6511 


. 3489 


5.358 


. 6525 


5.325 


.0021 


. 0212 


. 9788 


16 


45 


.06540 


.93460 


15.290 


.06554 


15.257 


1.0021 


.00214 


.99786 


15 


46 


. 6569 


. 3431 


5.222 


. 6583 


5.189 


.0022 


. 0216 


. 9784 


14 


47 


. 6598 


. 3402 


5.155 


. 6613 


5.122 


.0022 


. 0218 


. 9782 


13 


48 


. 6627 


. 3373 


5.089 


. 6642 


5.056 


.0022 


. 0220 


. 9780 


12 


49 


. 6656 


. 3343 


5.023 


. 6671 


4.990 


.0022 


. 0222 


. 9778 


11 


50 


.06685 


.93314 


14.958 


.06700 


14.924 


1.0022 


.00224 


.99776 


10 


51 


. 6714 


. 3285 


4.893 


. 6730 


4.860 


.0023 


. 0226 


. 9774 


9 


52 


. 6743 


. 3256 


4.829 


. 6759 


4.795 


.0023 


. 0228 


. 9772 


8 


53 


. 6772 


. 3227 


4.765 


. 6788 


4.732 


.0023 


. 0230 


. 9770 


7 


54 


. 6801 


. 3198 


4.702 


. 6817 


4.668 


.0023 


. 0231 


. 9768 


6 


55 


.06830 


.93169 


14.640 


.06846 


14.606 


1.0023 


.00233 


.99766 


5 


56 


. 6859 


. 3140 


4.578 


. 6876 


4.544 


.0024 


. 0235 


. 9764 


4 


57 


. 6888 


. 3111 


4.517 


. 6905 


4.482 


.0024 


. 0237 


. 9762 


3 


58 


. 6918 


. 3082 


4.456 


. 6934 


4.421 


.0024 


. 0239 


. 9760 


2 


59 


. 6947 


. 3053 


4.395 


. 6963 


4.361 


.0024 


. 0241 


. 9758 


1 


60 


. 6976 


. 3024 


4.335 


. 6993 


4.301 


.0024 


. 0243 


. 9756 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 





93° 



86° 



138 



Natural Functions. 



4° 




Natural Trigonometrical Functions. 


i: 


^5° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 
1.0024 


Vrs. sin. 


Cosine. 


M. 





.06976 


.93024 


14.335 


.06993 


14.301 


.00243 


.99756 


60 


1 


. 7005 


. 2995 


4.276 


. 7022 


4.241 


.0025 


. 0246 


. 9754 


59 


2 


. 7034 


. 2966 


4.217 


. 7051 


4.182 


.0025 


. 0248 


. 9752 


58 


3 


. 7063 


. 2937 


4.159 


. 7080 


4.123 


.0025 


. 0250 


. 9750 


57 


4 


. 7092 


. 2908 


4.101 


. 7110 


4.065 


.0025 


. 0252 


. 9748 


56 


5 


.07121 


.92879 


14.043 


.07139 


14.008 


1.0025 


.00254 


.99746 


55 


6 


. 7150 


. 2850 


3.986 


. 7168 


3.951 


.0026 


. 0256 


. 9744 


54 


7 


. 7179 


. 2821 


3.930 


. 7197 


3.894 


.0026 


. 0258 


. 9742 


53 


8 


. 7208 


. 2792 


3.874 


. 7226 


3.838 


.0026 


. 0260 


. 9740 


52 


9 


. 7237 


. 2763 


3.818 


. 7256 


3.782 


.0026 


. 0262 


. 9738 


51 


10 


.07266 


.92734 


13.763 


.07285 


13.727 


1.0026 


.00264 


.99736 


50 


11 


. 7295 


. 2705 


3.708 


. 7314 


3.672 


.0027 


. 0266 


. 9733 


49 


12 


. 7324 


. 2676 


3.654 


. 7343 


3.617 


.0027 


. 0268 


. 9731 


48 


13 


. 7353 


. 2647 


3.600 


. 7373 


3.563 


.0027 


. 0271 


. 9729 


47 


14 


. 7382 


. 2618 


3.547 


. 7402 


3.510 


.0027 


. 0273 


. 9727 


46 


15 


.07411 


.92589 


13.494 


.07431 


13.457 


1.0027 


.00275 


.99725 


45 


16 


. 7440 


. 2560 


3.441 


. 7460 


3.404 


.0028 


. 0277 


. 9723 


44 


17 


. 7469 


. 2531 


3.389 


. 7490 


3.351 


.0028 


. 0279 


. 9721 


43 


18 


. 7498 


. 2502 


3.337 


. 7519 


3.299 


.0028 


. 0281 


. 9718 


42 


19 


. 7527 


. 2473 


3.286 


. 7548 


3.248 


.0028 


. 0284 


. 9716 


41 


20 


.07556 


.92444 


13.235 


.07577 


13.197 


1.0029 


.00286 


.99714 


40 


21 


. 7585 


. 2415 


3.184 


. 7607 


3.146 


.0029 


. 0288 


. 9712 


39 


22 


. 7614 


. 2386 


3.134 


. 7636 


3.096 


.0029 


. 0290 


. 9710 


38 


23 


. 7643 


. 2357 


3.084 


. 7665 


3.046 


.0029 


. 0292 


. 9707 


37 


24 


. 7672 


. 2328 


3.034 


. 7694 


2.996 


.0029 


. 0295 


. 9705 


36 


25 


.07701 


.92299 


12.985 


.07724 


12.947 


1.0030 


.00297 


.99703 


35 


26 


. 7730 


. 2270 


2.937 


. 7753 


2.898 


.0030 


. 0299 


. 9701 


34 


27 


. 7759 


. 2241 


2.888 


. 7782 


2.849 


.0030 


. 0301 


. 9698 


33 


28 


. 7788 


. 2212 


2.840 


. 7812 


2.801 


.0030 


. 0304 


. 9696 


32 


29 


. 7817 


. 2183 


2.793 


. 7841 


2.754 


.0031 


. 0306 


. 9694 


31 


30 


.07846 


.92154 


12.745 


.07870 


12.706 


1.0031 


.00308 


.99692 


30 


31 


. 7875 


. 2125 


2.698 


. 7899 


2.659 


.0031 


. 0310 


. 9689 


29 


32 


. 7904 


. 2096 


2.652 


. 7929 


2.612 


.0031 


. 0313 


. 9687 


28 


33 


. 7933 


. 2067 


2.606 


. 7958 


2.566 


.0032 


. 0315 


. 9685 


27 


34 


. 7962 


. 2038 


2.560 


. 7987 


2.520 


.0032 


. 0317 


. 9682 


26 


35 


.07991 


.92009 


12.514 


.08016 


12.474 


1.0032 


.00320 


.99680 


25 


36 


. 8020 


. 1980 


2.469 


. 8046 


2.429 


.0032 


. 0322 


. 9678 


24 


37 


. 8049 


. 1951 


2.424 


. 8075 


2.384 


.0032 


. 0324 


. 9675 


23 


38 


. 8078 


. 1922 


2.379 


. 8104 


2.339 


.0033 


. 0327 


. 9673 


22 


39 


. 8107 


. 1893 


2.335 


. 8134 


2.295 


.0033 


. 0329 


. 9671 


21 


40 


.08136 


.91864 


12.291 


.08163 


12.250 


1.0033 


.00331 


.99668 


20 


41 


. 8165 


. 1835 


2.248 


. 8192 


2.207 


.0033 


. 0334 


. 9666 


19 


42 


. 8194 


. 1806 


2.204 


. 8221 


2.163 


.0034 


. 0336 


. 9664 


18 


43 


. 8223 


. 1777 


2.161 


. 8251 


2.120 


.0034 


. 0339 


. 9661 


17 


44 


. 8252 


. 1748 


2.118 


. 8280 


2.077 


.0034 


. 0341 


. 9659 


16 


45 


.08281 


.91719 


12.076 


.0S309 


12.035 


1.0034 


.00343 


.99656 


15 


46 


. 8310 


. 1690 


2.034 


. 8339 


1.992 


.0035 


. 0346 


. 9654 


14 


47 


. 8339 


. 1661 


1.992 


. 8368 


1.950 


.0035 


. 0348 


. 9652 


13 


48 


. 8368 


. 1632 


1.950 


. 8397 


1.909 


.0035 


. 0351 


. 9649 


12 


49 


. 8397 


. 1603 


1.909 


. 8426 


1.867 


.0035 


. 0353 


. 9647 


11 


50 


.08426 


.91574 


11.868 


.08456 


11.826 


1.0036 


.00356 


.99644 


10 


51 


. 8455 


. 1545 


1.828 


. 8485 


1.785 


.0036 


. 0358 


. 9642 


9 


52 


. 8484 


. 1516 


1.787 


. 8514 


1.745 


.0036 


. 0360 


. 9639 


8 


53 


. 85 13 


. 1487 


1.747 


. 8544 


1.704 


.0036 


. 0363 


. 9637 


7 


54 


. 8542 


. 1458 


1.707 


. 8573 


1.664 


.0037 


. 0365 


. 9634 


6 


55 


.08571 


.91429 


11.668 


.08602 


11.625 


1.0037 


.00368 


.99632 


5 


56 


. 8600 


. 1400 


1.628 


. 8632 


1.585 


.0037 


. 0370 


. 9629 


4 


57 


. 8629 


. 1371 


1.589 


. 8661 


1.546 


.0037 


. 0373 


. 9627 


3 


58 


. 8658 


. 1342 


1.550 


. 8690 


1.507 


.0038 


. 0375 


. 9624 


2 


59 


. 8687 


. 1313 


1.512 


. 8719 


1.468 


.0038 


. 0378 


. 9622 


1 


60 


. 8715 


. 1284 


1.171 


. 8749 


1.430 


.0038 


. 0380 


. 9619 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Co tang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



i° 



85° 



Natural Functions. 



139 



5° 




Natural Trigonometrical Functions. 


174° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.08715 


.91284 


11.474 


.08749 


11.430 


1.0038 


.00380 


.99619 


60 


1 


. 8744 


. 1255 


1.436 


. 8778 


1.392 


.0038 


. 0383 


. 9617 


59 


2 


. 8773 


. 1226 


1.398 


. 8807 


1.354 


.0039 


. 0386 


. 9614 


58 


3 


. 8802 


. 1197 


1.360 


. 8837 


1.316 


.0039 


. 0388 


. 9612 


57 


4 


. 8831 


. 1168 


1.323 


. 8866 


1.279 


.0039 


. 0391 


. 9609 


56 


5 


.08860 


.91139 


11.286 


.08895 


11.242 


1.0039 


.00393 


.99607 


55 


6 


. 8889 


. 1110 


1.249 


. 8925 


1.205 


.0040 


. 0396 


. 9604 


54 


7 


. 8918 


. 1082 


1.213 


. 8954 


1.168 


.0040 


. 0398 


. 9601 


53 


8 


. 8947 


. 1053 


1.176 


. 8983 


1.132 


.0040 


. 0401 


. 9599 


52 


9 


. 8976 


. 1024 


1.140 


. 9013 


1.095 


.0040 


. 0404 


. 9596 


51 


10 


.09005 


.90995 


11.104 


.09042 


11.059 


1.0041 


.00406 


.99594 


50, 


11 


. 9034 


. 0966 


1.069 


. 9071 


1.024 


.0041 


. 0409 


. 9591 


49 


12 


. 9063 


. 0937 


1.033 


. 9101 


0.988 


.0041 


. 0411 


. 9588 


48 


13 


. 9092 


. 0908 


0.998 


. 9130 


0.953 


.0041 


. 0414 


. 9586 


47 


14 


. 9121 


. 0879 


0.963 


. 9159 


0.918 


.0042 


. 0417 


. 9583 


46 


15 


.09150 


.90850 


10.929 


.09189 


10.883 


1.0042 


.00419 


.99580 


45 


16 


. 9179 


. 0821 


0.894 


. 9218 


0.848 


.0042 


. 0422 


. 9578 


44 


17 


. 9208 


. 0792 


0.860 


. 9247 


0.814 


.0043 


. 0425 


. 9575 


43 


18 


. 9237 


. 0763 


0.826 


. 9277 


0.780 


.0043 


. 0427 


. 9572 


42 


19 


. 9266 


. 0734 


0.792 


. 9306 


0.746 


.0043 


. 0430 


. 9570 


41 


20 


.09295 


.90705 


10.758 


.09335 


10.712 


1.0043 


.00433 


.99567 


40 


21 


. 9324 


. 0676 


0.725 


. 9365 


0.678 


.0044 


. 0436 


. 9564 


39 


22 


. 9353 


. 0647 


0.692 


. 9394 


0.645 


.0044 


. 0438 


. 9562 


38 


23 


. 9382 


. 0618 


0.659 


. 9423 


0.612 


.0044 


. 0441 


. 9559 


37 


24 


. 9411 


. 0589 


0.626 


. 9453 


0.579 


.0044 


. 0444 


. 9556 


36 


25 


.09440 


.90560 


10.593 


.09482 


10.546 


1.0045 


.00446 


.99553 


35 


26 


. 9469 


. 0531 


0.561 


. 9511 


0.514 


.0045 


. 0449 


. 9551 


34 


27 


. 9498 


. 0502 


0.529 


. 9541 


0.481 


.0045 


. 0452 


. 9548 


33 


28 


. 9527 


. 0473 


0.497 


. 9570 


0.449 


.0046 


. 0455 


. 9545 


32 


29 


. 9556 


. 0444 


0.465 


. 9599 


0.417 


.0046 


. 0458 


. 9542 


31 


30 


.09584 
. 9613 


.90415 
. 0386 


10.433 
0.402 


.09629 


10.385 


1.0046 
.0046 


.00460 
. 0463 


.99540 
. 9537 


30 


31 


. 9658 


0.354 


29 


32 


. 9642 


. 0357 


0.371 


. 9688 


0.322 


.0047 


. 0466 


. 9534 


28 


33 


. 9671 


. 0328 


0.340 


. 9717 


0.291 


.0047 


. 0469 


. 9531 


27 


34 


. 9700 


. 0300 


0.309 


. 9746 


0.260 


.0047 


. 0472 


. 9528 


26 


35 


.09729 


.90271 


10.278 


.09776 


10.229 


1.0048 


.00474 


.99525 


25 


36 


. 9758 


. 0242 


0.248 


. 9805 


0.199 


.0048 


. 0477 


. 9523 


24 


37 


. 9787 


. 0213 


0.217 


. 9834 


0.168 


.0048 


. 0480 


. 9520 


23 


38 


. 9816 


. 0184 


0.187 


. 9864 


0.138 


.0048 


. 0483 


. 9517 


22 


39 


. 9845 


. 0155 


0.157 


. 9893 


0.108 


.0049 


. 0486 


. 9514 


21 


40 


.09874 


.90126 


10.127 


.09922 


10.078 


1.0049 


.00489 


.99511 


20 


41 


. 9903 


. 0097 


0.098 


. 9952 


0.048 


.0049 


. 0491 


. 9508 


19 


42 


. 9932 


. 0068 


0.068 


. 9981 


0.019 


.0050 


. 0494 


. 9505 


18 


43 


. 9961 


. 0039 


0.039 


.10011 


9.9893 


.0050 


. 0497 


. 9503 


17 


44 


. 9990 


. 0010 


0.010 


.0040 


.9601 


.0050 


. 0500 


. 9500 


16 


45 


.10019 


.89981 


9.9812 


.10069 


9.9310 


1.0050 


.00503 


.99497 


15 


46 


. 0048 


. 9952 


.9525 


. 0099 


.9021 


.0051 


. 0506 


. 9494 


14 


47 


. 0077 


. 9923 


.9239 


. 0128 


.8734 


.0051 


. 0509 


. 9491 


13 


48 


. 0106 


. 9894 


.8955 


. 0158 


.8448 


.0051 


. 0512 


. 9488 


12 


49 


. 0134 


. 9865 


.8672 


. 0187 


.8164 


.0052 


. 0515 


. 9485 


11 


50 


.10163 


.89836 


9.8391 


.10216 


9.7882 


1.0052 


.00518 


.99482 


10 


51 


. 0192 


. 9807 


.8112 


. 0246 


.7601 


.0052 


. 0521 


. 9479 


9 


52 


. 0221 


. 9779 


.7834 


. 0275 


.7322 


.0053 


. 0524 


. 9476 


8 


53 


. 0250 


. 9750 


.7558 


. 0305 


.7044 


.0053 


. 0527 


. 9473 


7 


54 


. 0279 


. 9721 


.7283 


. 0334 


.6768 


.0053 


. 0530 


. 9470 


6 


55 


.10308 


.89692 


9.7010 


.10363 


9.6493 


1.0053 


.00533 


.99467 


5 


56 


. 0337 


. 9663 


.6739 


. 0393 


.6220 


.0054 


. 0536 


. 9464 


4 


57 


. 0366 


. 9634 


.6469 


. 0422 


.5949 


.0054 


. 0539 


. 9461 


3 


58 


. 0395 


. 9605 


.6200 


, 0452 


.5679 


.0054 


. 0542 


. 9458 


2 


59 


. 0424 


. 9576 


.5933 


. 0481 


.5411 


.0055 


. 0545 


. 9455 


1 


60 


. 0453 


. 9547 


.5668 


. 0510 


.5144 


.0055 


. 0548 


. 9452 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Co tang. 


1 Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



140 



Natural Functions. 



6° 



Natural Trigonometrical Functions. 



173° 



M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Co tang. 


Secant. 


Vrs. siu. 


Cosine. 


M. 





.10453 


.89547 


9.5668 


.10510 


9.5144 


1.0055 


.00548 


.99452 


60 


1 


. 0482 


. 9518 


.5404 


. 0540 


.4878 


.0055 


. 0051 


. 9449 


59 


2 


. 0511 


. 9489 


.5141 


. 0569 


.4614 


.0056 


. 0t>54 


. 9446 


58 


3 


. 0540 


. 9460 


.4880 


. 0599 


.4351 


.0056 


. 0o57 


. 9443 


57 


4 


. 0568 


. 9431 


.4620 


. 0628 


.4090 


.0056 


. 0o60 


. 9440 


:->6 


5 


.10597 


.89402 


9.4362 


.10657 


9.3831 


1.0057 


.00o63 


.99437 


55 


6 


. 0626 


. 9373 


.4105 


. 0687 


.3572 


.0057 


. 0566 


. 9434 


54 


7 


. 0655 


. 9345 


.3850 


. 0716 


.3315 


.0057 


. 0569 


. 9431 


53 


8 


. 0684 


. 9316 


.3596 


. 0746 


.3060 


.0057 


. 0572 


. 9428 


52 


9 


. 0713 


. 9287 


.3343 


. 0775 


.2806 


.0058 


. 0575 


. 9424 


51 


10 


.10742 


.89258 


9.3092 


.10805 


9.2553 


1.0058 


.00579 


.99421 


50 


11 


. 0771 


. 9229 


.2842 


. 0834 


.2302 


.0058 


. 0582 


. 9418 


49 


12 


. 0800 


. 9200 


.2593 


. 0863 


.2051 


.0059 


. 0585 


. 9415 


48 


13 


. 0829 


. 9171 


.2346 


. 0893 


.1803 


.0059 


. 0588 


. 9412 


47 


14 


. 0858 


. 9142 


.2100 


.0922 


.1555 


.0059 


. 0591 


. 9409 


46 


15 


.10887 


.89113 


9.1855 


.10952 


9.1309 


1.0060 


.00594 


.99406 


45 


16 


. 0916 


. 9084 


.1612 


. 0981 


.1064 


.0060 


. 0597 


. 9402 


44 


17 


. 0944 


. 9055 


.1370 


. 1011 


.0821 


.0060 


. 0601 


. 9399 


43 


18 


. 0973 


. 9026 


.1129 


. 1040 


.0579 


.0061 


. 0604 


. 9396 


42 


19 


. 1002 


. 8998 


.0890 


. 1069 


.0338 


.0061 


. 0607 


. 9393 


41 


20 


.11031 


.88969 


9.0651 


.11099 


9.0098 


1.0061 


.00610 


.99390 


40 


21 


. 1060 


. 8940 


.0414 


. 1128 


8.9860 


.0062 


. 0613 


. 9386 


39 


22 


. 1089 


. 8911 


.0179 


. 1158 


.9623 


.0062 


. 0617 


. 9383 


38 


23 


. 1118 


. 8882 


8.9944 


. 1187 


.9387 


.0062 


. 0620 


. 9380 


37 


24 


. 1147 


. 8853 


.9711 


. 1217 


.9152 


.0063 


. 0623 


. 9377 


36 


25 


.11176 


.88824 


8.9479 


.11246 


8.8918 


1.0063 


.00626 


.99373 


35 


26 


. 1205 


. 8795 


.9248 


. 1276 


.8686 


.0063 


. 0630 


. 9370 


34 


27 


. 1234 


. 8766 


.9018 


. 1305 


.8455 


.0064 


. 0633 


. 9367 


33 


28 


. 1262 


. 8737 


.8790 


. 1335 


.8225 


.0064 


. 0636 


. 9364 


32 


29 


. 1291 


. 8708 


.8563 


. 1364 


.7996 


.0064 


. 0639 


. 9360 


31 


30 


.11320 


.88680 


8.8337 


.11393 


8.7769 


1.0065 


.00643 


.99357 


30 


31 


. 1349 


. 8651 


.8112 


. 1423 


.7542 


.0065 


. 0646 


. 9354 


29 


32 


. 1378 


. 8622 


.7888 


. 1452 


.7317 


.0065 


. 0649 


. 9350 


28 


33 


. 1407 


. 8593 


.7665 


. 1482 


.7093 


.0066 


. 0653 


. 9347 


27 


34 


. 1436 


. 8564 


.7444 


. 1511 


.6870 


.0066 


. 0656 


. 9344 


26 


35 


.11465 


.88535 


8.7223 


.11541 


8.6648 


1.0066 


.00659 


.99341 


25 


36 


.1494 


. 8506 


.7004 


. 1570 


.6427 


.0067 


. 0663 


. 9337 


24 


37 


. 1523 


. 8477 


.6786 


. 1600 


.6208 


.0067 


. 0666 


. 9334 


23 


38 


. 1551- 


. 8448 


.6569 


. 1629 


.5989 


.0067 


. 0669 


. 9330 


22 


39 


. 1580 


. 8420 


.6353 


. 1659 


.5772 


.0068 


. 0673 


. 9327 


21 


40 


.11609 


.88391 


8.6138 


.11688 


8.5555 


1.0068 


.00676 


.99324 


20 


41 


. 1638 


. 8362 


.5924 


. 1718 


.5340 


.0068 


. 0679 


. 9320 


19 


42 


. 1667 


. 8333 


.5711 


. 1747 


.5126 


.0069 


. 0683 


. 9317 


18 


43 


. 1696 


. 8304 


.5499 


. 1777 


.4913 


.0069 


. 0686 


. 9314 


17 


44 


. 1725 


. 8272 


.5289 


. 1806 


.4701 


.0069 


. 0690 


. 9310 


16 


45 


.11754 


.88246 


8.5079 


.11836 


8.4489 


1.0070 


.00693 


.99307 


15 


46 


. 1783 


. 8217 


.4871 


. 1865 


.4279 


.0070 


. 0696 


. 9303 


14 


47 


. 1811 


. 8188 


.4663 


. 1895 


.4070 


.0070 


. 0700 


. 9300 


13 


48 


. 1840 


. 8160 


.4457 


. 1924 


.3862 


.0071 


. 0703 


. 9296 


12 


49 


. 1869 


. 8131 


.4251 


. 1954 


.3655 


.0071 


. 0707 


. 9293 


11 


50 


.11898 


.88102 


8.4046 


.11983 


8.3449 


1.0071 


.00710 


.99290 


10 


51 


. 1927 


. 8073 


.3843 


. 2013 


.3244 


.0072 


. 0714 


. 9286 


9 


52 


. 1956 


. 8044 


.3640 


. 2042 


.3040 


.0072 


. 0717 


. 9283 


8 


53 


. 1985 


. 8015 


.3439 


. 2072 


.2837 


.0073 


. 0721 


. 9279 


7 


54 


. 2014 


. 7986 


.3238 


. 2101 


.2635 


.0073 


. 0724 


. 9276 


6 


55 


,12042 


.87957 


8.3039 


12131 


8.2434 


1.0073 


.00728 


.99272 


5 


56 


. 2071 


. 7928 


.2840 


. 2160 


.2234 


.0074 


. 0731 


. 9269 


4 


57 


. 2100 


. 7900 


.2642 


. 2190 


.2035 


.0074 


. 0735 


. 9265 


3 


58 


. 2129 


. 7871 


.2446 


. 2219 


.1837 


.0074 


. 0738 


. 9262 


2 


59 


. 2158 


. 7842 


.2250 


. 2219 


.1640 


.0075 


. 0742 


. 9258 


1 


60 


. 2 1ST 


. 7813 


.2055 


. 2278 


.1443 


.0075 


. 0745 


. 9255 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



96° 



83° 



Natural Functions. 



141 



7° 




Natural Trigonometrical Functions. 


172° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.12187 


.87813 


8.2055 


.12278 


8.1443 


1.0075 


.00745 


.99255 


60 


1 


. 2216 


. 7787 


.1861 


. 2308 


.1248 


.0075 


. 0749 


. 9251 


59 


2 


. 2245 


. 7755 


.1668 


. 2337 


.1053 


.0076 


. 0752 


. 9247 


58 


3 


. 2273 


. 7726 


.1476 


. 2367 


.0860 


.0076 


. 0756 


. 9244 


57 


4 


. 2302 


. 7697 


.1285 


. 2396 


.0667 


.0076 


. 0760 


. 9240 


56 


5 


.12331 


.87669 


8.1094 


.12426 


8.0476 


1.0077 


.00763 


.99237 


55 


6 


. 2360 


. 7640 


.0905 


. 2456 


.0285 


.0077 


. 0767 


. 9233 


54 


7 


. 2389 


. 7611 


.0717 


. 2485 


.0095 


.0078 


. 0770 


. 9229 


53 


8 


. 2418 


. 7582 


.0529 


. 2515 


7.9906 


.0078 


. 0774 


. 9226 


52 


9 


. 2447 


. 7553 


.0342 


. 2544 


.9717 


.0078 


. 0778 


. 9222 


51 


10 


.12476 


.87524 


8.0156 


.12574 


7.9530 


1.0079 


.00781 


.99219 


50 


11 


. 2504 


. 7495 


7.9971 


. 2603 


.9344 


.0079 


. 0785 


. 9215 


49 


12 


. 2533 


. 7467 


.9787 


. 2633 


.9158 


.0079 


. 0788 


. 9211 


48 


13 


. 2562 


. 7438 


.9604 


. 2662 


.8973 


.0080 


. 0792 


. 9208 


47 


14 


. 2591 


. 7409 


.9421 


. 2692 


.8789 


.0080 


. 0796 


. 9204 


46 


15 


.12620 


.87380 


7.9240 


.12722 


7.8606 


1.0080 


.00799 


.99200 


45 


16 


. 2649 


. 7351 


.9059 


. 2751 


.8424 


.0081 


. 0803 


. 9197 


44 


17 


. 2678 


. 7322 


.8879 


. 2781 


.8243 


.0081 


. 0807 


. 9193 


43 


18 


. 2706 


. 7293 


.8700 


. 2810 


.8062 


.0082 


. 0810 


. 9189 


42 


19 


. 2735 


. 7265 


.8522 


. 2840 


.7882 


.0082 


. 0814 


. 9186 


41 


20 


.12764 


.87236 


7.8344 


.12869 


7.7703 


1.0082 


.00818 


.99182 


40 


21 


. 2793 


. 7207 


.8168 


. 2899 


.7525 


.0083 


. 0822 


. 9178 


39 


22 


. 2822 


. 7178 


.7992 


. 2928 


.7348 


.0083 


. 0825 


. 9174 


38 


23 


. 2851 


. 7149 


.7817 


. 2958 


.7171 


.0084 


. 0829 


. 9171 


37 


24 


. 2879 


. 7120 


.7642 


. 2988 


.6996 


.0084 


. 0833 


. 9167 


36 


25 


.12908 


.87091 


7.7469 


.13017 


7.6821 


1.0084 


.00837 


.99163 


35 


26 


. 2937 


. 7063 


.7296 


. 3047 


.6646 


.0085 


. 0840 


. 9160 


34 


27 


. 2966 


. 7034 


.7124 


. 3076 


.6473 


.0085 


. 0844 


. 9156 


33 


28 


. 2995 


. 7005 


.6953 


. 3106 


.6300 


.0085 


. 0848 


. 9152 


32 


29 


. 3024 


. 6976 


.6783 


. 3136 


.6129 


.0086 


. 0852 


. 9148 


31 


30 


.13053 


.86947 


7.6613 


.13165 


7.5957 


1.0086 


.00855 


.99144 


30 


31 


. 3081 


. 6918 


.6444 


. 3195 


.5787 


.0087 


. 0859 


. 9141 


29 


32 


. 3110 


. 6890 


.6276 


. 3224 


.5617 


.0087 


. 0863 


. 9137 


28 


33 


. 3139 


. 6861 


.6108 


. 3254 


.5449 


.0087 


. 0867 


. 9133 


27 


34 


. 3168 


. 6832 


.5942 


. 3284 


.5280 


.0088 


. 0871 


. 9129 


26 


35 


.13197 


.86803 


7.5776 


.13313 


7.5113 


1.0088 


.00875 


.99125 


25 


36 


. 3226 


. 6774 


.5611 


. 3343 


.4946 


.0089 


. 0878 


. 9121 


24 


37 


. 3254 


. 6745 


.5446 


. 3372 


.4780 


.0089 


. 0882 


. 9118 


23 


38 


. 3283 


. 6717 


.5282 


. 3402 


.4615 


.0089 


. 0886 


. 9114 


22 


39 


. 3312 


. 6688 


.5119 


. 3432 


.4451 


.0090 


. 0890 


. 9110 


21 


40 


.13341 


.86659 


7.4957 


.13461 


7.4287 


1.0090 


.00894 


.99106 


20 


41 


. 3370 


. 6630 


.4795 


. 3491 


.4124 


.0090 


. 0898 


. 9102 


19 


42 


. 3399 


. 6601 


.4634 


. 3520 


.3961 


.0091 


. 0902 


. 9098 


18 


43 


. 3427 


. 6572 


.4474 


. 3550 


.3800 


.0091 


. 0905 


. 9094 


17 


44 


. 3456 


. 6544 


.4315 


. 3580 


.3639 


.0092 


. 0909 


. 9090 


16 


45 


.13485 


.86515 


7.4156 


.13609 


7.3479 


1.0092 


.00913 


.99086 


15 


46 


. 3514 


. 6486 


.3998 


. 3639 


.3319 


.0092 


. 0917 


. 9083 


14 


47 


. 3543 


. 6457 


.3840 


. 3669 


.3160 


.0093 


. 0921 


. 9079 


13 


48 


. 3571 


. 6428 


.3683 


. 3698 


.3002 


.0093 


. 0925 


. 9075 


12 


49 


. 3600 


. 6400 


.3527 


. 3728 


.2844 


.0094 


. 0929 


. 9070 


11 


50 


.13629 


.86371 


7.3372 


.13757 


7.2687 


1.0094 


.00933 


.99067 


10 


51 


. 3658 


. 6342 


.3217 


. 3787 


.2531 


.0094 


. 0937 


. 9063 


9 


52 


. 3687 


. 6313 


.3063 


. 3817 


.2375 


.0095 


. 0941 


. 9059 


8 


53 


. 3716 


. 6284 


.2909 


. 3846 


.2220 


.0095 


. 0945 


. 9055 


7 


54 


. 3744 


. 6255 


.2757 


. 3876 


.2066 


.0096 


. 0949 


. 9051 


6 


55 


.13773 


.86227 


7.2604 


.13906 


7.1912 


1.0096 


.00953 


.99047 


5 


56 


. 3802 


. 6198 


.2453 


. 3935 


.1759 


.0097 


. 0957 


. 9043 


4 


57 


. 3831 


. 6169 


.2302 


. 3965 


.1607 


.0097 


. 0961 


. 9039 


3 


58 


. 3860 


. 6140 


.2152 


. 3995 


.1455 


.0097 


. 0965 


. 9035 


2 


59 


. 3888 


. 6111 


.2002 


. 4024 


.1304 


.0098 


. 0969 


. 9031 


1 


60 


. 3917 


. 6083 


.1853 


. 4054 


.1154 


.0098 


. 0973 


. 9027 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



142 



Natural Functions. 



8° 




Natural Trigonometrical Functions. 


171° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.13917 


.86083 


7.1853 


.14054 


7.1154 


1.0098 


.00973 


.99027 


60 


1 


. 3946 


. 6054 


.1704 


. 4084 


.1004 


.0099 


. 0977 


. 9023 


59 


2 


. 3975 


. 6025 


.1557 


. 4113 


.0854 


.0099 


. 0981 


. 9019 


58 


3 


. 4004 


. 5996 


.1409 


. 4143 


.0706 


.0099 


. 0985 


. 9015 


57 


4 


. 4032 


. 5967 


.1263 


. 4173 


.0558 


.0100 


. 0989 


. 9010 


56 


5 


.14061 


.85939 


7.1117 


.14202 


7.0410 


1.0100 


.00993 


.99006 


55 


6 


. 4090 


. 5910 


.0972 


. 4232 


.0264 


.0101 


. 0998 


. 9002 


54 


7 


. 4119 


. 5881 


.0827 


. 4262 


.0117 


.0101 


. 1002 


. 8998 


53 


8 


. 4148 


. 5852 


.0683 


. 4291 


6.9972 


.0102 


. 1006 


. 8994 


52 


9 


. 4176 


. 5823 


.0539 


. 4321 


.9827 


.0102 


. 1010 


. 8990 


51 


10 


.14205 


.85795 


7.0396 


.14351 


6.9682 


1.0102 


.01014 


.98986 


50 


11 


. 4234 


. 5766 


.0254 


. 4380 


.9538 


.0103 


. 1018 


. 8^82 


49 


12 


. 4263 


. 5737 


.0112 


. 4410 


.9395 


.0103 


. 1022 


. 8978 


48 


13 


. 4292 


. 5708 


6.9971 


. 4440 


.9252 


.0104 


. 1026 


. 8973 


47 


14 


. 4320 


. 5679 


.9830 


. 4470 


.9110 


.0104 


. 1031 


. 8969 


46 


15 


.14349 


.85651 


6.9690 


.14499 


6.8969 


1.0104 


.01035 


.98965 


45 


16 


. 4378 


. 5622 


.9550 


. 4529 


.8828 


.0105 


. 1039 


. 8961 


44 


17 


. 4407 


. 5593 


.9411 


. 4559 


.8687 


.0105 


. 1043 


. 8957 


43 


18 


. 4436 


. 5564 


.9273 


. 4588 


.8547 


.0106 


. 1047 


. 8952 


42 


19 


. 4464 


. 5536 


.9135 


. 4618 


.8408 


.0106 


. 1052 


. 8948 


41 


20 


.14493 


.85507 


6.8998 


.14648 


6.8269 


1.0107 


.01056 


.98944 


40 


21 


. 4522 


. 5478 


.8861 


. 4677 


.8131 


.0107 


. 1060 


. 8940 


39 


22 


. 4551 


. 5449 


.8725 


. 4707 


.7993 


.0107 


. 1064 


. 8936 


38 


23 


. 4579 


. 5420 


.8589 


. 4737 


.7856 


.0108 


. 1068 


. 8931 


37 


24 


. 4608 


. 5392 


.8454 


. 4767 


.7720 


.0108 


. 1073 


. 8927 


36 


25 


.14637 


.85363 


6.8320 


.14796 


6.7584 


1.0109 


.01077 


.98923 


35 


20 


. 4666 


. 5334 


.8185 


. 4826 


.7448 


.0109 


. 1081 


. 8919 


34 


27 


. 4695 


. 5305 


.8052 


. 4856 


.7313 


.0110 


. 1085 


. 8914 


33 


28 


. 4723 


. 5277 


.7919 


. 4886 


.7179 


.0110 


. 1090 


. 8910 


32 


29 


. 4752 


. 5248 


.7787 


. 491.5 


.7045 


.0111 


. 1094 


. 8906 


31 


30 


.14781 


.85219 


6.7655 


.14945 


6.6911 


1.0111 


.01098 


.98901 


30 


31 


. 4810 


. 5190 


.7523 


. 4975 


.6779 


.01.11 


. 1103 


. 8897 


29 


32 


. 4838 


. 5161 


.7392 


. 5004 


.6646 


.0112 


. 1107 


. 8893 


28 


33 


. 4867 


. 5133 


.7262 


. 5034 


.6514 


.0112 


. 1111 


. 8889 


27 


34 


. 4896 


. 5104 


.7132 


. 5064 


.6383 


.0113 


. 1116 


. 8884 


26 


35 


.14925 


.85075 


6.7003 


.15094 


6.6252 


1.0113 


.01120 


.98880 


25 


36 


. 4953 


. 5046 


.6874 


. 5123 


.6122 


.0114 


. 1124 


. 8876 


24 


37 


. 4982 


. 5018 


.6745 


. 5153 


.5992 


.0114 


. 1129 


. 8871 


23 


38 


. 5011 


. 4989 


.6617 


. 5183 


.5863 


.0115 


. 1133 


. 8867 


22 


39 


. 5040 


. 4960 


.6490 


. 5213 


.5734 


.0115 


. 1137 


. 8862 


21 


40 


.15068 


.84931 


6.6363 


.152*3 


6.5605 


1.0115 


.01142 


.98858 


20 


41 


. 5097 


. 4903 


.6237 


. 5272 


.5478 


.0116 


. 1146 


. 8854 


19 


42 


. 5126 


. 4874 


.6111 


. 5302 


.5350 


.0116 


. 1151 


. 8849 


18 


43 


. 5155 


. 4845 


.5985 


. 5332 


.5223 


.0117 


. 1155 


. 8845 


17 


44 


. 5183 


. 4816 


.5860 


. 5362 


.5097 


.0117 


. 1159 


. 8840 


16 


45 


.15212 


.84788 


6.5736 


.15391 


6.4971 


1.0118 


.01164 


.98836 


15 


46 


. 5241 


. 4759 


.5612 


. 5421 


.4845 


.0118 


. 1168 


. 8832 


14 


47 


. 5270 


. 4730 


.5488 


. 5451 


.4720 


.0119 


. 1173 


. 8827 


13 


48 


. 5298 


. 4701 


.5365 


. 5481 


.4596 


.0119 


. 1177 


. 8823 


12 


49 


. 5328 


. 4672 


.5243 


. 5511 


.4472 


.0119 


. 1182 


. 8818 


11 


50 


.15356 


.84644 


6.5121 


.15540 


6.4348 


1.0120 


.01186 


.98814 


10 


51 


. 5385 


. 4615 


.4999 


. 5570 


.4225 


.0120 


. 1190 


. 8809 


9 


52 


. 5413 


. 4586 


.4878 


. 5600 


.4103 


.0121 


. 1195 


. 8805 


8 


53 


. 5442 


. 4558 


.4757 


. 5630 


.3980 


.0121 


. 1199 


. 8800 


7 


54 


. r.iTl 


. 4529 


.4637 


. 5659 


.3859 


.0122 


. 1204 


. 8796 


6 


55 


.15500 


.84500 


6.4517 


.15689 


6.3737 


1.0122 


.01208 


.98791 


5 


56 




. 4471 


.4398 


. 5719 


.3616 


.0123 


. 1213 


. 8787 


4 


57 


. 5557 


. 4448 


.4279 


. 5749 


.3496 


.0123 


. 1217 


. 8782 


3 


58 


. 5586 


. 4414 


.4160 


. 5779 


.3376 


.0124 


. 1222 


. 8778 


2 


59 


. 5615 


. 4385 


.4042 


. 5809 


.3257 


.0124 


. 1227 


. 8773 


1 


60 


. 5643 


. 4356 


.892 1 


. 5838 


.3137 


.0125 


. 1231 


. 8769 





M. 


Cosine. 


Vrs. sin.j 


l Secant. 


Cotang. 


Tang. 


Cosec'nt 


Sine. 


Vrs. cos. 


M. 



98° 



81° 



Natural Functions. 



143 



9° 




Natural Trigonometrical Functions. 


170° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. < 


Cosine. 


M. 





.15643 


.84356 


6.3924 


.15838 


6.3137 


1.0125 


.01231 


98769 


60 


1 


. 5672 


. 4328 


.3807 


. 5868 


.3019 


.0125 


. 1236 


8764 


59 


2 


. 5701 


. 4299 


.3690 


. 5898 


.2901 


.0125 


. 1240 


8760 


58 


3 


. 5730 


. 4270 


.3574 


. 5928 


.2783 


.0126 


. 1245 


8755 


57 


4 


. 5758 


. 4242 


.3458 


. 5958 


.2665* 


.0126 


. 1249 


8750 


56 


5 


.15787 


.84213 


6.3343 


.15987 


6.2548 


1.0127 


.01254 


98746 


55 


6 


. 5816 


. 4184 


.3228 


. 6017 


.2432 


.0127 


. 1259 


8741 


54 


7 


. 5844 


. 4155 


.3113 


. 6047 


.2316 


.0128 


. 1263 


8737 


53 


8 


. 5873 


. 4127 


.2999 


. 6077 


.2200 


.0128 


. 1268 


8732 


52 


9 


. 5902 


. 4098 


.2885 


. 6107 


.2085 


.0129 


. 1272 


8727 


51 


10 


.15931 


.84069 


6.2772 


.16137 


6.1970 


1.0129 


.01277 


98723 


50 


11 


. 5959 


. 4041 


.2659 


. 6167 


.1856 


.0130 


. 1282 


8718 


49 


12 


. 5988 


. 4012 


.2546 


. 6196 


.1742 


.0130 


. 1286 


8714 


48 


13 


. 6017 


. 3983 


.2434 


. 6226 


.1628 


.0131 


. 1291 


8709 


47 


14 


. 6045 


. 3954 


.2322 


. 6256 


.1515 


.0131 


. 1296 


8704 


46 


15 


.16074 


.83926 


6.2211 


.16286 


6.1402 


1.0132 


.01300 


98700 


45 


16 


. 6103 


. 3897 


.2100 


. 6316 


.1290 


.0132 


. 1305 


8695 


44 


17 


. 6132 


. 3868 


.1990 


. 6346 


.1178 


.0133 


. 1310 


8690 


43 


18 


. 6160 


. 3840 


.1880 


. 6376 


.1066 


.0133 


. 1314 


8685 


42 


19 


. 6189 


. 3811 


.1770 


. 6405 


.0955 


.0134 


. 1319 


8681 


41 


20 


.16218 


.83782 


6.1661 


.16435 


6.0844 


1.0134 


.01324 


98676 


40 


21 


. 6246 


. 3753 


.1552 


. 6465 


.0734 


.0135 


. 1328 


8671 


39 


22 


. 6275 


. 3725 


.1443 


. 6495 


.0624 


.0135 


. 1333 


8667 


38 


23 


. 6304 


. 3696 


.1335 


. 6525 


.0514 


.0136 


. 1338 


8662 


37 


24 


. 6333 


. 3667 


.1227 


. 6555 


.0405 


.0136 


. 1343 


8657 


36 


25 


.16361 


.83639 


6.1120 


.16585 


6.0296 


1.0136 


.01347 


98652 


35 


26 


. 6390 


. 3610 


.1013 


. 6615 


.0188 


.0137 


. 1352 


8648 


34 


27 


. 6419 


. 3581 


.0906 


. 6644 


.0080 


.0137 


. 1357 


8643 


33 


28 


. 6447 


. 3553 


.0800 


. 6674 


5.9972 


.0138 


. 1362 • 


8638 


32 


29 


. 6476 


. 3524 


.0694 


. 6704 


.9865 


.0138 


. 1367 


8633 


31 


30 


.16505 


.83495 


6.0588 


.16734 


5.9758 


1.0139 


.01371 


98628 


30 


31 


. 6533 
. 6562 


. 3466 
. 3438 


.0483 
.0379 


. 6764 


.9651 


.0139 
.0140 


. 1376 
. 1381 


8624 
8619 


29 


32 


. 6794 


.9545 


28 


33 


. 6591 


. 3409 


.0274 


. 6824 


.9439 


.0140 


. 1386 


8614 


27 


34 


. 6619 


. 3380 


.0170 


. 6854 


.9333 


.0141 


. 1391 


8609 


26 


35 


.16648 


.83352 


6.0066 


.16884 


5.9228 


1.0141 


.01395 


98604 


25 


36 


. 6677 


. 3323 


5.9963 


. 6914 


.9123 


.0142 


. 1400 


8600 


24 


37 


. 6705 


. 3294 


.9860 


. 6944 


.9019 


.0142 


. 1405 


8595 


23 


38 


. 6734 


. 3266 


.9758 


. 6973 


.8915 


.0143 


. 1410 


8590 


22 


39 


. 6763 


. 3237 


.9655 


. 7003 


.8811 


.0143 


. 1415 


8585 


21 


40 


.16791 


.83208 


5.9554 


.17033 


5.8708 


1.0144 


.01420 


98580 


20 


41 


. 6820 


. 3180 


.9452 


. 7063 


.8605 


.0144 


. 1425 


8575 


19 


42 


. 6849 


. 3151 


.9351 


. 7093 


.8502 


.0145 


. 1430 


8570 


18 


43 


. 6878 


. 3122 


.9250 


. 7123 


.8400 


.0145 


. 1434 


8565 


17 


44 


. 6906 


. 3094 


.9150 


. 7153 


.8298 


.0146 


. 1439 


8560 


16 


45 


.16935 


.83065 


5.9049 


.17183 


5.8196 


1.0146 


.01444 


98556 


15 


46 


. 6964 


. 3036 


.8950 


. 7213 


.8095 


.0147 


. 1449 


8551 


14 


47 


. 6992 


. 3008 


.8850 


. 7243 


.7994 


.0147 


. 1454 


8546 


13 


48 


. 7021 


. 2979 


.8751 


. 7273 


.7894 


.0148 


. 1459 


8541 


12 


49 


. 7050 


. 2950 


.8652 


. 7303 


.7793 


.0148 


. 1464 


8536 


11 


50 


.17078 


.82922 


5.8554 


.17333 


5.7694 


1.0149 


.01469 


98531 


10 


51 


. 7107 


. 2893 


.8456 


. 7363 


.7594 


.0150 


. 1474 


8526 


9 


52 


. 7136 


. 2864 


.8358 


. 7393 


.7495 


.0150 


. 1479 


8521 


8 


53 


. 7164 


. 2836 


.8261 


. 7423 


.7396 


.0151 


. 1484 


8516 


7 


54 


. 7193 


. 2807 


.8163 


. 7453 


.7297 


.0151 


. 1489 


8511 


6 


55 


.17221 


.82778 


5.8067 


.17483 


5.7199 


1.0152 


.01494 


98506 


5 


56 


. 7250 


. 2750 


.7970 


. 7513 


.7101 


.0152 


. 1499 


8501 


4 


57 


. 7279 


. 2721 


.7874 


. 7543 


. .7004 


.0153 


. 1504 


8496 


3 


58 


. 7307 


. 2692 


.7778 


. 7573 


.6906 


.0153 


. 1509 


8491 


2 


59 


. 7336 


. 2664 


.7683 


. 7603 


.6809 


.0154 


. 1514 


8486 


1 


60 


. 7365 


. 2635 


.7588 


. 7633 


.6713 


.0154 


. 1519 


8481 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


|Vrs. cos. 


Sine. 


M. 



144 



Natural Functions. 



10° 




Natural Trigonometrical Functions. 


169° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.17365 


.82635 


5.7588 


.17633 


5.6713 


1.0154 


.01519 


.98481 


60 


1 


. 7393 


. 2606 


.7493 


. 7663 


.6616 


.0155 


. 1524 


. 8476 


59 


2 


. 7422 


. 2578 


.7398 


. 7693 


.6520 


.0155 


. 1529 


. 8471 


58 


3 


. 7451 


. 2549 


.7304 


. 7723 


.6425 


.0156 


. 1534 


. 8465 


57 


4 


. 7479 


. 2521 


.7210 


. 7753 


.6329 


.0156 


. 1539 


. 8460 


56 


5 


.17508 


.82492 


5.7117 


.17783 


5.6234 


1.0157 


.01544 


.98455 


55 


6 


. 7537 


. 2463 


.7023 


. 7813 


.6140 


.0157 


. 1550 


. 8450 


54 


7 


. 7565 


. 2435 


.6930 


. 7843 


.6045 


.0158 


. 1555 


. 8445 


53 


8 


. 7594 


. 2406 


.6838 


. 7873 


.5951 


.0158 


. 1560 


. 8440 


52 


9 


. 7622 


. 2377 


.6745 


. 7903 


.5857 


.0159 


. 1565 


. 8435 


51 


10 


.17651 


.82349 


5.6653 


.17933 


5.5764 


1.0159 


.01570 


.98430 


50 


11 


. 7680 


. 2320 


.6561 


. 7963 


.5670 


.0160 


. 1575 


. 8425 


49 


12 


. 7708 


. 2291 


.6470 


. 7993 


.5578 


.0160 


. 1580 


. 8419 


48 


13 


. 7737 


. 2263 


.6379 


. 8023 


.5485 


.0161 


. 1585 


. 8414 


47 


14 


. 7766 


. 2234 


.6288 


. 8053 


.5393 


.0162 


. 1591 


. 8409 


46 


15 


.17794 


.82206 


5.6197 


.18083 


5.5301 


1.0162 


.01596 


.98404 


45 


16 


. 7823 


. 2177 


.6107 


. 8113 


.5209 


.0163 


. 1601 


. 8399 


44 


17 


. 7852 


. 2148 


.6017 


. 8143 


.5117 


.0163 


. 1606 


. 8394 


43 


18 


. 7880 


. 2120 


.5928 


. 8173 


.5026 


.0164 


. 1611 


. 8388 


42 


19 


. 7909 


. 2091 


.5838 


. 8203 


.4936 


.0164 


. 1617 


. 8383 


41 


20 


.17937 


.82062 


5.5749 


.18233 


5.4845 


1.0165 


.01622 


.98378 


40 


21 


. 7966 


. 2034 


.5660 


. 8263 


.4755 


.0165 


. 1627 


. 8373 


39 


22 


. 7995 


. 2005 


.5572 


. 8293 


.4665 


.0166 


. 1632 


. 8368 


38 


23 


. 8023 


. 1977 


.5484 


. 8323 


.4575 


.0166 


. 1638 


. 8362 


37 


24 


. 8052 


. 1948 


.5396 


. 8353 


.4486 


.0167 


. 1643 


. 8357 


36 


25 


.18080 


.81919 


5.5308 


.18383 


5.4396 


1.0167 


.01648 


.98352 


35 


26 


. 8109 


. 1891 


.5221 


. 8413 


.4308 


.0168 


. 1653 


. 8347 


34 


27 


. 8138 


. 1862 


.5134 


. 8444 


.4219 


.0169 


. 1659 


. 8341 


33 


28 


. 8166 


. 1834 


.5047 


. 8474 


.4131 


.0169 


. 1664 


. 8336 


32 


29 


. 8195 


. 1805 


.4960 


. 8504 


.4043 


.0170 


. 1669 


. 8331 


31 


30 


.18223 


.81776 


5.4874 


.18534 


5.3955 


1.0170 


.01674 


.98325 


30 


31 


. 8252 


. 1748 


.4788 


. 8564 


.3868 


.0171 


. 1680 


. 8320 


29 


32 


. 8281 


. 1719 


.4702 


. 8594 


.3780 


.0171 


. 1685 


. 8315 


28 


33 


. 8309 


. 1691 


.4617 


. 8624 


.3694 


.0172 


. 1690 


. 8309 


27 


34 


. 8338 


. 1662 


.4532 


. 8654 


.3607 


.0172 


. 1696 


. 8304 


•26 


35 


.18366 


.81633 


5.4447 


.18684 


5.3521 


1.0173 


.01701 


.98299 


25 


36 


. 8395 


. 1605 


.4362 


. 8714 


.3434 


.0174 


. 1706 


. 8293 


24 


37 


. 8424 


. 1576 


.4278 


. 8745 


.3349 


.0174 


. 1712 


. 8288 


23 


38 


. 8452 


. 1548 


.4194 


. 8775 


.3263 


.0175 


. 1717 


. 8283 


22 


39 


. 8481 


. 1519 


.4110 


. 8805 


.3178 


.0175 


. 1722 


. 8277 


21 


40 


.18509 


.81490 


5.4026 


.18835 


5.3093 


1.0176 


.01728 


.98272 


20 


41 


. 8538 


. 1462 


.3943 


. 8865 


.3008 


.0176 


. 1733 


. 8267 


19 


42 


. 8567 


. 1433 


.3860 


. 8895 


.2923 


.0177 


. 1739 


. 8261 


18 


43 


. 8595 


. 1405 


.3777 


. 8925 


.2839 


.0177 


. 1744 


. 8256 


17 


44 


. 8624 


. 1376 


.3695 


. 8955 


.2755 


.0178 


. 1749 


. 8250 


16 


45 


.18652 


.81348 


1 5.3612 


.18985 


5.2671 


1.0179 


.01755 


.98245 


15 


46 


. 8681 


. 1319 


! .3530 


. 9016 


.2588 


.0179 


. 1760 


. 8240 


14 


47 


. 8709 


. 1290 


.3449 


. 9046 


.2505 


.0180 


. 1766 


. 8234 


13 


48 


. 8738 


. 1262 


.3367 


. 9076 


.2422 


.0180 


. 1771 


. 8229 


12 


49 


. 8767 


. 1233 


.3286 


. 9106 


.2339 


.0181 


. 1777 


. 8223 


11 


50 


.18795 


.81205 


5.3205 


.19136 


5.2257 


1.0181 


.01782 


.98218 


10 


51 


. 8824 


. 1176 


.3124 


. 9166 


.2174 


.0182 


. 1788 


. 8212 


9 


52 


. 8852 


. 1147 


.3044 


. 9197 


.2092 


.0182 


. 1793 


. 8207 


8 


53 


. 8881 


. 1119 


.2963 


. 9227 


.2011 


.0183 


. 1799 


. 8201 


7 


54 


. 8909 


. 1090 


i .2883 


. 9257 


.1929 


.0184 


. 1804 


. 8196 


6 


55 


.18988 


.81 062 


i 5.2803 


.19287 


5.1848 


1.0184 


.01810 


.98190 


5 


56 


. 8967 


. 1033 


, .2724 


. 9317 


.1767 


.0185 


. 1815 


. 8185 


4 


57 


. 8995 


. 1005 


1 .2645 


. 9347. 


.1686 


.0185 


. 1821 


. 8179 


3 


58 


. 9024 


. 0976 


.2566 


. 9378 


.1606 


.0186 


. 1826 


. 8174 


2 


59 


. 9052 


. 0948 


.2487 


. 9408 


.1525 


.0186 


. 1832 


. 8168 


1 


60 


. 9081 


. 0919 


.2408 


. 9438 


.1445 


.0187 


. 1837 


. 8163 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



100° 



79° 



Natural Functions. 



145 



11° 


Natural Trigonometrical Functions. 


168° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.19081 


.80919 


5.2408 


.19438 


5.1445 


1.0187 


.01837 


.98163 


60 


1 


. 9109 


. 0890 


.2330 


. 9468 


.1366 


.0188 


. 1843 


. 8157 


59 


2 


. 9138 


. 0862 


.2252 


. 9498 


.1286 


.0188 


. 1848 


. 8152 


58 


3 


. 9166 


. 0833 


.2174 


. 9529 


.1207 


.0189 


. 1854 


. 8146 


57 


4 


. 9195 


. 0805 


.2097 


. 9559 


.1128 


.0189 


. 1859 


. 8140 


56 


5 


.19224 


.80776 


5.2019 


.19589 


5.1049 


1.0190 


.01865 


.98135 


55 


6 


. 9252 


. 0748 


.1942 


. 9619 


.0970 


.0191 


. 1871 


. 8129 


54 


7 


. 9281 


. 0719 


.1865 


. 9649 


.0892 


.0191 


. 1876 


. 8124 


53 


8 


. 9309 


. 0691 


.1788 


. 9680 


.0814 


.0192 


. 1882 


. 8118 


52 


9 


. 9338 


. 0662 


.1712 


. 9710 


.0736 


.0192 


. 1887 


. 8112 


51 


10 


.19366 


.80634 


5.1636 


.19740 


5.0658 


1.0193 


.01893 


.98107 


50 


11 


. 9395 


. 0605 


.1560 


. 9770 


.0581 


.0193 


. 1899 


. 8101 


49 


12 


. 9423 


. 0576 


.1484 


. 9800 


.0504 


.0194 


. 1904 


. 8095 


48 


13 


. 9452 


. 0548 


.1409 


. 9831 


.0427 


.0195 


. 1910 


. 8090 


47 


14 


. 9480 


. 0519 


.1333 


. 9861 


.0350 


.0195 


. 1916 


. 8084 


46 


15 


.19509 


.80491 


5.1258 


.19891 


5.0273 


1.0196 


.01921 


.98078 


45 


16 


. 9537 


. 0462 


.1183 


. 9921 


.0197 


.0196 


. 1927 


. 8073 


44 


17 


. 9566 


. 0434 


.1109 


. 9952 


.0121 


.0197 


. 1933 


. 8067 


43 


18 


. 9595 


. 0405 


.1034 


. 9982 


.0045 


.0198 


. 1938 


. 8061 


42 


19 


. 9623 


. 0377 


.0960 


.20012 


4.9969 


.0198 


. 1944 


. 8056 


41 


20 


.19652 


.80348 


5.0886 


.20042 


4.9894 


1.0199 


.01950 


.98050 


40 


21 


. 9680 


. 0320 


.0812 


. 0073 


.9819 


.0199 


. 1956 


. 8044 


39 


22 


. 9709 


. 0291 


.0739 


. 0103 


.9744 


.0200 


. 1961 


. 8039 


38 


23 


. 9737 


. 0263 


.0666 


. 0133 


.9669 


.0201 


. 1967 


. 8033 


37 


24 


. 9766 


. 0234 


.0593 


. 0163 


.9594 


.0201 


. 1973 


. 8027 


36 


25 


.19794 


.80206 


5.0520 


.20194 


4.9520 


1.0202 


.01979 


.98021 


35 


26 


. 9823 


. 0177 


.0447 


. 0224 


.9446 


.0202 


. 1984 


. 8016 


34 


27 


. 9851 


. 0149 


.0375 


. 0254 


.9372 


.0203 


. 1990 


. 8010 


33 


28 


. 9880 


. 0120 


.0302 


. 0285 


.9298 


.0204 


. 1996 


. 8004 


32 


29 


. 9908 


. 0092 


.0230 


. 0315 


.9225 


.0204 


. 2002 


. 7998 


31 


30 


.19937 


.80063 


5.0158 


.20345 


4.9151 


1.0205 


.02007 


.97992 


30 


31 


. 9965 


. 0035 


.0087 


. 0375 


.9078 


.0205 


. 2013 


. 7987 


29 


32 


. 9994 


. 0006 


.0015 


. 0106 


.9006 


.0206 


. 2019 


. 7981 


28 


33 


.20022 


.79978 


4.9944 


. 0436 


.8933 


.0207 


. 2025 


. 7975 


27 


34 


. 0051 


. 9949 


.9873 


. 0466 


.8860 


.0207 


. 2031 


. 7969 


26 


35 


.20079 


.79921 


4.9802 


.20497 


4.8788 


1.0208 


.02037 


.97963 


25 


36 


. 0108 


. 9892 


.9732 


. 0527 


.8716 


.0208 


. 2042 


. 7957 


24 


37 


. 0136 


. 9863 


.9661 


. 0557 


.8644 


.0209 


. 2048 


. 7952 


23 


38 


. 0165 


. 9835 


.9591 


. 0588 


.8573 


.0210 


. 2054 


. 7946 


22 


39 


. 0193 


. 9807 


.9521 


. 0618 


.8501 


.0210 


. 2060 


. 7940 


21 


40 


.20222 


.79778 


4.9452 


.20648 


4.8430 


1.0211 


.02066 


.97934 


20 


41 


. 0250 


. 9750 


.9382 


. 0679 


.8359 


.0211 


. 2072 


. 7928 


19 


42 


. 0279 


. 9721 


.9313 


. 0709 


.8288 


.0212 


. 2078 


. 7922 


18 


43 


. 0307 


. 9693 


.9243 


. 0739 


.8217 


.0213 


. 2084 


. 7916 


17 


44 


. 0336 


. 9664 


.9175 


. 0770 


.8147 


.0213 


. 2089 


. 7910 


16 


45 


.20364 


.79636 


4.9106 


.20800 


4.8077 


1.0214 


.02095 


.97904 


15 


46 


. 0393 


. 9607 


.9037 


. 0830 


.8007 


.0215 


. 2101 


. 7899 


14 


47 


. 0421 


. 9579 


.8969 


. 0861 


.7937 


.0215 


. 2107 


. 7893 


13 


48 


. 0450 


. 9550 


.8901 


. 0891 


.7867 


.0216 


. 2113 


. 7887 


12 


49 


. 0478 


. 9522 


.8833 


. 0921 


.7798 


.0216 


. 2119 


. 7881 


11 


50 


.20506 


.79493 


4.8765 


.20952 


4.7728 


1.0217 


.02125 


.97875 


10 


51 


. 0535 


. 9465 


.8697 


. 0982 


.7659 


.0218 


. 2131 


. 7869 


9 


52 


. 0563 


. 9436 


.8630 


. 1012 


.7591 


.0218 


. 2137 


. 7863 


8 


53 


. 0592 


. 9408 


.8563 


. 1043 


.7522 


.0219 


. 2143 


. 7857 


7 


54 


. 0620 


. 9379 


.8496 


. 1073 


.7453 


.0220 


. 2149 


. 7851 


6 


55 


.20649 


.79351 


4.8429 


.21104 


4.7385 


1.0220 


.02155 


.97845 


5 


56 


. 0677 


. 9323 


.8362 


. 1134 


.7317 


.0221 


. 2161 


. 7839 


4 


57 


. 0706 


. 9294 


.8296 


. 1164 


.7249 


.0221 


. 2167 


. 7833 


3 


58 


. 0734 


. 9266 


.8229 


. 1195 


.7181 


.0222 


. 2173 


. 7827 


2 


59 


. 0763 


. 9237 


.8163 


. 1225 


.7114 


.0223 


. 2179 


. 7821 


1 


60 


. 0791 


. 9209 


.8097 


. 1256 


.7046 


.0223 


. 2185 


. 7815 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt | 


Vrs. cos. 


Sine. 


M. 



101° 



78° 



10 



146 



Natural Functions. 



12 





Natural Trigonometrical Functions. 


167° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin 


Cosine. 


M. 





.20791 


.79209 


4.8097 


.21256 


4.7046 


1.0223 


.02185 


.97815 


60, 


1 


. 0820 


. 9180 


.8~032 


. 1286 


.6979 


.0224 


. 2191 


. 7809 


59 


2 


. 0848 


. 9152 


.7966 


. 1316 


.6912 


.0225 


. 2197 


. 7803 


58 


3 


. 0876 


. 9123 


.7901 


. 1347 


.6845 


.0225 


. 2203 


. 7806 


57 


4 


. 0905 


. 9105 


.7835 


. 1377 


.6778 


.0226 


. 2209 


. 7790 


56 


5 


.20933 


.79066 


4.7770 


.21408 


4.6712 


1.0226 


.02215 


.97784 


55 


6 


. 0962 


. 9038 


.7706 


. 1438 


.6646 


.0227 


. 2222 


. 7778 


54 


7 


. 0990 


. 9010 


.7641 


. 1468 


.6580 


.0228 


. 2228 


. 7772 


53 


8 


. 1019 


. 8981 


.7576 


. 1499 


.6514 


.0228 


. 2234 


. 7766 


52 


9 


. 1047 


. 8953 


.7512 


. 1529 


.6448 


.0229 


. 2240 


. 7760 


51 


10 


.21076 


.78924 


4.7448 


.21560 


4.6382 


1.0230 


.02246 


.97754 


50 


11 


. 1104 


. 8896 


.7384 


. 1590 


.6317 


.0230 


. 2252 


. 7748 


49 


12 


. 1132 


. 8867 


.7320 


. 1621 


.6252 


.0231 


. 2258 


. 7741 


48 


13 


. 1161 


. 8839 


.7257 


. 1651 


.6187 


.0232 


. 2264 


. 7735 


47 


14 


. 1189 


. 8811 


.7193 


. 1682 


.6122 


.0232 


. 2271 


. 7729 


46 


15 


.21218 


.78782 


4.7130 


.21712 


4.6057 


1.0233 


.02277 


.97723 


45 


16 


. 1246 


. 8754 


.7067 


. 1742 


.5993 


.0234 


. 2283 


. 7717 


44 


17 


. 1275 


. 8725 


.7004 


. 1773 


.5928 


.0234 


. 2289 


. 7711 


43 


18 


. 1303 


. 8697 


.6942 


. 1803 


.5864 


.0235 


. 2295 


. 7704 


42 


19 


. 1331 


.'8668 


.6879 


. 1834 


.5800 


.0235 


. 2302 


. 7698 


41 


20 


.21360 


.78640 


4.6817 


.21864 


4.5736 


1.0236 


.02308 


.97692 


40 


21 


. 1388 


. 8612 


.6754 


. 1895 


.5673 


.0237 


. 2314 


. 7686 


39 


22 


. 1417 


. 8583 


.6692 


. 1925 


.5609 


.0237 


. 2320 


. 7680 


38 


23 


. 1445 


. 8555 


.6631 


. 1956 


.5546 


.0238 


. 2326 


. 7673 


37 


24 


. 1473 


. 8526 


.6569 


. 1986 


.5483 


.0239 


. 2333 


. 7667 


36 


25 


.21502 


.78508 


4.6507 


.22017 


4.5420 


1.0239 


.02339 


.97661 


35 


26 


. 1530 


. 8470 


.6446 


. 2047 


.5357 


.0240 


. 2345 


. 7655 


34 


27 


. 1559 


. 8441 


.6385 


. 2078 


.5294 


.0241 


. 2351 


. 7648 


33 


28 


. 1587 


. 8413 


.6324 


. 2108 


.5232 


.0241 


. 2358 


. 7642 


32 


29 


. 1615 


. 8384 


.6263 


. 2139 


.5169 


.0242 


. 2364 


. 7636 


31 


30 


.21644 


.78356 


4.6202 


.22169 


4.5107 


1.0243 


.02370 


.97630 


30 


31 


. 1672 


. 8328 


.6142 


. 2200 


.5045 


.0243 


. 2377 


. 7623 


29 


32 


. 1701 


. 8299 


.6081 


. 2230 


.4983 


.0244 


. 2383 


. 7617 


28 


33 


. 1729 


. 8271 


.6021 


. 2261 


.4921 


.0245 


. 2389 


. 7611 


27 


34 


. 1757 


. 8242 


.5961 


. 2291 


.4860 


.0245 


. 2396 


. 7604 


26 


35 


.21786 


.78214 


4.5901 


.22322 


4.4799 


1.0246 


.02402 


.97598 


25 


36 


. 1814 


. 8186 


.5841 


. 2353 


.4737 


.0247 


. 2408 


. 7592 


24 


37 


. 1843 


. 8154 


.5782 


. 2383 


.4676 


.0247 


. 2415 


. 7585 


23 


38 


. 1871 


. 8129 


.5722 


. 2414 


.4615 


.0248 


. 2421 


. 7579 


22 


39 


. 1899 


. 8100 


.5663 


. 2444 


.4555 


.0249 


. 2427 


. 7573 


21 


40 


.21928 


.78072 


4.5604 


.22475 


4.4494 


1.0249 


.02434 


.97566 


20 


41 


. 1956 


. 8043 


.5545 


. 2505 


.4434 


.0250 


. 2440 


. 7560 


19 


42 


. 1985 


. 8015 


.5486 


. 2536 


.4373 


.0251 


. 2446 


. 7553 


18 


43 


. 2013 


. 7987 


.5428 


. 2566 


.4313 


.0251 


. 2453 


. 7547 


17 


44 


. 2041 


. 7959 


.5369 


. 2597 


.4253 


.0252 


. 2459 


. 7541 


16 


45 


.22070 


.77930 


4.5311 


.22628 


4.4194 


1.0253 


.02466 


.97534 


15 


46 


. 2098 


. 7902 


.5253 


. 2658 


.4134 


.0253 


. 2472 


. 7528 


14 


47 


. 2126 


. 7873 


.5195 


. 2689 


.4074 


.0254 


. 2479 


. 7521 


13 


48 


. 2155 


. 7845 


.5137 


. 2719 


.4015 


.0255 


. 2485 


. 7515 


12 


49 


. 21 S3 


. 7817 


.5079 


. 2750 


.3956 


.0255 


. 2491 


. 7508 


11 


50 


.22211 


.77788 


4.5021 


.22781 


4.3897 


1.0256 


.02498 


.97502 


10 


51 


. 2240 


. 7760 


.4964 


. 2811 


.3838 


.0257 


. 2504 


. 7495 


9 


52 


. 2268 


. 7732 


.4907 


. 2842 


.3779 


.0257 


. 2511 


. 7489 


8 


53 


. 2297 


. 7703 


.4850 


. 2872 


.3721 


.0258 


. 2517 


. 7483 


7 


54 


. 2825 


. 7675 


.4793 


. 2903 


.3662 


.0259 


. 2524 


. 7476 


6 


55 


.22353 


.77647 


4.4736 


.22934 


4.3604 


1.0260 


.02530 


.97470 


5 


56 


. 2382 


. 7618 


.4679 


. 2964 


.3546 


.0260 


. 2537 


. 7463 


4 


57 


. 2110 


. 7590 


.4623 


. 2995 


.3488 


.0261 


. 2543 


. 7457 


3 


58 


. 2438 


. 7561 


.4566 


. 3025 


.3430 


.0262 


. 2550 


. 7450 


2 


59 


. 2167 


. 75:;:; 


.4510 


. 3056 


.3372 


.0262 


. 2556 


. 7443 


1. 


60 


. 2495 


. 7505 


.4454 


. 3087 


.3315 


.0263 


. 2563 


. 7437 


C 


M. 


( M-il|C. 


Vrs. sin.: 


Secant. 


Co tang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


Rl. 



102° 



77° 



Natural Functions. 



147 



13 c 




Natural Trigonometrical Functions. 


166° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.22495 


.77505 


4.4454 


.23087 


4.3315 


1.0263 


.02563 


.97437 


60 


1 


. 2523 


. 7476 


.4398 


. 3117 


.3257 


.0264 


. 2569 


. 7430 


59 


2 


. 2552 


. 7448 


.4342 


. 3148 


.3200 


.0264 


. 2576 


. 7424 


58 


3 


. 2580 


. 7420 


.4287 


. 3179 


.3143 


.0265 


. 2583 


. 7417 


57 


4 


. 2608 


. 7391 


.4231 


. 3209 


.3086 


.0266 


. 2589 


. 7411 


56 


5 


.22637 


.77363 


4.4176 


.23240 


4.3029 


1.0266 


.02596 


.97404 


55 


6 ! . 2665 


. 7335 


.4121 


. 3270 


.2972 


.0267 


. 2602 


. 7398 


54 


7 1 . 2693 


. 7306 


.4065 


. 3301 


.2916 


.0268 


. 2609 


. 7391 


53 


8 . 2722 


. 7278 


.4011 


. 3332 


.2859 


.0268 


. 2616 


. 7384 


52 


9 . 2750 


. 7250 


.3956 


. 3363 


.2803 


.0269 


. 2622 


. 7378 


51 


10 .22778 


.77221 


4.3901 


.23393 


4.2747 


1.0270 


.02629 


.97371 


50 


11 . 2807 


. 7193 


.3847 


. 3424 


.2691 


.0271 


. 2635 


. 7364 


49 


12 . 2835 


. 7165 


.3792 


. 3455 


.2635 


.0271 


. 2642 


. 7358 


48 


13 . 2863 


. 7136 


.3738 


. 3485 


.2579 


.0272 


. 2649 


. 7351 


47 


14 : . 2892 


. 7108 


.3684 


. 3516 


.2524 


.0273 


. 2655 


. 7344 


46 


15 .22920 


.77080 


4.3630 


.23547 


4.2468 


1.0273 


.02662 


.97338 


45 


16 . 2948 


. 7052 


.3576 


. 3577 


.2413 


.0274 


. 2669 


. 7331 


44 


17 . 2977 


. 7023 


.3522 


. 3608 


.2358 


.0275 


. 2675 


. 7324 


43 


18 . 3005 


. 6995 


.3469 


. 3639 


.2303 


.0276 


. 2682 


. 7318 


42 


19 . 3033 


. 6967 


.3415 


. 3670 


.2248 


.0276 


. 2689 


. 7311 


41 


20 .23061 


.76938 


4.3362 


.23700 


4.2193 


1.0277 


.02695 


.97304 


40 


21 . 3090 


. 6910 


.3309 


. 3731 


.2139 


.0278 


. 2702 


. 7298 


39 


22 . 3118 


. 6882 


.3256 


. 3762 


.2084 


.0278 


. 2709 


. 7291 


38 


23 . 3146 


. 6853 


.3203 


. 3793 


.2030 


.0279 


. 2716 


. 7284 


37 


24 . 3175 


. 6825 


.3150 


. 3823 


.1976 


.0280 


. 2722 


. 7277 


86 


25 .23203 


.76797 


4.3098 


.23854 


4.1921 


1.0280 


.02729 


.97271 


35 


26 . 3231 


. 6769 


.3045 


. 3885 


.1867 


.0281 


. 2736 


. 7264 


34 


27 . 3260 


. 6740 


.2993 


. 3916 


.1814 


.0282 


. 2743 


. 7257 


33 


28 . 3288 


. 6712 


.2941 


. 3946 


.1760 


.0283 


. 2749 


. 7250 


32 


29 . 3316 


. 6684 


.2888 


. 3977 


.1706 


.0283 


. 2756 


. 7244 


31 


30 .23344 


.76655 


4.2836 


.24008 


4.1653 


1.0284 


.02763 


.97237 


30 


31 


. 3373 


. 6627 


.2785 


. 4039 


.1600 


.0285 


. 2770 


. 7230 


29 


32 


. 3401 


. 6599 


.2733 


. 4069 


.1546 


.0285 


. 2777 


. 7223 


28 


33 


. 3429 


. 6571 


.2681 


. 4100 


.1493 


.0286 


. 2783 


. 7216 


27 


34 


. 3458 


. 6542 


.2630 


. 4131 


.1440 


.0287 


. 2790 


. 7210 


26 


35 .23486 


.76514 


4.2579 


.24162 


4.1388 


1.0288 


.02797 


.97203 


25 


36 . 3514 


. 6486 


.2527 


. 4192 


.1335 


.0288 


. 2804 


. 7196 


24 


37 . 3542 


. 6457 


.2476 


. 4223 


.1282 


.0289 


. 2811 


. 7189 


23 


38 . 3571 


. 6429 


.2425 


. 4254 


.1230 


.0290 


. 2818 


. 7182 


22 


39 . 3599 


. 6401 


.2375 


. 4285 


.1178 


.0291 


. 2824 


. 7175 


21 


40 .23627 


.76373 


4.2324 


.24316 


4.1126 


1.0291 


.02831 


.97169 


20 


41 . 3655 


. 6344 


.2273 


. 4346 


.1073 


.0292 


. 2838 


. 7162 


19 


42 . 3684 


. 6316 


.2223 


. 4377 


.1022 


.0293 


. 2845 


. 7155 


18 


43 


. 3712 


. 6288 


.2173 


. 4408 


.0970 


.0293 


. 2852 


. 7148 


17 


44 


. 3740 


. 6260 


.2122 


. 4439 


.0918 


.0294 


. 2859 


. 7141 


16 


45 


.23768 


.76231 


4.2072 


.24470 


4.0867 


1.0295 


.02866 


.97134 


15 


46 


. 3797 


. 6203 


.2022 


. 4501 


.0815 


.0296 


. 2873 


. 7127 


14 


47 


. 3825 


. 6175 


.1972 


. 4531 


.0764 


.0296 


. 2880 


. 7120 


13 


48 


. 3853 


. 6147 


.1923 


. 4562 


.0713 


.0297 


. 2886 


. 7113 


12 


49 


. 3881 


. 6118 


.1873 


. 4593 


.0662 


.0298 


. 2893 


. 7106 


11 


50 


.23910 


.76090 


4.1824 


.24624 


4.0611 


1.0299 


.02900 


.97099 


10 


51 


. 3938 


. 6062 


.1774 


. 4655 


.0560 


.0299 


. 2907 


. 7092 


9 


52 


. 3966 


. 6034 


.1725 


. 4686 


.0509 


.0300 


. 2914 


. 7086 


8 


53 


. 3994 


. 6005 


.1676 


. 4717 


.0458 


.0301 


. 2921 


. 7079 


7 


54 


. 4023 


. 5977 


.1627 


. 4747 


.0408 


.0302 


. 2928 


. 7072 


6 


55 


.24051 


.75949 


4.1578 


.24778 


4.0358 


1.0302 


.02935 


.97065 


5 


56 


. 4079 


. 5921 


.1529 


. 4809 


.0307 


.0303 


. 2942 


. 7058 


4 


57 


. 4107 


. 5892 


.1481 


. 4840 


.0257 


.0304 


. 2949 


. 7051 


3 


58 


. 4136 


. 5864 


.1432 


. 4871 


.0207 


.0305 


. 2956 


. 7044 


2 


59 


. 4164 


. 5836 


.1384 


. 4902 


.0157 


.0305 


. 2963 


. 7037 


1 


60 


. 4192 


. 5808 


.1336 


. 4933 


.0108 


.0306 


. 2970 


. 7029 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



148 



Natural Functions. 



14° 



Natural Trigonometrical Functions. 



165° 



M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.24192 


.75808 


4.1336 


.24933 


4.0108 


1.0306 


.02970 


.97029 


60 


1 


. 4220 


. 5779 


.1287 


. 4964 


.0058 


.0307 


. 2977 


. 7022 


59 


2 


. 4249 


. 5751 


.1239 


. 4995 


.0009 


.0308 


. 2984 


. 7015 


58 


3 


. 4277 


. 5723 


.1191 


. 5025 


3.9959 


.0308 


. 2991 


. 7008 


57 


4 


. 4305 


. 5695 


.1144 


. 5056 


.9910 


.0309 


. 2999 


. 7001 


56 


5 


.24333 


.75667 


4.1096 


.25087 


3.9861 


1.0310 


.03006 


.96994 


55 


6 


. 4361 


. 5638 


.1048 


. 5118 


.9812 


.0311 


. 3013 


. 6987 


54 


7 


. 4390 


. 5610 


.1001 


. 5149 


.9763 


.0311 


. 3020 


. 6980 


53 


8 


. 4418 


. 5582 


.0953 


. 5180 


.9714 


.0312 


. 3027 


. 6973 


52 


9 


. 4446 


. 5554 


.0906 


. 5211 


.9665 


.0313 


. 3034 


. 6966 


51 


10 


.24474 


.75526 


4.0859 


.25242 


3.9616 


1.0314 


.03041 


.96959 


50 


11 


. 4502 


. 5497 


.0812 


. 5273 


.9568 


.0314 


. 3048 


. 6952 


49 


12 


. 4531 


. 5469 


.0765 


. 5304 


.9520 


.0315 


. 3055 


. 6944 


48 


13 


. 4559 


. 5441 


.0718 


. 5335 


.9471 


.0316 


. 3063 


. 6937 


47 


14 


. 4587 


. 5413 


.0672 


. 5366 


.9423 


.0317 


. 3070 


. 6930 


46 


15 


.24615 


.75385 


4.0625 


.25397 


3.9375 


1.0317 


.03077 


.96923 


45 


16 


. 4643 


. 5356 


.0579 


. 5428 


.9327 


.0318 


. 3084 


. 6916 


44 


17 


. 4672 


. 5328 


.0532 


. 5459 


.9279 


.0319 


. 3091 


. 6909 


43 


18 


. 4700 


. 5300 


.0486 


. 5490 


.9231 


.0320 


. 3098 


. 6901 


42 


19 


. 4728 


. 5272 


.0440 


. 5521 


.9184 


.0320 


. 3106 


. 6894 


41 


20 


.24756 


.75244 


4.0394 


.25552 


3.9136 


1.0321 


.03113 


.96887 


40 


21 


. 4784 


. 5215 


.0348 


. 5583 


.9089 


.0322 


. 3120 


. 6880 


39 


22 


. 4813 


. 5187 


.0302 


. 5614 


.9042 


.0323 


. 3127 


. 6873 


38 


23 


. 4841 


. 5159 


.0256 


. 5645 


.8994 


.0323 


. 3134 


. 6865 


37 


24 


. 4869 


. 5131 


.0211 


. 5676 


.8947 


.0324 


. 3142 


. 6858 


36 


25 


.24897 


.75103 


4.0165 


.25707 


3.8900 


1.0325 


.03149 


.96851 


35 


26 


. 4925 


. 5075 


.0120 


. 5738 


.8853 


.0326 


. 3156 


. 6844 


34 


27 


. 4953 


. 5046 


.0074 


. 5769 


.8807 


.0327 


. 3163 


. 6836 


33 


28 


. 4982 


. 5018 


.0029 


. 5800 


.8760 


.0327 


. 3171 


. 6829 


32 


29 


. 5010 


. 4990 


3.9984 


. 5831 


.8713 


.0328 


. 3178 


. 6822 


31 


30 


.25038 


.74962 


3.9939 


.25862 


3.8667 


1.0329 


.03185 


.96815 


30 


31 


. 5066 


. 4934 


.9894 


. 5893 


.8621 


.0330 


. 3192 


. 6807 


29 


32 


. 5094 


. 4906 


.9850 


. 5924 


.8574 


.0330 


. 3200 


. 6800 


28 


33 


. 5122 


. 4877 


.9805 


. 5955 


.8528 


.0331 


. 3207 


. 6793 


27 


34 


. 5151 


. 4849 


.9760 


. 5986 


.8482 


.0332 


. 3214 


. 6785 


26 


35 


.25179 


.74821 


3.9716 


.26017 


3.8436 


1.0333 


.03222 


.96778 


25 


36 


. 5207 


. 4793 


.9672 


. 6048 


.8390 


.0334 


. 3229 


. 6771 


24 


37 


. 5235 


. 4765 


.9627 


. 6079 


.8345 


.0334 


. 3236 


. 6763 


23 


38 


. 5263 


. 4737 


.9583 


. 6110 


.8299 


.0335 


. 3244 


. 6756 


22 


39 


. 5291 


. 4709 


.9539 


. 6141 


.8254 


.0336 


. 3251 


. 6749 


21 


40 


.25319 


.74680 


3.9495 


.26172 


3.8208 


1.0337 


.03258 


.96741 


20 


41 


. 5348 


. 4652 


.9451 


. 6203 


.8163 


.0338 


. 3266 


. 6734 


19 


42 


. f):;7G 


. 4624 


.9408 


. 6234 


.8118 


.0338 


. 3273 


. 6727 


18 


43 


. 5404 


. 4596 


.9364 


. 6266 


.8073 


.0339 


. 3281 


. 6719 


17 


44 


. 5432 


. 4568 


.9320 


. 6297 


.8027 


.0340 


. 3288 


. 6712 


16 


45 


.25460 


.74540 


3.9277 


.26328 


3.7983 


1.0341 


.03295 


.96704 


15 


46 


. 5488 


. 4512 


.9234 


. 6359 


.7938 


.0341 


. 3303 


. 6697 


14 


47 


. 5516 


. 4483 


.9190 


. 6390 


.7893 


.0342 


. 3310 


. 6690 


13 


48 


. 6544 


. 4455 


.9147 


. 6421 


.7848 


.0343 


. 3318 


. 6682 


12 


49 


. 5573 


. 1127 


.9104 


. 6452 


.7804 


.0344 


. 3325 


. 6675 


11 


50 


.25601 


.74399 


3.9061 


.26483 


3.7759 


1.0345 


.03332 


.96667 


10 


51 


. 5629 


. 1371 


.9018 


. 651 1 


.7715 


.0345 


. 3340 


. 6660 


9 


52 


. 5657 


. 4344 


.8976 


. 6546 


.7671 


.0346 


. 3347 


. 6652 


8 


53 


. 5685 


. 4315 


.8933 


. 6577 


.7627 


.0347 


. 3355 


. 6645 


7 


54 


. 5713 


. 4287 


.8890 


. 6608 


.7583 


.0348 


. 3362 


. 6638 


6 


55 


.25741 


.74259 


3.8848 


.26639 


3.7539 


1.0349 


.03370 


.96630 


5 


66 


. 5769 


. 4230 


i .8805 


. 6670 


.7495 


.0349 


. 3377 


. 6623 


4 


57 


. 5798 


. 4202 


.8763 


. 6701 


.7451 


.0350 


. 3385 


. 6615 


3 


58 


. 6626 


. 4174 


.872] 


. 6732 


.7107 


.0351 


. 3392 


. 6608 


2 


59 


. 6854 


. 4146 


.8679 


. 6764 


.7364 


.0352 


. 3400 


. 6600 


1 


60 


. 5882 


. Ills 




. 6795 


.7320 


.0353 


. 3407 


. 6592 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Co tang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



104° 



75° 



Natural Functions. 



149 



15° 


Natural Trigonometrical Functions. 


164° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


[Vrs. sin. 


Cosine. 


M. 





.25882 


.74118 


3.8637 


.26795 


3.7320 


1.0353 


.03407 


.96592 


60 


1 


. 5910 


. 4090 


.8595 


. 6826 


.7277 


.0353 


. 3415 


. 6585 


59 


2 


. 5938 


. 4062 


.8553 


. 6857 


.7234 


.0354 


. 3422 


. 6577 


58 


3 


. 5966 


. 4034 


.8512 


. 6888 


.7191 


.0355 


. 3430 


. 6570 


57 


4 


. 5994 


. 4006 


.8470 


. 6920 


.7147 


.0356 


. 3438 


. 6562 


56 


5 


.26022 


.73978 


3.8428 


.26951 


3.7104 


1.0357 


.03445 


.96555 


55 


6 


. 6050 


. 3949 


.8387 


. 6982 


.7062 


.0358 


. 3453 


. 6547 


54 


7 


. 6078 


. 3921 


.8346 


. 7013 


.7019 


.0358 


. 3460 


. 6540 


53 


8 


. 6107 


. 3893 


.8304 


. 7044 


.6976 


.0359 


. 3468 


. 6532 


52 


9 


. 6135 


. 3865 


.8263 


. 7076 


.6933 


.0360 


. 3475 


. 6524 


51 


10 


.26163 


.73837 


3.8222 


.27107 


3.6891 


1.0361 


.03483 


.96517 


50 


11 


. 6191 


. 3809 


.8181 


. 7138 


.6848 


.0362 


. 3491 


. 6509 


49 


12 


. 6219 


. 3781 


.8140 


. 7169 


.6806 


.0362 


. 3498 


. 6502 


48 


13 


. 6247 


. 3753 


.8100 


. 7201 


.6764 


.0363 


. 3506 


. 6494 


47 


14 


. 6275 


. 3725 


.8059 


. 7232 


.6722 


.0364 


. 3514 


. 6486 


46 


15 


.26303 


.73697 


3.8018 


.27263 


3.6679 


1.0365 


.03521 


.96479 


45 


16 


. 6331 


. 3669 


.7978 


. 7294 


.6637 


.0366 


. 3529 


. 6471 


44 


17 


. 6359 


. 3641 


.7937 


. 7326 


.6596 


.0367 


. 3536 


. 6463 


43 


18 


. 6387 


. 3613 


.7897 


. 7357 


.6554 


.0367 


. 3544 


. 6456 


42 


19 


. 6415 


. 3585 


.7857 


. 7388 


.6512 


.0368 


. 3552 


. 6448 


41 


20 


.26443 


.73556 


3.7816 


.27419 


3.6470 


1.0369 


.03560 


.96440 


40 


21 


. 6471 


. 3528 


.7776 


. 7451 


.6429 


.0370 


. 3567 


. 6433 


39 


22 


. 6499 


. 3500 


.7736 


. 7482 


.6387 


.0371 


. 3575 


. 6425 


38 


23 


. 6527 


. 3472 


.7697 


. 7513 


.6346 


.0371 


. 3583 


. 6417 


37 


24 


. 6556 


. 3444 


.7657 


. 7544 


.6305 


.0372 


. 3590 


. 6409 


36 


25 


.26584 


.73416 


3.7617 


.27576 


3.6263 


1.0373 


1 .03598 


.96402 


35 


26 


. 6612 


. 3388 


.7577 


. 7607 


.6222 


.0374 


i . 3606 


. 6394 


34 


27 


. 6640 


. 3360 


.7538 


. 7638 


.6181 


.0375 


. 3614 


. 6386 


33 


28 


. 6668 


. 3332 


.7498 


. 7670 


.6140 


.0376 


! . 3621 


. 6378 


32 


29 


. 6696 


. 3304 


.7459 


. 7701 


.6100 


.0376 


i . 3629 


. 6371 


31 


30 


.26724 


.73276 


3.7420 


.27732 


3.6059 


1.0377 


.03637 


.96363 


30 


31 


. 6752 


. 3248 


.7380 


. 7764 


.6018 


.0378 


: . 3645 


. 6355 


29 


32 


. 6780 


. 3220 


.7341 


. 7795 


.5977 


.0379 


. 3652 


. 6347 


28 


33 


. 6808 


. 3192 


.7302 


. 7826 


.5937 


.0380 


. 3660 


. 6340 


27 


34 


. 6836 


. 3164 


.7263 


. 7858 


.5896 


.0381 


. 3668 


. 6332 


26 


35 


.26864 


.73136 


3.7224 


.27889 


3.5856 


1.0382 


.03676 


.96324 


25 


36 


. 6892 


. 3108 


.7186 


. 7920 


.5816 


.0382 


. 3684 


. 6316 


24 


37 


. 6920 


. 3080 


.7147 


. 7952 


.5776 


.0383 


. 3691 


. 6308 


23 


38 


. 6948 


. 3052 


.7108 


. 7983 


.5736 


.0384 


. 3699 


. 6301 


22 


39 


. 6976 


. 3024 


.7070 


. 8014 


.5696 


.0385 


. 3707 


. 6293 


21 


40 


.27004 


.72996 


3.7031 


.28046 


3.5656 


1.0386 


.03715 


.96285 


20 


41 


. 7032 


. 2968 


.6993 


. 8077 


.5616 


.0387 


. 3723 


. 6277 


19 


42 


. 7060 


. 2940 


.6955 


. 8109 


.5576 


.0387 


. 3731 


. 6269 


18 


43 


. 7088 


. 2912 


.6917 


. 8140 


.5536 


.0388 


. 3739 


. 6261 


17 


44 


. 7116 


. 2884 


.6878 


. 8171 


.5497 


.0389 


. 3746 


. 6253 


16 


45 


.27144 


.72856 


3.6840 


.28203 


3.5457 


1.0390 


.03754 


.96245 


15 


46 


. 7172 


. 2828 


.6802 


. 8234 


.5418 


.0391 


. 3762 


. 6238 


14 


47 


. 7200 


. 2800 


.6765 


. 8266 


.5378 


.0392 


. 3770 


. 6230 


13 


48 


. 7228 


. 2772 


.6727 


. 8297 


.5339 


.0393 


. 3778 


. 6222 


12 


49 


. 7256 


. 2744 


.6689 


. 8328 


.5300 


.0393 


. 3786 


. 6214 


11 


50 


.27284 


.72716 


3.6651 


.28360 


3.5261 


1.0394 


.03794 


.96206 


10 


51 


. 7312 


. 2688 


.6614 


. 8391 


.5222 


.0395 


. 3802 


. 6198 


9 


52 


. 7340 


. 2660 


.6576 


. 8423 


.5183 


.0396 


. 3810 


. 6190 


8 


53 


. 7368 


. 2632 


.6539 


. 8454 


.5144 


.0397 


. 3818 


. 6182 


7 


54 


. 7396 


. 2604 


.6502 


. 8486 


.5105 


.0398 


. 3826 


. 6174 


6 


55 


.27424 


.72576 


3.6464 


.28517 


3.5066 


1.0399 


.03834 


.96166 


5 


56 


. 7452 


. 2548 


.6427 


. 8549 


.5028 


.0399 


. 3842 


. 6158 


4 


57 


. 7480 


. 2520 


.6390 


. 8580 


.4989 


.0400 


. 3850 


. 6150 


3 


58 


. 7508 


. 2492 


.6353 


. 8611 


.4951 


.0401 


. 3858 


. 6142 


2 


59 


. 7536 


. 2464 


.6316 


. 8643 


.4912 


.0402 


. 3866 


. 6134 


1 


60 


. 7564 


. 2436 


.6279 


. 8674 


.4874 


.0403 


. 3874 


. 6126 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 


101 


>° 














7 


4° 



150 



Natural Functions. 



16 c 




Natural Trigonometrical Functions. 


163° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.27564 


.72436 


3.6279 


.28674 


3.4874 


1.0403 


.03874 


.96126 


60 


1 


. 7592 


. 2408 


.6243 


. 8706 


.4836 


.0404 


. 3882 


. 6118 


59 


2 


. 7620 


. 2380 


.6206 


. 8737 


.4798 


.0405 


. 3890 


. 6110 


58 


3 


. 7648 


. 2352 


.6169 


. 8769 


.4760 


.0406 


. 3898 


. 6102 


57 


4 


. 7675 


. 2324 


.6133 


. 8800 


.4722 


.0406 


. 3906 


. 6094 


56 


5 


.27703 


.72296 


3.6096 


.28832 


3.4684 


1.0407 


.03914 


.96086 


55 


6 


. 7731 


. 2268 


.6060 


. 8863 


.4646 


.0408 


. 3922 


. 6078 


54 


7 


. 7759 


. 2240 


.6024 


. 8895 


.4608 


.0409 


. 3930 


. 6070 


53 


8 


. 7787 


. 2213 


.5987 


. 8926 


.4570 


.0410 


. 3938 


. 6062 


52 


9 


. 7815 


. 2185 


.5951 


. 8958 


.4533 


.0411 


. 3946 


. 6054 


51 


10 


.27843 


.72157 


3.5915 


.28990 


3.4495 


1.0412 


.03954 


.96045 


50 


11 


. 7871 


. 2129 


.5879 


. 9021 


.4458 


.0413 


. 3962 


. 6037 


49 


12 


. 7899 


. 2101 


.5843 


. 9053 


.4420 


.0413 


. 3971 


. 6029 


48 


13 


. 7927 


. 2073 


.5807 


. 9084 


.4383 


.0414 


. 3979 


. 6021 


47 


14 


. 7955 


. 2045 


.5772 


. 9116 


.4346 


.0415 


. 3987 


. 6013 


46 


15 


.27983 


.72017 


3.5736 


.29147 


3.4308 


1.0416 


.03995 


.96005 


45 


16 


. 8011 


. 1989 


.5700 


. 9179 


.4271 


.0417 


. 4003 


. 5997 


44 


17 


. 8039 


. 1961 


.5665 


. 9210 


.4234 


.0418 


. 4011 


. 5989 


43 


18 


. 8067 


. 1933 


.5629 


. 9242 


.4197 


.0419 


. 4019 


. 5980 


42 


19 


. 8094 


. 1905 


.5594 


. 9274 


.4160 


.0420 


. 4028 


. 5972 


41 


20 


.28122 


.71877 


3.5559 


.29305 


3.4124 


1.0420 


.04036 


.95964 


40 


21 


. 8150 


. 1849 


.5523 


. 9337 


.4087 


.0421 


. 4044 


. 5956 


39 


22 


. 8178 


. 1822 


.5488 


. 9368 


.4050 


.0422 


. 4052 


. 5948 


38 


23 


. 8206 


. 1794 


.5453 


. 9400 


.4014 


.0423 


. 4060 


. 5940 


37 


24 


. 8234 


. 1766 


.5418 


. 9432 


.3977 


.0424 


. 4069 


. 5931 


36 


25 


.28262 


.71738 


3.5383 


.29463 


3.3941 


1.0425 


.04077 


.95923 


35 


26 


. 8290 


. 1710 


.5348 


. 9495 


.3904 


.0426 


. 4085 


. 5915 


34 


27 


. 8318 


. 1682 


.5313 


. 9526 


.3868 


.0427 


. 4093 


. 5907 


33 


28 


. 8346 


. 1654 


.5279 


. 9558 


.3832 


.0428 


. 4101 


. 5898 


32 


29 


. 8374 


. 1626 


.5244 


. 9590 


.3795 


.0428 


. 4110 


. 5890 


31 


30 


.28401 


.71608 


3.5209 


.29621 


3.3759 


1.0429 


.04118 


.95882 


30 


31 


. 8429 


. 1570 


.5175 


. 9653 


.3723 


.0430 


. 4126 


. 5874 


29 


32 


. 8457 


. 1543 


.5140 


. 9685 


.3687 


.0431 


. 4134 


. 5865 


28 


33 


. 8185 


. 1515 


.5106 


. 9716 


.3651 


.0432 


. 4143 


. 5857 


27 


34 


. 8513 


. 1487 


.5072 


. 9748 


.3616 


.0433 


. 4151 


. 5849 


26 


35 


.28541 


.71459 


3.5037 


.29780 


3.3580 


1.0434 


.04159 


.95840 


25 


36 


. 8569 


. 1431 


.5003 


. 9811 


.3544 


.0435 


. 4168 


. 5832 


24 


37 


. 8597 


. 1403 


.4969 


. 9843 


.3509 


.0436 


. 4176 


. 5824 


23 


38 


. 8624 


. 1375 


.4935 


. 9875 


.3473 


.0437 


. 4184 


. 5816 


22 


39 


. 8652 


. 1347 


.4901 


. 9906 


.3438 


.0438 


. 4193 


. 5807 


21 


40 


.28680 


.71320 


3.4867 


.29938 


3.3402 


1.0438 


.04201 


.95799 


20 


41 


. 8708 


. 1292 


.4833 


. 9970 


.3367 


.0439 


. 4209 


. 5791 


19 


42 


. 8736 


. 1264 


.4799 


.30001 


.3332 


.0440 


. 4218 


. 5782 


18 


43 


. 8764 


. 1236 


.4766 


. 0033 


.3296 


.0441 


. 4226 


. 5774 


17 


44 


. 8792 


. 1208 


.4732 


. 0065 


.3261 


.0442 


. 4234 


. 5765 


16 


45 


.28820 


.71180 


3.4698 


.30096 


3.3226 


1.0443 


.04243 


.95757 


15 


46 


. 8847 


. 1152 


.4665 


. 0128 


.3191 


.0444 


. 4251 


. 5749 


14 


47 


. 8875 


. 1125 


.4632 


. 0160 


.3156 


.0445 


. 4260 


. 5740 


13 


48 


. 8903 


. 1097 


.4598 


. 0192 


.3121 


.0446 


. 4268 


. 5732 


12 


49 


. 8931 


. 1069 


.4565 


. 0223 


.3087 


.0447 


. 4276 


. 5723 


11 


50 


.28959 


.71041 


3.4532 


.30255 


3.3052 


1.0448 


.04285 


.95715 


10 


51 


. 8987 


. 1013 


.4498 


. 0287 


.3017 


.0448 


. 4293 


. 5707 


9 


52 


. 9014 


. 0985 


.4465 


. 0319 


.2983 


.0449 


. 4302 


. 5698 


8 


63 


. 9042 


. 0958 


. 1 132 


. 0350 


.2948 


.0450 


. 4310 


. 5690 


7 


54 


. 9070 


. 0930 


.4399 


. 0382 


.2914 


.0451 


. 4319 


. 5681 


6 


55 


.29098 


.70902 


3.4366 


.30414 


3.2879 


1.0452 


.04327 


.95673 


5 


56 


. 9126 


. 0874 


.4334 


. 0446 


.2845 


.0453 


. 4335 


. 5664 


4 


57 


. 9154 


. 0846 


.4301 


. 0178 


.2811 


.0454 


. 4344 


. 5656 


3 


58 


. 9181 


. 0818 


.4268 


. 0509 


.2777 


.0455 


. 4352 


. 5647 


2 


59 


. 9209 


. 0791 


.4236 


. 0541 


.2712 


.0456 


. 4361 


. 5639 


1 


60 


. 9237 


. 0763 


.4203 


. 0573 


.2708 


.0457 


. 4369 


. 5630 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Co tang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



106° 



73° 



Natukal Functions. 



151 



17° 


Natural Trigonometrical Functions. 


162° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.29237 


.70763 


3.4203 


.30573 


3.2708 


1.0457 


.04369 


.95630 


60 


1 


. 9265 


. 0735 


,4170 


. 0605 


.2674 


.0458 


. 4378 


. 5622 


59 


2 


. 9293 


. 0707 


.4138 


. 0637 


.2640 


.0459 


. 4386 


. 5613 


58 


3 


. 9321 


. 0679 


.4106 


. 0668 


.2607 


.0460 


. 4395 


. 5605 


57 


4 


. 9348 


. 0651 


.4073 


. 0700 


.2573 


.0461 


. 4404 


. 5596 


56 


5 


.29376 


.70624 


3.4041 


.30732 


3.2539 


1.0461 


.04412 


.95588 


55 


6 


. 9404 


. 0596 


.4009 


. 0764 


.2505 


.0462 


. 4421 


. 5579 


54 


7 


. 9432 


. 0568 


.3977 


. 0796 


.2472 


.0463 


. 4426 


. 5571 


53 


8 


. 9460 


. 0540 


.3945 


. 0828 


.2438 


.0464 


. 4438 


. 5562 


52 


9 


. 9487 


. 0512 


.3913 


. 0859 


.2405 


.0465 


. 4446 


. 5554 


51 


10 


.29515 


.70485 


3.3881 


.30891 


3.2371 


1.0466 


.04455 


.95545 


50 


11 


. 9543 


. 0457 


.3849 


. 0923 


.2338 


.0467 


. 4463 


. 5536 


49 


12 


. 9571 


. 0429 


.3817 


. 0955 


.2305 


.0468 


. 4472 


. 5528 


48 


13 


. 9598 


. 0401 


.3785 


. 0987 


.2271 


.0469 


. 4481 


. 5519 


47 


14 


. 9626 


. 0374 


.3754 


. 1019 


.2238 


.0470 


. 4489 


. 5511 


46 


15 


.29654 


.70346 


3.3722 


.31051 


3.2205 


1.0471 


.04498 


.95502 


45 


16 


. 9682 


. 0318 


.3690 


. 1083 


.2172 


.0472 


. 4507 


. 5493 


44 


17 


. 9710 


. 0290 


.3659 


. 1115 


.2139 


.0473 


. 4515 


. 5485 


43 


18 


. 9737 


. 0262 


.3627 


. 1146 


.2106 


.0474 


. 4524 


. 5476 


42 


19 


. 9765 


. 0235 


.3596 


. 1178 


.2073 


.0475 


. 4532 


. 5467 


41 


20 


.29793 


.70207 


3.3565 


.31210 


3.2041 


1.0476 


.04541 


.95459 


40 


21 


. 9821 


. 0179 


.3534 


. 1242 


.2008 


.0477 


. 4550 


. 5450 


39 


22 


. 9848 


. 0151 


.3502 


. 1274 


.1975 


.0478 


. 4558 


. 5441 


38 


23 


. 9876 


. 0124 


.3471 


. 1306 


.1942 


.0478 


. 4567 


. 5433 


37 


24 


. 9904 


. 0096 


.3440 


. 1338 


.1910 


.0479 


. 4576 


. 5424 


36 


25 


.29932 


.70068 


3.3409 


.31370 


3.1877 


1.0480 


.04585 


.95415 


35 


26 


. 9959 


. 0040 


.3378 


. 1402 


.1845 


.0481 


. 4593 


. 5407 


34 


27 


. 9987 


. 0013 


.3347 


. 1434 


.1813 


.0482 


. 4602 


. 5398 


33 


28 


.30015 


.69982 


.3316 


. 1466 


.1780 


.0483 


. 4611 


. 5389 


32 


29 


. 0043 


. 9957 


.3286 


. 1498 


.1748 


.0484 


. 4619 


. 5380 


31 


30 


.30070 


.69929 


3.3255 


.31530 


3.1716 


1.0485 


.04628 


.95372 


30 


31 


. 0098 


. 9902 


.3224 


. 1562 


.1684 


.0486 


. 4637 


. 5363 


29 


32 


. 0126 


. 9874 


.3194 


. 1594 


.1652 


.0487 


. 4646 


. 5354 


28 


33 


. 0154 


. 9846 


.3163 


. 1626 


.1620 


.0488 


. 4654 


. 5345 


27 


34 


. 0181 


. 9818 


.3133 


. 1658 


.1588 


.0489 


. 4663 


. 5337 


26 


35 


.30209 


.69791 


3.3102 


.31690 


3.1556 


1.0490 


.04672 


.95328 


25 


36 


. 0237 


. 9763 


.3072 


. 1722 


.1524 


.0491 


. 4681 


. 5319 


24 


37 


. 0265 


. 9735 


.3042 


. 1754 


.1492 


.0192 


. 4690 


. 5310 


23 


38 


. 0292 


. 9707 


.3011 


. 1786 


.1460 


.0493 


. 4698 


. 5301 


22 


39 


. 0320 


. 9680 


.2981 


. 1818 


.1429 


.0494 


. 4707 


. 5293 


21 


40 


.30348 


.69652 


3.2951 


.31850 


3.1397 


1.0495 


.04716 


.95284 


20 


41 


. 0375 


. 9624 


.2921 


. 1882 


.1366 


.0496 


. 4725 


. 5275 


19 


42 


. 0403 


. 9597 


.2891 


. 1914 


.1334 


.0497 


. 4734 


. 5266 


18 


43 


. 0431 


. 9569 


.2861 


. 1946 


.1303 


.0498 


. 4743 


. 5257 


17 


44 


. 0459 


. 9541 


.2831 


. 1978 


.1271 


.0499 


. 4751 


. 5248 


16 


45 


.30486 


.69513 


3.2801 


.32010 


3.1240 


1.0500 


.04760 


.95239 


15 


46 


. 0514 


. 9486 


.2772 


. 2042 


.1209 


.0501 


. 4769 


. 5231 


14 


47 


. 0542 


. 9458 


.2742 


. 2074 


.1177 


.0502 


. 4778 


. 5222 


13 


48 


. 0569 


. 9430 


.2712 


. 2106 


.1146 


.0503 


. 4787 


. 5213 


12 


49 


. 0597 


. 9403 


.2683 


. 2138 


.1115 


.0504 


. 4796 


. 5204 


11 


50 


.30625 


.69375 


3.2653 


.32171 


3.1084 


1.0505 


.04805 


.95195 


10 


51 


. 0653 


. 9347 


.2624 


. 2203 


.1053 


.0506 


. 4814 


. 5186 


9 


52 


. 0680 


. 9320 


.2594 


. 2235 


.1022 


.0507 


. 4823 


. 5177 


8 


53 


. 0708 


. 9292 


.2565 


. 2267 


.0991 


.0508 


. 4832 


. 5168 


7 


54 


. 0736 


. 9264 


.2535 


. 2299 


.0960 


.0509 


. 4840 


. 5159 


6 


55 


.30763 


.69237 


3.2506 


.32331 


3.0930 


1.0510 


.04849 


.95150 


5 


56 


. 0791 


. 9209 


.2477 


. 2363 


.0899 


.0511 


. 4858 


. 5141 


4 


57 


. 0819 


. 9181 


.2448 


. 2395 


.0868 


.0512 


. 4867 


. 5132 


3 


58 


. 0846 


. 9154 


.2419 


. 2428 


.0838 


.0513 


. 4876 


. 5124 


2 


59 


. 0874 


. 9126 


.2390 


. 2460 


.0807 


.0514 


. 4885 


. 5115 


1 


60 


. 0902 


. 9098 


.2361 


. 2492 


.0777 


.0515 


. 4894 


. 5106 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



107° 



72° 



152 



Natural Functions. 



18 


D 


Natural Trigonometrical 


Functions. 


161° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.30902 


.69098 


3.2361 


.32492 


3.0777 


1.0515 


.04894 


.95106 


60 


1 


. 0929 


. 9071 


.2332 


. 2524 


.0746 


.0516 


. 4903 


. 5097 


59 


2 


. 0957 


. 9043 


.2303 


. 2556 


.0716 


.0517 


. 4912 


. 5088 


58 


3 


. 0985 


. 9015 


.2274 


. 2588 


.0686 


.0518 


. 4921 


. 5079 


57 


4 


. 1012 


. 8988 


.2245 


. 2621 


.0655 


.0519 


. 4930 


. 5070 


56 


5 


.31040 


.68960 


3.2216 


.32653 


3.0625 


1.0520 


.04939 


.95061 


55 


6 


. 1068 


. 8932 


.2188 


. 2685 


.0595 


.0521 


. 4948 


. 5051 


54 


7 


. 1095 


. 8905 


.2159 


. 2717 


.0565 


.0522 


. 4957 


. 5042 


53 


8 


. 1123 


. 8877 


.2131 


. 2749 


.0535 


.0523 


. 4966 


. 5033 


52 


9 


. 1150 


. 8849 


.2102 


. 2782 


.0505 


.0524 


. 4975 


. 5024 


51 


10 


.31178 


.68822 


3.2074 


.32814 


3.0475 


1.0525 


.04985 


.95015 


50 


11 


. 1206 


. 8794 


.2045 


. 2846 


.0445 


.0526 


. 4994 


. 5006 


49 


12 


. 1233 


. 8766 


.2017 


. 2878 


.0415 


.0527 


. 5003 


. 4997 


48 


13 


. 1261 


. 8739 


.1989 


. 2910 


.0385 


.0528 


. 5012 


. 4988 


47 


14 


. 1289 


. 8711 


.1960 


. 2943 


.0356 


.0529 


. 5021 


. 4979 


46 


15 


.31316 


.68684 


3.1932 


.32975 


3.0326 


1.0530 


.05030 


.94970 


45 


16 


. 1344 


. 8656 


.1904 


. 3007 


.0296 


.0531 


. 5039 


. 4961 


44 


17 


. 1372 


. 8628 


.1876 


. 3039 


.0267 


.0532 


. 5048 


. 4952 


43 


18 


. 1399 


. 8601 


.1848 


. 3072 


.0237 


.0533 


. 5057 


. 4942 


42 


19 


. 1427 


. 8573 


.1820 


. 3104 


.0208 


.0534 


. 5066 


. 4933 


41 


20 


.31454 


.68545 


3.1792 


.33136 


3.0178 


1.0535 


.05076 


.94924 


40 


21 


. 1482 


. 8518 


.1764 


. 3169 


.0149 


.0536 


. 5085 


. 4915 


39 


22 


. 1510 


. 8490 


.1736 


. 3201 


.0120 


.0537 


. 5094 


. 4906 


38 


23 


. 1537 


. 8463 


.1708 


. 3233 


.0090 


.0538 


. 5103 


. 4897 


37 


24 


. 1565 


. 8435 


.1681 


. 3265 


.0061 


.0539 


. 5112 


. 4888 


36 


25 


.31592 


.68407 


3.1653 


.33298 


3.0032 


1.0540 


.05121 


.94878 


35 


26 


. 1620 


. 8380 


.1625 


. 3330 


.0003 


.0541 


. 5131 


. 4869 


34 


27 


. 1648 


. 8352 


.1598 


. 3362 


2.9974 


.0542 


. 5140 


. 4860 


33 


28 


. 1675 


. 8325 


.1570 


. 3395 


.9945 


.0543 


. 5149 


. 4851 


32 


29 


. 1703 


. 8297 


.1543 


. 3427 


.9916 


.0544 


. 5158 


. 4841 


31 


30 


.31730 


.68269 


3.1515 


.33459 


2.9887 


1.0545 


.05168 


.94832 


30 


31 


. 1758 


. 8242 


.1488 


. 3492 


.9858 


.0546 


. 5177 


. 4823 


29 


32 


. 1786 


. 8214 


.1461 


. 3524 


.9829 


.0547 


. 5186 


. 4814 


28 


33 


. 1813 


. 8187 


.1433 


. 3557 


.9800 


.0548 


. 5195 


. 4805 


27 


34 


. 1841 


. 8159 


.1106 


. 3589 


.9772 


.0549 


. 5205 


. 4795 


26 


35 


.31868 


.68132 


3.1379 


.33621 


2.9743 


1.0550 


.05214 


.94786 


25 


36 


. 1896 


. 8104 


.1352 


. 3654 


.9714 


.0551 


. 5223 


. 4777 


24 


37 


. 1923 


. 8076 


.1325 


. 3686 


.9686 


.0552 


. 5232 


. 4767 


23 


38 


. 1951 


. 8049 


.1298 


. 3718 


.9657 


.0553 


. 5242 


. 4758 


22 


39 


. 1978 


. 8021 


.1271 


. 3751 


.9629 


.0554 


. 5251 


. 4749 


21 


40 


.32006 


.67994 


3.1244 


.33783 


2.9600 


1.0555 


.05260 


.94740 


20 


41 


. 2034 


. 7966 


.1217 


. 3816 


.9572 


.0556 


. 5270 


. 4730 


19 


42 


. 2061 


. 7939 


.1190 


. 3848 


.9544 


.0557 


. 5279 


. 4721 


18 


43 


. 2089 


. 7911 


.1163 


. 3880 


.9515 


.0558 


. 5288 


. 4712 


17 


44 


. 2116 


. 7SS4 


.1137 


. 3913 


.9487 


.0559 


. 5297 


. 4702 


16 


45 


.32144 


.67856 


3.1110 


.33945 


2.9459 


1.0560 


.05307 


.94693 


15 


46 


. 2171 


. 7828 


.1083 


. 3978 


.9431 


.0561 


. 5316 


. 4684 


14 


47 


. 2199 


. 7801 


.1057 


. 4010 


.9403 


.0562 


. 5326 


. 4674 


13 


48 


. 2226 


. 7773 


.1030 


. 4043 


.9375 


.0563 


. 5335 


. 4665 


12 


49 


. 2254 


. 7746 


.1004 


. 4075 


.9347 


.0565 


. 5344 


. 4655 


11 


50 


.32282 


.67718 


3.0977 


.34108 


2.9319 


1.0566 


.05354 


.94646 


10 


51 


. 2309 


. 7691 


.0951 


. 4140 


.9291 


.0567 


. 5363 


. 4637 


9 


52 


. 2337 


. 7663 


.0925 


. 4173 


.9263 


.0568 


. 5373 


. 4627 


8 


53 


. 2364 


. 7636 


.0898 


. 4205 


.9235 


.0569 


. 5382 


. 4618 


7 


54 


. 2392 


. 7 cos 


.0872 


. 4238 


.9208 


.0570 


. 5391 


. 4608 


6 


55 


.32419 


.67581 


3.0846 


.34270 


2.9180 


1.0571 


.05401 


.94599 


5 


56 


. 2447 


. 7553 


.0820 


. 4303 


.9152 


.0572 


. 5410 


. 4590 


4 


57 


. 2474 


. 7526 


.0793 


. 4335 


.9125 


.0573 


. 5420 


. 4580 


3 


68 


. 2502 


. 7498 


.0767 


. 4368 


.9097 


.0574 


. 5429 


. 4571 


2 


50 


. 2529 


. 7171 


.0741 


. 4400 


.9069 


.0575 


. 5439 


. 4561 


1 


60 


. 2557 


. 7443 


.0715 


. 4433 


.9042 


.0576 


. 5448 


. 4552 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



108° 



71° 



Natueal Functions. 



153 



19° 


Natural Trigonom 


etrical Functions. 


160° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.32557 


.67443 


3.0715 


.34433 


2.9042 


1.0576 


.05448 


.94552 


60 


1 


. 2584 


. 7416 


.0690 


. 4465 


.9015 


.0577 


. 5458 


. 4542 


59 


2 


. 2612 


. 7388 


.0664 


. 4498 


.8987 


.0578 


. 5467 


. 4533 


58 


3 


. 2639 


. 7361 


.0638 


. 4530 


.8960 


.0579 


. 5476 


. 4523 


57 


4 


. 2667 


. 7333 


.0612 


. 4563 


.8933 


.0580 


. 5486 


. 4514 


56 


5 


.32694 


.67306 


3.0586 


.34595 


2.8905 


1.0581 


.05495 


.94504 


55 


6 


. 2722 


. 7278 


.0561 


. 4628 


.8878 


.0582 


. 5505 


. 4495 


54 


7 


. 2749 


. 7251 


.0535 


. 4661 


.8851 


.0584 


. 5515 


. 4485 


53 


8 


. 2777 


. 7223 


.0509 


. 4693 


.8824 


.0585 


. 5524 


. 4476 


52 


9 


. 2804 


. 7196 


.0484 


. 4726 


.8797 


.0586 


. 5534 


. 4466 


51 


10 


.32832 


.67168 


3.0458 


.34758 


2.8770 


1.0587 


.05543 


.94457 


50 


11 


. 2859 


. 7141 


.0433 


. 4791 


.8743 


.0588 


. 5553 


. 4447 


49 


12 


. 2887 


. 7113 


.0407 


. 4824 


.8716 


.0589 


. 5562 


. 4438 


48 


13 


. 2914 


. 7086 


.0382 


. 4856 


.8689 


.0590 


. 5572 


. 4428 


47 


14 


. 2942 


. 7058 


.0357 


. 4889 


.8662 


.0591 


. 5581 


. 4418 


46 


15 


.32969 


.67031 


3.0331 


.34921 


2.8636 


1.0592 


.05591 


.94409 


45 


16 


. 2996 


. 7003 


.0306 


. 4954 


.8609 


.0593 


. 5601 


. 4399 


44 


17 


. 3024 


. 6976 


.0281 


. 4987 


.8582 


.0594 


. 5610 


. 4390 


43 


18 


. 3051 


. 6948 


.0256 


. 5019 


.8555 


.0595 


. 5620 


. 4380 


42 


19 


. 3079 


. 6921 


.0231 


. 5052 


.8529 


.0596 


. 5629 


. 4370 


41 


20 


.33106 


.66894 


3.0206 


.35085 


2.8502 


1.0598 


.05639 


.94361 


40 


21 


. 3134 


. 6866 


.0181 


. 5117 


.8476 


.0599 


. 5649 


. 4351 


39 


22 


. 3161 


. 6839 


.0156 


. 5150 


.8449 


.0600 


. 5658 


. 4341 


38 


23 


. 3189 


. 6811 


.0131 


. 5183 


.8423 


.0601 


. 5668 


. 4332 


37 


24 


. 3216 


. 6784 


.0106 


. 5215 


.8396 


.0602 


. 5678 


. 4322 


36 


25 


.33243 


.66756 


3.0081 


.35248 


2.8370 


1.0603 


.05687 


.94313 


35 


26 


. 3271 


. 6729 


.0056 


. 5281 


.8344 


.0604 


. 5697 


. 4303 


34 


27 


. 3298 


. 6701 


.0031 


. 5314 


.8318 


.0605 


. 5707 


. 4293 


33 


28 


. 3326 


. 6674 


.0007 


. 5346 


.8291 


.0606 


. 5716 


. 4283 


32 


29 


. 3353 


. 6647 


2.9982 


. 5379 


.8265 


.0607 


. 5726 


. 4274 


31 


30 


.33381 


.66619 


2.9957 


.35412 


2.8239 


1.0608 


.05736 


.94264 


30 


31 


. 3408 


. 6592 


.9933 


. 5445 


.8213 


.0609 


. 5745 


. 4254 


29 


32 


. 3435 


. 6564 


.9908 


. 5477 


.8187 


.0611 


. 5755 


. 4245 


28 


33 


. 3463 


. 6537 


.9884 


. 5510 


.8161 


.0612 


. 5765 


. 4235 


27 


34 


. 3490 


. 6510 


.9859 


. 5543 


.8135 


.0613 


. 5775 


. 4225 


26 


35 


.33518 


.66482 


2.9835 


.35576 


2.8109 


1.0614 


.05784 


.94215 


25 


36 


. 3545 


. 6455 


.9810 


. 5608 


.8083 


.0615 


. 5794 


. 4206 


24 


37 


. 3572 


. 6427 


.9786 


. 5641 


.8057 


.0616 


. 5804 


. 4196 


23 


38 


. 3600 


. 6400 


.9762 


. 5674 


.8032 


.0617 


. 5814 


. 4186 


22 


39 


. 3627 


. 6373 


.9738 


. 5707 


.8006 


.0618 


. 5823 


. 4176 


21 


40 


.33655 


.66345 


2.9713 


.35739 


2.7980 


1.0619 


.05833 


.94167 


20 


41 


. 3682 


. 6318 


.9689 


. 5772 


.7954 


.0620 


. 5843 


. 4157 


19 


42 


. 3709 


. 6290 


.9665 


. 5805 


.7929 


.0622 


. 5853 


. 4147 


18 


43 


. 3737 


. 6263 


.9641 


. 5838 


.7903 


.0623 


. 5863 


. 4137 


17 


44 


. 3764 


. 6236 


.9617 


. 5871 


.7878 


.0624 


. 5872 


. 4127 


16 


45 


.33792 


.66208 


2.9593 


.35904 


2.7852 


1.0625 


.05882 


.94118 


15 


46 


. 3819 


. 6181 


.9569 


. 5936 


.7827 


.0626 


. 5892 


. 4108 


14 


47 


. 3846 


. 6153 


.9545 


. 5969 


.7801 


.0627 


. 5902 


. 4098 


13 


48 


. 3874 


. 6126 


.9521 


. 6002 


.7776 


.0628 


. 5912 


. 4088 


12 


49 


. 3901 


. 6099 


.9497 


. 6035 


.7751 


.0629 


. 5922 


. 4078 


11 


50 


.33928 


.66071 


2.9474 


.36068 


2.7725 


1.0630 


.05932 


.94068 


10 


51 


. 3956 


. 6044 


.9450 


. 6101 


.7700 


.0632 


. 5941 


. 4058 


9 


52 


. 3983 


. 6017 


.9426 


. 6134 


.7675 


.0633 


. 5951 


. 4049 


8 


53 


. 4011 


. 5989 


.9402 


. 6167 


.7650 


.0634 


. 5961 


. 4039 


7 


54 


. 4038 


. 5962 


.9379 


. 6199 


.7625 


.0635 


. 5971 


. 4029 


6 


55 


.34065 


.65935 


2.9355 


.36232 


2.7600 


1.0636 


.05981 


.94019 


5 


56 


. 4093 


. 5907 


.9332 


. 6265 


.7574 


.0637 


. 5991 


. 4009 


4 


57 


. 4120 


. 5880 


.9308 


. 6298 


.7549 


.0638 


. 6001 


. 3999 


3 


58 


. 4147 


. 5853 


.9285 


. 6331 


.7524 


.0639 


. 6011 


. 3989 


2 


59 


. 4175 


. 5825 


1 .9261 


. 6364 


.7500 


.0641 


. 6021 


. 3979 


1 


60 


. 4202 


. 5798 


' .9238 


. 6397 


.7475 


.0642 


. 6031 


. 3969 





M. 


Cosine. 


Vrs. sin.j 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



109° 



70° 



154 



Natural Functions. 



20° 




Natural Trigonometrical Functions. 


159° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Co tang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.34202 


.65798 


2.9238 


.36397 


2.7475 


1.0642 


.06031 


.93969 


60 


1 


. 4229 


. 5771 


.9215 


. 6430 


.7450 


.0643 


. 6041 


. 3959 


59 


2 


. 4257 


. 5743 


.9191 


. 6463 


.7425 


.0644 


. 6051 


. 3949 


58 


3 


. 4284 


. 5716 


.9168 


. 6496 


.7400 


.0645 


. 6061 


. 3939 


57 


4 


. 4311 


. 5689 


.91 15 


. 6529 


.7376 


.0646 


. 6071 


. 3929 


56 


5 


.34339 


.65661 


2.9122 


.36562 


2.7351 


1.0647 


.06080 


.93919 


55 


6 


. 4366 


. 5634 


.9098 


. 6595 


.7326 


.0648 


. 6090 


. 3909 


54 


7 


. 4393 


. 5607 


.9075 


. 6628 


.7302 


.0650 


. 6100 


. 3899 


53 


8 


. 4421 


. 5579 


.9052 


. 6661 


.7277 


.0651 


. 6110 


. 3889 


52 


9 


. ins 


. 5552 


.9029 


. 6694 


.7252 


.0652 


. 6121 


. 3879 


51 


10 


.34475 


.65525 


2.9006 


.36727 


2.7228 


1.0653 


.06)131 


.93869 


50 


11 


. 4502 


. 5497 


.8983 


. 6760 


.7204 


.0654 


. 6141 


. 3859 


19 


12 


. 4530 


. 5470 


.8960 


. 6793 


.7179 


.0655 


. 6151 


. 3849 


48 


13 


. 4557 


. 5443 


.8937 


. 6826 


.7155 


.0656 


. 6161 


. 3839 


17 


14 


. 4584 


. 5415 


.8915 


. 6859 


.7130 


.0658 


. 6171 


. 3829 


16 


15 


.34612 


.65388 


2.8892 


.36892 


2.7106 


1.0659 


.06181 


.93819 


45 


16 


. 4639 


. 5361 


.8869 


. 6925 


.7082 


.0660 


. 6191 


. 3809 


14 


17 


. 4666 


. 5334 


.8846 


. 6958 


.7058 


.0661 


. 6201 


. 3799 


13 


18 


. 4693 


. 5306 


.8824 


. 6991 


.7033 


.0662 


. 6211 


. 3789 


12 


19 


. 4721 


. 5-279 


.8801 


. 7024 


.7009 


.0663 


. 6221 


. 3779 


11 


20 


.34748 


.65252 


2.8778 


.37057 


2.6985 


1.0664 


.06231 


.93769 


40 


21 


. 4775 


. 5225 


.8756 


. 7090 


.6961 


.0666 


. 6241 


. 3758 


39 


22 


. 4803 


. 5197 


.8733 


. 7123 


.6937 


.0667 


. 6251 


. 3748 


38 


23 


. 4830 


. 5170 


.8711 


. 7156 


.6913 


.0668 


. 6262 


. 3738 


3,7 


24 


. 4857 


. 5143 


.8688 


. 7190 


.6889 


.0669 


. 0272 


. 3728 


36 


25 


.34884 


.65115 


2.8666 


.37223 


2.6865 


1.0670 


.00282 


.93718 


35 


26 


. 4912 


. 5088 


.8644 


. 7256 


.6841 


.0671 


. 6292 


. 3708 


34 


27 


. 4939 


. 5061 


.8621 


. 7289 


.6817 


.0673 


. 6302 


. 3698 


33 


28 


. 4966 


. 5034 


.8599 


. 7322 


.6794 


.0674 


. 6312 


. 3687 


32 


29 


. 4993 


. 5006 


.8577 


. 7355 


.6770 


.0675 


. 6323 


. 3677 


31 


30 


.35021 


.64979 


2.8554 


.37388 


2.0746 


1.0676 


.06333 


.93667 


30 


31 


. 5018 


. 4952 


.8532 


. 7422 


.6722 


.0677 


. 6343 


. 3657 


29 


32 


. 5075 


. 4925 


.8510 


. 7455 


.6699 


.0678 


. 6353 


. 3647 


28 


33 


. 5102 


. 4897 


.8488 


. 7 Ins 


.6675 


.0(579 


. 6363 


. 3637 


27 


34 


. 5130 


. 4870 


.8466 


. 7521 


.6652 


.0081 


. 6373 


. 3626 


26 


35 


.35157 


.648 13 


2.8444 


.37554 


2.6628 


1.0682 


.06384 


.93616 


25 


36 


. 5184 


. 4816 


.8422 


. 7587 


.6604 


.0683 


. 03,01 


. 3606 


24 


37 


. 5211 


. 4789 


.8400 


. 7621 


.6581 


.0681 


. 6104 


. 3596 


23 


38 


. 5239 


. 4761 


.8378 


. 7654 


.6558 


.0685 


. 6414 


. 3585 


22 


39 


. 5266 


. 4734 


.8356 


. 7687 


.6534 


.0686 


. 0125 


. 3575 


21 


40 


.35293 


.61707 


2.8334 


.3,7720 


2.6511 


1.0688 


.00135 


.93565 


20 


41 


. 5320 


. 1680 


.S3 12 


. 7754 


.6487 


.0689 


. our, 


. 3555 


19 


42 


. 5347 


. 4652 


.8290 


. 7787 


.6464 


.0690 


. 6456 


. 3544 


18 


43 


. 5375 


. 4625 


.8269 


. 7820 


.6441 


.0691 


. 0100 


. 3534 


17 


44 


. 5102 


. 4598 


.8247 


. 7853 


.6418 


.0692 


. 6176 


. 3524 


16 


45 


.35429 


.64571 


2.S225 


.37887 


2.6394 


1.0694 


.00186 


.93513 


15 


46 


. 5456 


. 1544 


.8204 


. 7920 


.6371 


.0695 


. 6197 


. 3503 


14 


47 


. 5483 


. 1516 


.8182 


. 7953 


.6348 


. 009(5 


. 6507 


. 3,193 


13 


48 


. 5511 


. lis'.) 


.8160 


. 7986 


.6325 


.0697 


. 6517 


. 3482 


12 


49 




. 4462 


.8139 


. 8020 


.03,02 


.0698 


. 6528 


. 3,172 


11 


50 


.35565 


.64435 


2.8117 


.38053 


2.0279 


1.0699 


.06538 


.93462 


10 


51 


. 5592 


. 4408 


.8096 


. B086 


.6256 


.0701 


. 6548 


. 3451 


9 


52 


. 5619 


. 4380 


.8074 


. 8120 


.0233 


.0702 


. 6559 


. 3441 


8 


53 


. 5647 


. 1353 


.8053 


. 8153 


.0210 


.0703 


. 6569 


. 3431 


7 


54 


. 5674 


. 4326 


.8032 


. 8186 


.6187 


.0704 


. 0579 


. 3420 


6 




.35701 


.64299 


2. SO 10 




2.6164 


1.0705 


.06590 


.03,110 


5 




. 6728 


. 4272 


.7989 


. 8253 


.01 12 


.0707 


. 0000 


. 3,100 


4 


:>7 


. 5755 


. 4245 


.7 '.'tis 


. 8286 


.6119 


.0708 


. 6611 


. 3389 


3 


58 




. 4217 


.7947 


. 8320 


.6096 


.0709 


. 0021 


. 3,3,79 


2 




. 5810 


. 1190 


.7925 




.6073 


.0710 


. 003,1 


. 3,3,08 


1 




. 6837 


. 1163 


.7904 




.6051 


.0711 


. 0012 


. :\:\:^ 





M. 


Cosine. 


Vn. -in. 


Secant 


Cotang. 


Tan K . 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 


11 


0° 














< 


>o° 



Natural Functions. 



155 



21° 


Natural Trigonometrical Functions. 


158° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 
1.0711 


Vrs. sin. 


Cosine. 


M. 





.35837 


.64163 


2.7904 


.38386 


2.6051 


.06642 


.93358 


60 


1 


. 5864 


. 4136 


.7883 


. 8420 


.6028 


.0713 


. 6652 


. 3348 


59 


2 


. 5891 


. 4109 


.7862 


. 8453 


.6006 


.0714 


. 6663 


. 3337 


58 


3 


. 5918 


. 4082 


.7841 


. 8486 


.5983 


.0715 


. 6673 


. 3327 


57 


4 


. 5945 


. 4055 


.7820 


. 8520 


.5960 


.0716 


. 6684 


. 3316 


56 


5 


.35972 


.64027 


2.7799 


.38553 


2.5938 


1.0717 


.06694 


.93306 


55 


6 


. 6000 


. 4000 


.7778 


. 8587 


.5916 


.0719 


. 6705 


. 3295 


54 


7 


. 6027 


. 3973 


.7757 


. 8620 


.5893 


.0720 


. 6715 


. 3285 


53 


8 


. 6054 


. 3946 


.7736 


. 8654 


.5871 


.0721 


. 6726 


. 3274 


52 


9 


. 6081 


. 3919 


.7715 


. 8687 


.5848 


.0722 


. 6736 


. 3264 


51 


10 


.36108 


.63892 


2.7694 


.38720 


2.5826 


1.0723 


.06747 


.93253 


50 


11 


. 6135 


. 3865 


.7674 


. 8754 


.5804 


.0725 


. 6757 


. 3243 


49 


12 


. 6162 


. 3837 


.7653 


. 8787 


.5781 


.0726 


. 6768 


. 3232 


48 


13 


. 6189 


. 3810 


.7632 


. 8821 


.5759 


.0727 


. 6778 


. 3222 


47 


U 


. 6217 


. 3783 


.7611 


. 8854 


.5737 


.0728 


. 6789 


. 3211 


46 


15 


.36244 


.63756 


2.7591 


.38888 


2.5715 


1.0729 


.06799 


.93201 


45 


16 


. 6271 


. 3729 


.7570 


. 8921 


.5693 


.0731 


. 6810 


. 3190 


44 


17 


. 6298 


. 3702 


.7550 


. 8955 


.5671 


.0732 


. 6820 


. 3180 


43 


18 


. 6325 


. 3675 


.7529 


. 8988 


.5610 


.0733 


. 6831 


. 3169 


42 


19 


. 6352 


. 3618 


.7509 


. 9022 


.5627 


.0734 


. 6841 


. 3158 


41 


'20 


.36379 


.63621 


2.7488 


.39055 


2.5605 


1.0736 


.06852 


.93148 


40 


21 


. 6406 


. 3593 


.7468 


. 9089 


.5583 


.0737 


. 6863 


. 3137 


39 


22 


. 6433 


. 3566 


.7447 


. 9122 


.5561 


.0738 


. 6873 


. 3127 


38 


23 


. 6460 


. 3539 


.7427 


. 9156 


.5539 


.0739 


. 6884 


. 3116 


37 


21 


. 6488 


. 3512 


.7406 


. 9189 


.5517 


.0710 


. 6894 


. 3105 


36 


25 


.36515 


.63485 


2.7386 


.39223 


2.5495 


1.0742 


.06905 


.93095 


35 


26 


. 6542 


. 3458 


.7366 


. 9257 


.5473 


.0743 


. 6916 


. 3084 


34 


27 


. 6569 


. 3431 


.7346 


. 9290 


.5451 


.0744 


. 6926 


. 3074 


33 


2S 


. 6596 


. 3404 


.7325 


. 9324 


.5430 


.0745 


. 6937 


. 3063 


32 


29 


. 6623 


. 3377 


.7305 


. 9357 


.5408 


.0747 


. 6947 


. 3052 


31 


30 


.36650 


.63350 


2.7285 


.39391 


2.5386 


1.0748 


.06958 


.93042 


30 


31 


. 6677 


. 3323 


.7265 


. 9125 


.5365 


.0749 


. 6969 


. 3031 


29 


32 


. 6704 


. 3296 


.7245 


. 9458 


.5343 


.0750 


. 6979 


. 3020 


28 


33 


. 6731 


. 3269 


.7225 


. 9492 


.5322 


.0751 


. 6990 


. 3010 


27 


31 


. 6758 


. 3242 


.7205 


. 9525 


.5300 


.0753 


. 7001 


. 2999 


26 


35 


.36785 


.63214 


2.7185 


.39559 


2.5278 


1.0754 


.07012 


.92988 


25 


36 


. 6812 


. 3187 


.7165 


. 9593 


.5257 


.0755 


. 7022 


. 2978 


24 


37 


. 6839 


. 3160 


.7145 


. 9626 


.5236 


.0756 


. 7033 


. 2967 


23 


38 


. 6866 


. 3133 


.7125 


. 9660 


.5214 


.0758 


. 7044 


. 2956 


22 


39 


. 6893 


. 3106 


.7105 


. 9694 


.5193 


.0759 


. 7054 


. 2915 


21 


40 


.36921 


.63079 


2.7085 


.39727 


2.5171 


1.0760 


.07065 


.92935 


20 


41 


. 6948 


. 3052 


.7065 


. 9761 


.5150 


.0761 


. 7076 


. 2924 


19 


42 


. 6975 


. 3025 


.7045 


. 9795 


.5129 


.0763 


. 7087 


. 2913 


18 


43 


. 7002 


. 2998 


.7026 


. 9828 


.5108 


.0764 


. 7097 


. 2902 


17 


44 


. 7029 


. 2971 


.7006 


. 9862 


.5086 


.0765 


. 7108 


. 2892 


16 


45 


.37056 


.62944 


2.6986 


.39896 


2.5065 


1.0766 


.07119 


.92881 


15 


46 


i 7083 


. 2917 


.6967 


. 9930 


.5044 


.0768 


. 7130 


. 2870 


14 


47 


. 7110 


. 2890 


.6947 


. 9963 


.5023 


.0769 


. 7141 


. 2859 


13 


48 


. 7137 


. 2863 


.6927 


. 9997 


.5002 


.0770 


. 7151 


. 2848 


12 


49 


. 7164 


. 2836 


.6908 


.40031 


.4981 


.0771 


. 7162 


. 2838 


11 


50 


.37191 


.62809 


2.6888 


.40065 


2.4960 


1.0773 


.07173 


.92827 


10 


51 


. 7218 


. 2782 


.6869 


. 0098 


.4939 


.0774 


. 7184 


. 2816 


9 


52 


. 7245 


. 2755 


.6849 


. 0132 


.4918 


.0775 


. 7195 


. 2805 


8 


53 


. 7272 


. 2728 


.6830 


. 0166 


.4897 


.0776 


. 7205 


. 2794 


7 


54 


. 7299 


. 2701 


.6810 


. 0200 


.4876 


.0778 


. 7216 


. 2784 


6 


55 


.37326 


.62674 


2.6791 


.10233 


2.4855 


1.0779 


.07227 


.92773 


5 


56 


. 7353 


. 2647 


.6772 


. 0267 


.4834 


.0780 


. 7238 


. 2762 


4 


57 


. 7380 


. 2620 


.6752 


. 0301 


.4813 


.0781 


. 7249 


. 2751 


3 


58 


. 7407 


. 2593 


.6733 


. 0335 


.4792 


.0783 


. 7260 


. 2740 


2 


59 


. 7434 


. 2566 


.6714 


. 0369 


.4772 


.0784 


. 7271 


. 2729 


1 


60 


. 7461 


. 2539 


.6695 
Secant. 


. 0403 


.4751 


.0785 


. 7282 


. 2718 





M. 


Cosine. 


Vrs. sin. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



111° 



68° 



156 



Natural Functions. 



22 c 




Natural Trigonometrical Functions. 


157° 


M. 


Sine. 


Vrs. cos. 


Cosee'nt 


Tan S . 


Cotang. 


Secant. 


Vrs. 6in. 


Cosine. 


M. 





.37461 


.62539 


2.6695 


.40403 


2.4751 


1.0785 


.07282 


.92718 


60 


1 


. 7488 


. 2512 


.6675 


. 0436 


.4730 


.0787 


. 7292 


. 2707 


59 


2 


. 7514 


. 2485 


.6656 


. 0470 


.4709 


.0788 


. 7303 


. 2696 


58 


3 


. 7541 


. 2458 


.6637 


. 0504 


.4689 


.0789 


. 7314 


. 2686 


57 


4 


. 7568 


. 2431 


.6618 


. 0538 


.4668 


.0790 


. 7325 


. 2675 


56 


5 


.37595 


.62404 


2.6599 


.40572 


2.4647 


1.0792 


.07336 


.92664 


55 


6 


. 7622 


. 2377 


.6580 


. 0606 


.4627 


.0793 


. 7347 


. 2653 


54 


7 


. 7649 


. 2351 


.6561 


. 0640 


.4606 


.0794 


. 7358 


. 2642 


53 


8 


. 7676 


. 2324 


.6542 


. 0673 


.4586 


.0795 


. 7369 


. 2631 


52 


9 


. 7703 


. 2297 


.6523 


. 0707 


.4565 


.0797 


. 7380 


. 2620 


51 


10 


.37730 


.62270 


2.6504 


.40741 


2.4545 


1.0798 


.07391 


.92609 


50 


11 


. 7757 


. 2243 


.6485 


. 0775 


.4525 


.0799 


. 7402 


. 2598 


19 


12 


. 7784 


. 2216 


.6466 


. 0809 


.4504 


.0801 


. 7413 


. 2587 


48 


13 


. 7811 


. 2189 


.6447 


. 0843 


.4484 


.0802 


. 7424 


. 2576 


47 


14 


. 7838 


. 2162 


.6428 


. 0877 


.4463 


.0803 


. 7435 


. 2565 


46 


15 


.37865 


.62135 


2.6410 


.40911 


2.4443 


1.0804 


.07446 


.92554 


45 


16 


. 7892 


. 2108 


.6391 


. 0945 


.4423 


.0806 


. 7457 


. 2543 


14 


17 


. 7919 


. 2081 


.6372 


. 0979 


.4403 


.0807 


. 7468 


. 2532 


43 


18 


. 7946 


. 2054 


.6353 


. 1013 


.4382 


.0808 


. 7479 


. 2521 


12 


19 


. 7972 


. 2027 


.6335 


. 1047 


.4362 


.0810 


. 7490 


. 2510 


11 


20 


.37999 


.62000 


2.6316 


.41081 


2.4342 


1.0811 


.07501 


.92499 


10 


21 


. 8026 


. 1974 


.6297 


. 1115 


.4322 


.0812 


. 7512 


. 2488 


39 


22 


. 8053 


. 1947 


.6279 


. 1149 


.4302 


.0813 


. 7523 


. 2477 


38 


23 


. 8080 


. 1920 


.6260 


. 1183 


.4282 


.0815 


. 7534 


. 2466 


37 


24 


. 8107 


. 1893 


.62 J 2 


. 1217 


.4262 


.0816 


. 7545 


. 2455 


36 


25 


.38134 


.61866 


2.6223 


.41251 


2.4242 


1.0817 


.07556 


.02113 


35 


26 


. 8161 


. 1839 


.6205 


. 1285 


.4222 


.0819 


. 7567 


. 2432 


34 


27 


. 8188 


. 1812 


.6186 


. 1319 


.4202 


.0820 


. 7579 


. 2421 


33 


28 


. 8214 


. 1785 


.6168 


. 1353 


.4182 


.0821 


. 7590 


. 2410 


32 


29 


. 8241 


. 1758 


.6150 


. 1387 


.4162 


.0823 


. 7601 


. 2399 


31 


SO 


.38268 


.61732 


2.6131 


.41421 


2.4142 


1.0824 


.07612 


.92388 


30 


31 


. 8295 


. 1705 


.6113 


. 1455 


.4122 


.0825 


. 7623 


. 2377 


29 


32 


. 8322 


. 1678 


.6095 


. 1489 


.4102 


.0826 


. 7634 


. 2366 


28 


33 


. 8349 


. 1651 


.6076 


. 1524 


.4083 


.0828 


. 7645 


. 2354 


27 


34 


. 8376 


. 1624 


.6058 


. 1558 


.4063 


.0829 


. 7657 


. 2343 


26 


35 


.38403 


.61597 


2.6040 


.41592 


2.4043 


1.0830 


.07668 


.92332 


25 


36 


. 8429 


. 1570 


.6022 


. 1626 


.4023 


.0832 


. 7679 


. 2321 


24 


37 


. 8456 


. 1544 


.6003 


. 1660 


.4004 


.0833 


. 7690 


. 2310 


23 


38 


. 8483 


. 1517 


.5985 


. 1694 


.3984 


.0834 


. 7701 


. 2299 


22 


39 


. 8510 


. 1490 


.5967 


. 172S 


.3964 


.0836 


. 7712 


. 2287 


21 


40 


.38537 


.61463 


2.5949 


.41762 


2.3945 


1.0837 


.07724 


.92276 


20 


41 


. 8564 


. L436 


.5931 


. 1797 


.3925 


.0838 


. 7735 


. 2265 


19 


42 


. 8591 


. 1409 


.5913 


. 1831 


.3906 


.0840 


. 7746 


. 2254 


18 


43 


. 8617 


. L382 


.5895 


. 1865 


.3886 


.0841 


. 7757 


. 2242 


17 


44 


. still 


. L356 


.5877 


. 1899 


.3867 


.0842 


. 7769 


. 22:51 


16 


45 


.38671 


.61329 


2.5859 


.41933 


2.3847 


1.0844 


.07780 


.92220 


15 


46 


. 8698 


. L302 


.5841 


. 1968 


.3828 


.0845 


. 7701 


. 2209 


14 


47 


. 8725 


. 1275 


.5823 


. 2002 


.3808 


.0846 


. 7802 


. 2197 


13 


48 


. 875] 


. 1248 


.5805 


. 2036 


.3789 


.0847 


. 7814 


. 2186 


12 


49 


. 8778 


. L222 


.5787 


. 2()7() 


.3770 


.0849 


. 7825 


. 2175 


11 


50 


.38805 


.61195 


2.5770 


.42105 


2.3750 


1 .0850 


.07836 


.92164 


10 


51 


. 8832 


. 1168 


.5752 


. 2139 


.3731 


.0851 


. 7847 


. 2152 


9 


52 


. 8859 


. 1111 


.5734 


. 217:; 


.3712 


.0853 


. 7S59 


. 2141 


8 


53 


. 8886 


. mi 


.5716 


. 2207 


.3692 


.0854 


. 7S70 


. 2130 


7 


54 


. 8912 


. loss 


.5699 


. 2212 


.3673 


.0855 


. 788] 


. 2118 


6 


55 


.38939 


.61061 


2.5681 


.42276 


2.3654 


L.0857 


.07893 


.92107 


5 


56 


. 8966 


. L034 


.5663 


. 2310 


.3635 


.0858 


. 7904 


. 2096 


4 


57 


. 8993 


. 1007 


.5646 


. 2344 


.3616 


.0859 


. 7915 


. 2084 


3 


58 


. 9019 


. 0980 


.5628 


. 2379 


.8597 


.0861 


. 7927 


. 2073 


2 


59 


. •tun; 


. 0954 


.5610 


. 2413 


.:;:>: 7 


.0862 


. 7938 


. 2062 


1 


60 


. 9078 


. 0927 


.5593 


. 2447 


.3558 


.0864 


. 7919 


. 2050 





M. 


1 losine. 


Vrs. .Kin. 


Secant. 


Cotang. 


Tang. 


Cosee'nt 


Vrs. cos. 


Sine. 


M. 



112° 



67° 



Natural Functions. 



157 



23° 


Natural Trigonometrical Functions. 


156° 


M. 


Sine. 


Vrs. cos. 


Cosecant 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 


• 


.39073 


.60927 


2.5593 


.42447 


2.3558 


1.0864 


.07949 


.92050 


60 


1 


. 9100 


. 0900 


.5575 


. 2482 


.3539 


.0865 


. 7961 


. 2039 


59 


2 


. 9126 


. 0873 


.5558 


. 2516 


.3520 


.0866 


. 7972 


. 2028 


58 


3 


. 9153 


. 0846 


.5540 


. 2550 


.3501 


.0868 


. 7984 


. 2016 


57 


4 


. 9180 


. 0820 


.5523 


. 2585 


.3482 


.0869 


. 7995 


. 2005 


56 


5 


.39207 


.60793 


2.5506 


.42619 


2.3463 


1.0870 


.08006 


.91993 


55 


6 


. 9234 


. 0766 


.5488 


. 2654 


.3445 


.0872 


. 8018 


. 1982 


54 


7 


. 9260 


. 0739 


.5471 


. 2688 


.3426 


.0873 


. 8029 


. 1971 


53 


8 


. 9287 


. 0713 


.5453 


. 2722 


.3407 


.0874 


. 8041 


. 1959 


52 


9 


. 9314 


. 0686 


.5436 


. 2757 


.3388 


.0876 


. 8052 


. 1948 


51 


10 


.39341 


.60659 


2.5419 


.42791 


2.3369 


1.0877 


.08063 


.91936 


50 


11 


. 9367 


. 0632 


.5402 


. 2826 


.3350 


.0878 


. 8075 


. 1925 


49 


12 


. 9394 


. 0606 


.5384 


. 2860 


.3332 


.0880 


. 8086 


. 1913 


48 


13 


. 9421 


. 0579 


.5367 


. 2894 


.3313 


.0881 


. 8098 


. 1902 


47 


14 


". 9448 


. 0552 


.5350 


. 2929 


.3294 


.0882 


. 8109 


. 1891 


46 


15 


.39474 


.60526 


2.5333 


.42963 


2.3276 


1.0884 


.08121 


.91879 


15 


16 


. 9501 


. 0499 


.5316 


. 2998 


.3257 


.0885 


. 8132 


. 1868 


44 


17 


. 9528 


. 0472 


.5299 


. 3032 


.3238 


.0886 


. 8144 


. 1856 


43 


18 


. 9554 


. 0445 


.5281 


. 3067 


.3220 


.0888 


. 8155 


. 1845 


42 


19 


. 9581 


. 0419 


.5264 


. 3101 


.3201 


.0889 


. 8167 


. 1833 


41 


20 


.39608 


.60392 


2.5247 


.43136 


2.3183 


1.0891 


.08178 


.91822 


40 


21 


. 9635 


. 0365 


.5230 


. 3170 


.3164 


.0892 


. 8190 


. 1810 


39 


22 


. 9661 


. 0339 


.5213 


. 3205 


.3145 


.0893 


. 8201 


. 1798 


38 


23 


. 9688 


. 0312 


.5196 


. 3239 


.3127 


.0895 


. 8213 


. 1787 


37 


24 


. 9715 


. 0285 


.5179 


. 3274 


.3109 


.0896 


. 8224 


. 1775 


36 


25 


.39741 


.60258 


2.5163 


.43308 


2.3090 


1.0897 


.08236 


.91764 


35 


26 


. 9768 


. 0232 


.5146 


. 3343 


.3072 


.0899 


. 8248 


. 1752 


34 


27 


. 9795 


. 0205 


.5129 


. 3377 


.3053 


.0900 


. 8259 


. 1741 


33 


28 


. 9821 


. 0178 


.5112 


. 3412 


.3035 


.0902 


. 8271 


. 1729 


32 


29 


. 9848 


. 0152 


.5095 


. 3447 


.3017 


.0903 


. 8282 


. 1718 


31 


30 


.39875 


.60125 


2.5078 


.43481 


2.2998 


1.0904 


.08294 


.91706 


30 


31 


. 9901 


. 0098 


.5062 


. 3516 


.2980 


.0906 


. 8306 


. 1694 


29 


32 


. 9928 


. 0072 


.5045 


. 3550 


.2962 


.0907 


. 8317 


. 1683 


28 


33 


. 9955 


. 0045 


.5028 


. 3585 


.2944 


.0908 


. 8329 


. 1671 


27 


34 


. 9981 


. 0018 


.5011 


. 3620 


.2925 


.0910 


. 8340 


. 1659 


26 


35 


.40008 


.59992 


2.4995 


.43654 


2.2907 


1.0911 


.08352 


.91648 


25 


36 


. 0035 


. 9965 


.4978 


. 3689 


.2889 


.0913 


. 8364 


. 1636 


24 


37 


. 0061 


. 9938 


.4961 


. 3723 


.2871 


.0914 


. 8375 


. 1625 


23 


38 


. 0088 


. 9912 


.4945 


. 3758 


.2853 


.0915 


. 8387 


. 1613 


22 


39 


. 0115 


. 9885 


.4928 


. 3793 


.2835 


.0917 


. 8399 


. 1601 


21 


40 


.40141 


.59858 


2.4912 


.43827 


2.2817 


1.0918 


.08410 


.91590 


20 


41 


. 0168 


. 9832 


.4895 


. 3862 


.2799 


.0920 


. 8422 


. 1578 


19 


42 


. 0195 


. 9805 


.4879 


. 3897 


.2781 


.0921 


. 8434 


. 1566 


18 


43 


. 0221 


. 9778 


.4862 


. 3932 


.2763 


.0922 


. 8445 


. 1554 


17 


44 


. 0248 


. 9752 


.4846 


. 3966 


.2745 


.0924 


. 8457 


. 1543 


16 


45 


.40275 


.59725 


2.4829 


.44001 


2.2727 


1.0925 


.08469 


.91531 


15 


46 


. 0301 


. 9699 


.4813 


. 4036 


•2709 


.0927 


. 8480 


. 1519 


14 


47 


. 0328 


. 9672 


.4797 


. 4070 


.2691 


.0928 


. 8492 


. 1508 


13 


48 


. 0354 


. 9645 


.4780 


. 4105 


.2673 


.0929 


. 8504 


. 1496 


12 


49 


. 0381 


. 9619 


.4764 


. 4140 


.2655 


.0931 


. 8516 


. 1484 


11 


50 


.40408 


.59592 


2.4748 


.44175 


2.2637 


1.0932 


.08527 


.91472 


10 


51 


. 0434 


. 9566 


.4731 


. 4209 


.2619 


.0934 


. 8539 


. 1461 


9 


52 


. 0461 


. 9b39 


.4715 


. 4244 


.2602 


.0935 


. 8551 


. 1449 


8 


53 


. 0487 


. 9512 


.4699 


. 4279 


.2584 


.0936 


. 8563 


. 1437 


7 


54 


. 0514 


. 9486 


.4683 


. 4314 


.2566 


.0938 


. 8575 


. 1425 


6 


55 


.40541 


.59459 


2.4666 


.44349 


2.2548 


1.0939 


.08586 


.91414 


5 


56 


. 0o67 


. 9433 


.4650 


. 4383 


.2531 


.0941 


. 8598 


. 1402 


4 


57 


. 0594 


. 9406 


.4634 


. 4418 


.2513 


.0942 


. 8610 


. 1390 


3 


58 


. 0620 


. 9379 


.4618 


. 4453 


.2495 


.0943 


. 8622 


. 1378 


2 


59 


. 0647 


. 9353 


.4602 


. 4488 


.2478 


.0945 


. 8634 


. 1366 


1 


60 


. 0674 


. 9326 


.4586 


. 4523 


.2460 


.0946 


. 8645 


. 1354 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



113° 



66° 



158 



Natural Functions. 



24° 




Natural Trigonometrical Functions. 


155° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.40674 


.59326 


2.4586 


.44523 


2.2460 


1.0946 


.08645 


.91354 


60 


1 


. 0700 


. 9300 


.4o70 


. 45o8 


.24*3 


.0948 


. 8657 


. 1343 


59 


2 


. 0727 


. 9273 


.4554 


. 4593 


.2*25 


.0949 


. 8669 


. 1331 


58 


3 


. 0753 


. 9247 


.4538 


. 4627 


.2*08 


.0951 


. 8681 


. 1319 


57 


4 


. 0780 


. 9220 


.4o22 


. 4662 


.23*0 


.0952 


. 86<>3. 


. 1307 


56 


5 


.40806 


.59193 


2.4506 


.44697 


2.23"3 


1.0953 


.08705 


.91295 


55 


6 


. 0833 


. 9167 


.4190 


. 4732 


.2355 


.0955 


. 8716 


. 1283 


54 


7 


. 0860 


. 9140 


.4474 


. 4767 


.2338 


.0956 


. 8728 


. 1271 


53 


8 


. 0886 


. 9114 


.4458 


. 4802 


.2320 


.0958 


. 8740 


. 1260 


52 


9 


. 0913 


. 9087 


.4442 


. 4837 


.2303 


.0959 


. 8752 


. 1248 


51 


10 


.40939 


.59061 


2.4426 


.44872 


2.2286 


1.0961 


.08764 


.91236 


50 


11 


. 0966 


. 9034 


.4411 


. 4907 


.2268 


.0962 


. 8776 


. 1224 


49 


12 


. 0992 


. 9008 


.4395 


. 4942 


.2251 


.0963 


. 8788 


. 1212 


48 


13 


. 1019 


. 8981 


.4379 


. 4977 


.2234 


.0965 


. 8800 


. 1200 


47 


14 


. 1045 


. 8955 


.4363 


. 5012 


.2216 


.0966 


. 8812 


. 1188 


46 


15 


.41072 


.58928 


2.4347 


.45047 


2.2199 


1.0968 


.08824 


.91176 


45 


16 


. 1098 


. 8901 


.4332 


. 5082 


.2182 


.0969 


. 8836 


. 1164 


44 


17 


. 1125 


. 8875 


.4316 


. 5117 


.2165 


.0971 


. 8848 


. 1152 


43 


18 


. 1151 


. 8848 


.4300 


. 5152 


.2147 


.0972 


. 8860 


. 1140 


42 


19 


. 1178 


. 8822 


.4285 


. 5187 


.2130 


.0973 


. 8872 


. 1128 


41 


20 


.41204 


.58795 


2.4269 


.45222 


2.2113 


1.0975 


.08884 


.91116 


40 


21 


. 1231 


. 8769 


.4254 


. 5257 


.2096 


.0976 


. 8896 


. 1104 


39 


22 


. 1257 


. 8742 


.4238 


. 5292 


.2079 


.0978 


. 8908 


. 1092 


38 


23 


. 1284 


. 8716 


.4222 


. 5327 


.2062 


.0979 


. 8920 


. 1080 


37 


24 


.1310 


. 8689 


.4207 


. 5362 


.2045 


.0981 


. 8932 


. 1068 


36 


25 


.41337 


.58663 


2.4191 


.45397 


2.2028 


1.0982 


.08944 


.91056 


35 


26 


. 1363 


. 8636 


.4176 


. 5432 


.2011 


.0984 


. 8956 


. 1044 


34 


27 


. 1390 


. 8610 


.4160 


. 5467 


.1994 


.0985 


. 8968 


. 1032 


33 


28 


. 1416 


. 8584 


.4145 


. 5502 


.1977 


.0986 


. 8980 


. 1020 


32 


29 


. 1443 


. 8557 


.4130 


. 5537 


.1960 


.0988 


. 8992 


. 1008 


31 


30 


.41469 


.58531 


2.4114 


.45573 


2.1943 


1.0989 


.09004 


.90996 


30 


31 


. 1496 


. 8504 


.4099 


. 5608 


.1926 


.0991 


. 9016 


. 0984 


29 


32 


. 1522 


. 8478 


.4083 


. 5643 


.1909 


.0992 


. 9028 


. 0972 


28 


33 


. 1549 


. 8451 


.4068 


. 5678 


.1892 


.0994 


. 9040 


. 0960 


27 


34 


. 1575 


. 8425 


.4053 


. 5713 


.1875 


.0995 


. 9052 


. 0948 


26 


35 


41602 


.58398 


2.4037 


.45748 


2.1859 


1.0997 


.09064 


.90936 


25 


36 


. 1628 


. 8372 


.4022 


. 5783 


.1842 


.0998 


. 9076 


. 0924 


24 


37 


. 1654 


. 8345 


.4007 


. 5819 


.1825 


.1000 


. 9088 


. 0911 


23 


38 


. 1681 


. 8319 


.3992 


. 5854 


.1808 


.1001 


. 9101 


. 0899 


22 


39 


. 1707 


. 8292 


.3976 


. 5889 


.1792 


.1003 


. 9113 


. 0887 


21 


40 


.41734 


.58266 


2.3961 


.45924 


2.1775 


1.1004 


.09125 


.90875 


20 


41 


. 1760 


. 8240 


.3946 


. 5960 


.1758 


.1005 


. 9137 


. 0863 


19 


42 


. 1787 


. 8213 


.3931 


. 5995 


.1741 


.1007 


. 9149 


. 0851 


18 


43 


. 1813 


. 8187 


.3916 


. 6030 


.1725 


.1008 


. 9161 


. 0839 


17 


44 


. 1839 


. 8160 


.3901 


. 6065 


.1708 


.1010 


. 9173 


. 0826 


16 


45 


.41866 


.58134 


2.3886 


.46101 


2.1692 


1.1011 


.09186 


.90814 


15 


46 


. 1892 


. 8108 


.3871 


. 6136 


.1675 


.1013 


. 9198 


. 0802 


14 


47 


. 1919 


. 8081 


.3856 


. 6171 


.1658 


.1014 


. 9210 


. 0790 


13 


48 


. 1945 


. 8055 


.3811 


. 6206 


.1642 


.1016 


. 9222 


. 0778 


12 


49 


. 1972 


. 8028 


.3826 


. 6242 


.1625 


.1017 


. 9234 


. 0765 


11 


50 


.41998 


.58002 


2.3811 


.46277 


2.1609 


1.1019 


.09247 


.90753 


10 


51 


. 2024 


. 7975 


.3796 


. 6312 


.1592 


.1020 


. 9259 


. 0741 


9 


52 


. 2051 


. 7949 


.3781 


. 6348 


.1576 


.1022 


. 9271 


. 0729 


8 


53 


. 2077 


. 7923 


.3766 


. 6383 


.1559 


.1023 


. 9283 


. 0717 


7 


54 


. 2103 


. 7896 


.3751 


. 6418 


.1543 


.1025 


. 9296 


. 0704 


6 


55 


.42130 


.57870 


2.3736 


.464.54 


2.1527 


1.1026 


.09308 


.90692 


5 


56 


. 2156 


. 7844 


.3721 


. 6489 


.1510 


.1028 


. 9320 


. 0680 


4 


57 


. 2183 


. 7817 


.3706 


. 6524 


.1494 


.1029 


. 9332 


. 0668 


3 


58 


. 2209 


. 7791 


.3691 


. 6500 


.1478 


.1031 


. 9345 


. 0655 


2 


59 


. 2235 


. 7764 


.8677 


. 6595 


.1461 


.1032 


. 9357 


. 0643 


1 


60 


. 2262 


. 7738 


.8662 


. 6631 


.1445 


.1034 


. 9869 


. 0631 





M. 


Cosine. 


Vrs. sin. 


1 Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


WE. 



114° 



65° 



Natural Functions. 



159 



25° 




Natural Trigonom 


etrical Functions. 


154° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.42262 


.57738 


2.3662 


.46631 


2.1445 


1.1034 


.09369 


.90631 


60 


1 


. 2288 


. 7712 


.3647 


. 6666 


.1429 


.1035 


. 9381 


. 0618 


59 


2 


. 2314 


. 7685 


.3632 


. 6702 


.1412 


.1037 


. 9394 


. 0606 


58 


g 


. 2341 


. 7659 


.3618 


. 6737 


.1396 


.1038 


. 9406 


. 0594 


57 


4 


. 2367 


. 7633 


.3603 


. 6772 


.1380 


.1040 


. 9418 


. 0581 


56 


5 


.42394 


.57606 


2.3588 


.46808 


2.1364 


1.1041 


.09431 


.90569 


55 


6 


. 2420 


. 7580 


.3574 


. 6843 


.1348 


.1043 


. 9443 


. 0557 


54 


7 


. 2446 


. 7554 


.3559 


. 6879 


.1331 


.1044 


. 9455 


. 0544 


53 


8 


. 2473 


. 7527 


.3544 


. 6914 


.1315 


.1046 


. 9468 


. 0532 


52 


9 


. 2499 


. 7501 


.3530 


. 6950 


.1299 


.1047 


. 9480 


. 0520 


51 


10 


.42525 


.57475 


2.3515 


.46985 


2.1283 


1.1049 


.09492 


.90507 


50 


11 


. 2552 


. 7448 


.3501 


. 7021 


.1267 


.1050 


. 9505 


. 0495 


49 


12 


. 2578 


. 7422 


.3486 


. 7056 


.1251 


.1052 


. 9517 


. 0483 


48 


13 


. 2604 


. 7396 


.3472 


. 7092 


.1235 


.1053 


. 9530 


. 0470 


47 


14 


. 2630 


. 7369 


.3457 


. 7127 


.1219 


.1055 


. 9542 


. 0458 


46 


15 


.42657 


.57343 


2.3443 


.47163 


2.1203 


1.1056 


.09554 


.90445 


45 


16 


. 2683 


. 7317 


.3428 


. 7199 


.1187 


.1058 


. 9567 


. 0433 


44 


17 


. 2709 


. 7290 


.3414 


. 7234 


.1171 


.1059 


. 9579 


. 0421 


43 


18 


. 2736 


. 7264 


.3399 


. 7270 


.1155 


.1061 


. 9592 


. 0408 


42 


19 


. 2762 


. 7238 


.3385 


. 7305 


.1139 


.1062 


. 9604 


. 0396 


41 


20 


.42788 


.57212 


2.3371 


.47341 


2.1123 


1.1064 


.09617 


.90383 


40 


21 


. 2815 


. 7185 


.3356 


. 7376 


.1107 


.1065 


. 9629 


. 0371 


39 


22 


. 2841 


. 7159 


.3342 


. 7412 


.1092 


.1067 


. 9641 


. 0358 


38 


23 


. 2867 


. 7133 


.3328 


. 7448 


.1076 


.1068 


. 9654 


. 0346 


37 


24 


. 2893 


. 7106 


.3313 


. 7483 


.1060 


.1070 


. 9666 


. 0333 


36 


25 


.42920 


.57080 


2.3299 


.47519 


2.1044 


1.1072 


.09679 


.90321 


35 


26 


. 2946 


. 7054 


.3285 


. 7555 


.1028 


.1073 


. 9691 


. 0308 


34 


27 


. 2972 


. 7028 


.3271 


. 7590 


.1013 


.1075 


. 9704 


. 0296 


33 


28 


. 2998 


. 7001 


.3256 


. 7626 


.0997 


.1076 


. 9716 


. 0283 


32 


29 


. 3025 


. 6975 


.3242 


. 7662 


.0981 


.1078 


. 9729 


. 0271 


31 


30 


.43051 


.56949 


2.3228 


.47697 


2.0965 


1.1079 


.09741 


.90258 


30 


31 


. 3077 


. 6923 


.3214 


. 7733 


.0950 


.1081 


. 9754 


. 0246 


29 


32 


. 3104 


. 6896 


.3200 


. 7769 


.0934 


.1082 


. 9766 


. 0233 


28 


33 


. 3130 


. 6870 


.3186 


. 7805 


.0918 


.1084 


. 9779 


. 0221 


27 


34 


. 3156 


. 6844 


.3172 


. 7840 


.0903 


.1085 


. 9792 


. 0208 


26 


35 


.43182 


.56818 


2.3158 


.47876 


2.0887 


1.1087 


.09804 


.90196 


25 


36 


. 3208 


. 6791 


.3143 


. 7912 


.0872 


.1088 


. 9817 


. 0183 


24 


37 


. 3235 


. 6765 


.3129 


. 7948 


.0856 


.1090 


. 9829 


. 0171 


23 


38 


. 3261 


. 6739 


.3115 


. 7983 


.0840 


.1092 


. 9842 


. 0158 


22 


39 


. 3287 


. 6713 


.3101 


. 8019 


.0825 


.1093 


. 9854 


. 0145 


21 


40 


.43313 


.56686 


2.3087 


.48055 


2.0809 


1.1095 


.09867 


.90133 


20 


41 


. 3340 


. 6660 


.3073 


. 8091 


.0794 


.1096 


. 9880 


. 0120 


19 


42 


. 3366 


. 6634 


.3059 


. 8127 


.0778 


.1098 


. 9892 


. 0108 


18 


43 


. 3392 


. 6608 


.3046 


. 8162 


.0763 


.1099 


. 9905 


. 0095 


17 


44 


. 3418 


. 6582 


.3032 


. 8198 


.0747 


.1101 


. 9917 


. 0082 


16 


45 


.43444 


.56555 


2.3018 


.48234 


2.0732 


1.1102 


.09930 


.90070 


15 


46 


. 3471 


. 6529 


.3004 


. 8270 


.0717 


.1104 


. 9943 


. 0057 


14 


47 


. 3497 


. 6503 


.2990 


. 8306 


.0701 


.1106 


. 9955 


. 0044 


13 


48 


. 3523 


. 6477 


.2976 


. 8342 


.0686 


.1107 


. 9968 


. 0032 


12 


49 


. 3549 


. 6451 


.2962 


. 8378 


.0671 


.1109 


. 9981 


. 0019 


11 


50 


.43575 


.56424 


2.2949 


.48414 


2.0655 


1.1110 


.09993 


.90006 


10 


51 


. 3602 


. 6398 


.2935 


. 8449 


.0640 


.1112 


.10006 


!89994 


9 


52 


. 3628 


. 6372 


.2921 


. 8485 


.0625 


.1113 


. 0019 


. 9981 


8 


53 


. 3654 


. 6346 


.2907 


. 8521 


.0609 


.1115 


. 0031 


. 9968 


7 


54 


. 3680 


. 6320 


.2894 


. 8557 


.0594 


.1116 


. 0044 


. 9956 


6 


55 


.43706 


.56294 


2.2880 


.48593 


2.0579 


1.1118 


.10057 


.89943 


5 


56 


. 3732 


. 6267 


.2866 


. 8629 


.0564 


.1120 


. 0070 


. 9930 


4 


57 


. 3759 


. 6241 


.2853 


. 8665 


.0548 


.1121 


. 0082 


. 9918 


3 


58 


. 3785 


. 6215 


.2839 


. 8701 


.0533 


.1123 


. 0095 


. 9905 


2 


59 


. 3811 


. 6189 


.2825 


. 8737 


.0518 


.1124 


. 0108 


. 9892 


1 


60 


. 3837 


. 6163 


.2812 


. 8773 


.0503 


.1126 


. 0121 


. 9879 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



115° 



64° 



160 



Natural Functions. 



26° 




Natural Trigonometrical Functions. 


153° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.43837 


.56163 


2.2812 


.48773 


2.0503 


1.1126 


.10121 


.89879 


60 


1 


. 3863 


. 6137 


.2798 


. 8809 


.0488 


.1127 


. 0133 


. 9867 


59 


2 


. 3889 


. 6111 


.2784 


. 8845 


.0473 


.1129 


. 0146 


. 9854 


58 


3 


. 3915 


. 6084 


.2771 


. 8881 


.0458 


.1131 


. 0159 


. 9841 


57 


4 


. 3942 


. 6058 


.2757 


. 8917 


.0443 


.1132 


. 0172 


. 9828 


56 


5 


.43968 


.56032 


2.2744 


.48953 


2.0427 


1.1134 


.10184 


.89815 


55 


6 


. 3994 


. 6006 


.2730 


. 8989 


.0412 


.1135 


. 0197 


. 9803 


54 


7 


. 4020 


. 5980 


.2717 


. 9025 


.0397 


.1137 


. 0210 


. 9790 


53 


8 


. 4046 


. 5954 


.2703 


. 9062 


.0382 


.1139 


. 0223 


. 9777 


52 


9 


. 4072 


. 5928 


.2690 


. 9098 


.0367 


.1140 


. 0236 


. 9764 


51 


10 


.44098 


.55902 


2.2676 


.49134 


2.0352 


1.1142 


.10248 


.89751 


50 


11 


. 4124 


. 5875 


.2663 


. 9170 


.0338 


.1143 


. 0261 


. 9739 


49 


12 


. 4150 


. 5849 


.2650 


. 9206 


.0323 


.1145 


. 0274 


. 9726 


48 


13 


. 4177 


. 5823 


.2636 


. 9242 


.0308 


.1147 


. 0287 


. 9713 


47 


14 


. 4203 


. 5797 


.2623 


. 9278 


.0293 


.1148 


. 0300 


. 9700 


46 


15 


.44229 


.55771 


2.2610 


.49314 


2.0278 


1.1150 


.10313 


.89687 


45 


16 


. 4255 


. 5745 


.2596 


. 9351 


.0263 


.1151 


. 0326 


. 9674 


44 


17 


. 4281 


. 5719 


.2583 


. 9387 


.0248 


.1153 


. 0338 


. 9661 


43 


18 


. 4307 


. 5693 


.2570 


. 9423 


.0233 


.1155 


. 0351 


. 9649 


42 


19 


. 4333 


. 5667 


.2556 


. 9459 


.0219 


.1156 


. 0364 


. 9636 


41 


20 


.44359 


.55641 


2.2543 


.49495 


2.0204 


1.1158 


.10377 


.89623 


40 


21 


. 4385 


. 5615 


.2530 


. 9532 


.0189 


.1159 


. 0390 


. 9610 


39 


22 


. 4411 


. 5589 


.2517 


. 9568 


.0174 


.1161 


. 0403 


. 9597 


38 


23 


. 4437 


. 5562 


.2503 


. 9604 


.0159 


.1163 


. 0416 


. 9584 


37 


24 


. 4463 


. 5536 


.2490 


. 9640 


.0145 


.1164 


. 0429 


. 9571 


36 


25 


.44489 


.55510 


2.2477 


.49677 


2.0130 


1.1166 


.10442 


.89558 


35 


26 


. 4516 


. 5484 


.2464 


. 9713 


.0115 


.1167 


. 0455 


. 9545 


34 


27 


. 4542 


. 5458 


.2451 


. 9749 


.0101 


.1169 


. 0468 


. 9532 


33 


28 


. 4568 


. 5432 


.2438 


. 9785 


.0086 


.1171 


. 0481 


. 9519 


32 


29 


. 4594 


. 5406 


.2425 


. 9822 


.0071 


.1172 


. 0493 


. 9506 


31 


30 


.44620 


.55380 


2.2411 


.49858 


2.0058 


1.1174 


.10506 


.89493 


30 


o1 


. 4646 


. 5354 


.2398 


. 9894 


.0042 


.1176 


. 0519 


. 9480 


29 


32 


. 4672 


. 5328 


.2385 


. 9931 


.0028 


.1177 


. 0532 


. 9467 


28 


33 


. 4698 


. 5302 


.2372 


. 9967 


.0013 


.1179 


. 0545 


. 9454 


27 


34 


. 4724 


. 5276 


.2359 


.50003 


1.9998 


.1180 


. 0558 


. 9441 


26 


35 


.44750 


.55250 


2.2346 


.50040 


1.9984 


1.1182 


.10571 


.89428 


25 


36 


. 4776 


. 5224 


.2333 


. 0076 


.9969 


.1184 


. 0584 


. 9415 


24 


37 


. 4802 


. 5198 


.2320 


. 0113 


.9955 


.1185 


. 0598 


. 9402 


23 


38 


. 4828 


. 5172 


.2307 


. 0149 


.9940 


.1187 


. 0611 


. 9389 


22 


39 


. 4854 


. 5146 


.2294 


. 0185 


.9926 


.1189 


. 0624 


. 9376 


21 


40 


.44880 


.55120 


2.2282 


.50222 


1.9912 


1.1190 


.10637 


.89363 


20 


41 


. 4906 


. 5094 


.2269 


. 0258 


.9897 


.1192 


. 0650 


. 9350 


19 


42 


. 4932 


. 5068 


.2256 


. 0295 


.9883 


.1193 


. 0663 


. 9337 


18 


43 


. 4958 


. 5042 


.2243 


. 0331 


.9868 


.1195 


. 0676 


. 9324 


17 


44 


. 4984 


. 5016 


.2230 


. 0368 


.9854 


.1197 


. 0689 


. 9311 


16 


45 


.45010 


.54990 


2.2217 


.50404 


1.9840 


1.1198 


.10702 


.89298 


15 


46 


. 5036 


. 4964 


.2204 


. 0441 


.9825 


.1200 


. 0715 


. 9285 


14 


47 


. 5062 


. 4938 


.2192 


. 0477 


.9811 


.1202 


. 0728 


. 9272 


13 


48 


. 5088 


. 4912 


.2179 


. 0514 


.9797 


.1203 


. 0741 


. 9258 


12 


49 


. 5114 


. 4886 


.2166 


. 0550 


.9782 


.1205 


. 0754 


. 9245 


11 


50 


.45140 


.54860 


2.2153 


..50587 


1.9768 


1.1207 


.10768 


.89232 


10 


51 


. 5166 


. 4834 


.2141 


. 0623 


.9754 


.1208 


. 0781 


. 9219 


9 


52 


. 519] 


. 4808 


.2128 


. 0660 


.9739 


.1210 


. 0794 


. 9206 


8 


53 


. 5217 


. 4782 


.2115 


. 0696 


.9725 


.1212 


. 0807 


. 9193 


7 


54 


. 5243 


. 4756 


.2103 


. 0733 


.9711 


.1213 


. 0820 


. 9180 





55 


. 15269 


.54730 


2.2090 


.50769 


1.9697 


1.1215 


.10833 


.89166 


5 


56 


. 5295 


. 1705 


.2077 


. 0806 


.90S.", 


.1217 


. 0846 


. 9153 


4 


57 


. 6321 


. 1679 


.2065 


. 08 13 


.9668 


.1218 


. 0860 


. 9140 


3 


58 


. 5347 


. 1653 


.2052 


. 0879 


.9654 


.1220 


. 0873 


. 9127 


2 


59 


. 5373 


. 4627 


.2039 


. 0916 


.9640 


.1222 


. 0886 


. 9114 


1 


60 


. 5399 


. 4601 


.2027 


. 0952 


.9626 


.122:; 


. 0899 


. 9101 





M. 


< tadne. 


Vrs. sin. 


Secant. 


Ootang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



116° 



63° 



Natural Functions. 



161 



27° 


Natural Trigonometrical Functions. 


152° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Co tang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.45399 


.54601 


2.2027 


.50952 


1.9626 


1.1223 


.10899 


.89101 


60 


1 


. 5425 


. 4575 


.2014 


. 0989 


.9612 


.1225 


. 0912 


. 9087 


59 


2 


. 5451 


. 4549 


.2002 


. 1026 


.9598 


.1226 


. 0926 


. 9074 


58 


3 


. 5477 


. 4523 


.1989 


. 1062 


.9584 


.1228 


. 0939 


. 9061 


57 


4 


. 5503 


. 4497 


.1977 


. 1099 


.9570 


.1230 


. 0952 


. 9048 


56 


5 


.45528 


.54471 


2.1964 


.51136 


1.9556 


1.1231 


.10965 


.89034 


55 


8 


. 5554 


. 4445 


.1952 


. 1172 


.9542 


.1233 


. 0979 


. 9021 


54 


7 


. 5580 


. 4420 


.1939 


. 1209 


.9528 


.1235 


. 0992 


. 9008 


53 


8 


. 5606 


. 4394 


.1927 


. 1246 


.9514 


.1237 


. 1005 


. 8995 


52 


9 


. 5632 


. 4368 


.1914 


. 1283 


.9500 


.1238 


. 1018 


. 8981 


51 


10 


.45658 


.54342 


2.1902 


.51319 


1.9486 


1.1240 


.11032 


.88968 


50 


11 


. 5684 


. 4316 


.1889 


. 1356 


.9472 


.1242 


. 1045 


. 8955 


49 


12 


. 5710 


. 4290 


.1877 


. 1393 


.9458 


.1243 


. 1058 


. 8942 


48 


13 


. 5736 


. 4264 


.1865 


. 1430 


.9444 


.1245 


. 1072 


. 8928 


47 


14 


. 5761 


. 4238 


.1852 


. 1466 


.9430 


.1247 


. 1085 


. 8915 


46 


15 


.45787 


.54213 


2.1840 


.51503 


1.9416 


1.1248 


.11098 


.88902 


45 


16 


. 5813 


. 4187 


.1828 


. 1540 


.9402 


.1250 


. 1112 


. 8888 


44 


17 


. 5839 


. 4161 


.1815 


. 1577 


.9388 


.1252 


. 1125 


. 8875 


43 


18 


. 5865 


. 4135 


.1803 


. 1614 


.9375 


.1253 


. 1138 


. 8862 


42 


19 


. 5891 


. 4109 


.1791 


. 1651 


.9361 


.1255 


. 1152 


. 8848 


41 


20 


.45917 


.54083 


2.1778 


.51687 


1.9347 


1.1257 


.11165 


.88835 


40 


21 


. 5942 


. 4057 


.1766 


. 1724 


.9333 


.1258 


. 1178 


. 8822 


39 


22 


. 5968 


. 4032 


.1754 


. 1761 


.9319 


.1260 


. 1192 


. 8808 


38 


23 


. 5994 


. 4006 


.1742 


. 1798 


.9306 


.1262 


. 1205 


. 8795 


37 


24 


. 6020 


. 3980 


.1730 


. 1835 


.9292 


.1264 


. 1218 


. 8781 


36 


25 


.46046 


.53954 


2.1717 


.51872 


1.9278 


1.1265 


.11232 


.88768 


35 


26 


. 6072 


. 3928 


.1705 


. 1909 


.9264 


.1267 


. 1245 


. 8755 


34 


27 


. 6097 


. 3902 


.1693 


. 1946 


.9251 


.1269 


. 1259 


. 8741 


33 


28 


. 6123 


. 3877 


.1681 


. 1983 


.9237 


.1270 


. 1272 


. 8728 


32 


29 


. 6149 


. 3851 


.1669 


. 2020 


.9223 


.1272 


. 1285 


. 8714 


31 


30 


.46175 


.53825 


2.1657 


.52057 


1.9210 


1.1274 


.11299 


.88701 


30 


31 


. 6201 


. 3799 


.1645 


. 2094 


.9196 


.1275 


. 1312 


. 8688 


29 


32 


. 6226 


. 3773 


.1633 


. 2131 


.9182 


.1277 


. 1326 


. 8674 


28 


33 


. 6252 


. 3748 


.1620 


. 2168 


.9169 


.1279 


. 1339 


. 8661 


27 


34 


. 6278 


. 3722 


.1608 


. 2205 


.9155 


.1281 


. 1353 


. 8647 


26 


35 


.46304 


.53696 


2.1596 


.52242 


1.9142 


1.1282 


.11366 


.88634 


25 


36 


. 6330 


. 3670 


.1584 


. 2279 


.9128 


.1284 


. 1380 


. 8620 


24 


37 


. 6355 


. 3645 


.1572 


. 2316 


.9115 


.1286 


. 1393 


. 8607 


23 


38 


. 6381 


. 3619 


.1560 


. 2353 


.9101 


.1287 


. 1407 


. 8593 


22 


39 


. 6407 


. 3593 


.1548 


. 2390 


.9088 


.1289 


. 1420 


. 8580 


21 


40 


.46433 


.53567 


2.1536 


.52427 


1.9074 


1.1291 


.11434 


.88566 


20 


41 


. 6458 


. 3541 


.1525 


. 2464 


.9061 


.1293 


. 1447 


. 8553 


19 


42 


. 6484 


. 3516 


.1513 


. 2501 


.9047 


.1294 


. 1461 


. 8539 


18 


43 


. 6510 


. 3490 


.1501 


. 2538 


.9034 


.1296 


. 1474 


. 8526 


17 


44 


. 6536 


. 3464 


.1489 


. 2575 


.9020 


.1298 


. 1488 


. 8512 


16 


45 


.46561 


.53438 


2.1477 


.52612 


1.9007 


1.1299 


.11501 


.88499 


15 


46 


. 6587 


. 3413 


.1465 


. 2650 


.8993 


.1301 


. 1515 


. 8485 


14 


47 


. 6613 


. 3387 


.1453 


. 2687 


.8980 


.1303 


. 1528 


. 8472 


13 


48 


. 6639 


. 3361 


.1441 


. 2724 


.8967 


.1305 


. 1542 


. 8458 


12 


49 


. 6664 


. 3336 


.1430 


. 2761 


.8953 


.1306 


. 1555 


. 8444 


11 


50 


.46690 


.53310 


2.1418 


.52798 


1.8940 


1.1308 


.11569 


.88431 


10 


51 


. 6716 


. 3284 


.1406 


. 2836 


.8927 


.1310 


. 1583 


. 8417 


9 


52 


. 6741 


. 3258 


.1394 


. 2873 


.8913 


.1312 


. 1596 


. 8404 


8 


53 


. 6767 


. 3233 


.1382 


. 2910 


.8900 


.1313 


. 1610 


. 8390 


7 


54 


. 6793 


. 3207 


.1371 


. 2947 


.8887 


.1315 


. 1623 


. 8376 


6 


55 


.46819 


.53181 


2.1359 


.52984 


1.8873 


1.1317 


.11637 


.88363 


5 


56 


. 6844 


. 3156 


.1347 


. 3022 


.8860 


.1319 


. 1651 


. 8349 


4 


57 


. 6870 


. 3130 


.1335 


. 3059 


.8847 


.1320 


. 1664 


. 8336 


3 


58 


. 6896 


. 3104 


.1324 


. 3096 


.8834 


.1322 


. 1678 


. 8322 


2 


59 


. 6921 


. 3078 


.1312 


. 3134 


.8820 


.1324 


. 1691 


. 8308 


1 


60 


. 6947 


. 3053 


.1300 


. 3171 


.8807 


.1326 


. 1705 


. 8295 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



117° 



62° 



11 



162 



Natural Functions. 



28° 




Natural Trigonometrical Functions. 


151° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.46947 


.53053 


2.1300 


.53171 


1.8807 


1.1326 


.11705 


.88295 


60 


1 


. 6973 


. 3027 


.1289 


. 3208 


.8794 


.1327 


. 1719 


. 8281 


59 


2 


. 6998 


. 3001 


.1277 


. 3245 


.8781 


.1329 


. 1732 


. 8267 


58 


3 


. 7024 


. 2976 


.1266 


. 3283 


.8768 


.1331 


. 1746 


. 8254 


57 


4 


. 7050 


. 2950 


.1254 


. 3320 


.8754 


.1333 


. 1760 


. 8240 


56 


5 


.47075 


.52924 


2.1242 


.53358 


1.8741 


1.1334 


.11774 


.88226 


55 


6 


. 7101 


. 2899 


.1231 


. 3395 


.8728 


.1336 


. 1787 


. 8213 


54 


7 


. 7127 


. 2873 


.1219 


. 3432 


.8715 


.1338 


. 1801 


. 8199 


53 


8 


. 7152 


. 2847 


.1208 


. 3470 


.8702 


.1340 


. 1815 


. 8185 


52 


9 


. 7178 


. 2822 


.1196 


. 3507 


.8689 


.1341 


. 1828 


. 8171 


51 


10 


.47204 


.52796 


2.1185 


.53545 


1.8676 


1.1343 


.11842 


.88158 


50 


11 


. 7229 


. 2770 


.1173 


. 3582 


.8663 


.1345 


. 1856 


. 8144 


49 


12 


. 7255 


. 2745 


.1162 


. 3619 


.8650 


.1347 


. 1870 


. 8130 


48 


13 


. 7281 


. 2719 


.1150 


. 3657 


.8637 


.1349 


. 1883 


. 8117 


47 


14 


. 7306 


. 2694 


.1139 


. 3694 


.8624 


.1350 


. 1897 


. 8103 


46 


15 


.47332 


.52668 


2.1127 


.53732 


1.8611 


1.1352 


.11911 


•88089 


45 


16 


. 7357 


. 2642 


.1116 


. 3769 


.8598 


.1354 


. 1925 


. 8075 


44 


17 


. 7383 


. 2617 


.1104 


. 3807 


.8585 


.1356 


. 1938 


. 8061 


43 


18 


. 7409 


. 2591 


.1093 


. 3844 


.8572 


.1357 


. 1952 


. 8048 


42 


19 


. 7434 


. 2565 


.1082 


. 3882 


.8559 


.1359 


. 1966 


. 8034 


41 


20 


.47460 


.52540 


2.1070 


.53919 


1.8546 


1.1361 


.11980 


.88020 


40 


21 


. 7486 


. 2514 


.1059 


. 3957 


.8533 


.1363 


. 1994 


. 8006 


39 


22 


. 7511 


. 2489 


.1048 


. 3995 


.8520 


.1365 


. 2007 


. 7992 


38 


23 


. 7537 


. 2463 


.1036 


. 4032 


.8507 


.1366 


. 2021 


. 7979 


37 


24 


. 7562 


. 2437 


.1025 


. 4070 


.8495 


.1368 


. 2035 


. 7965 


36 


25 


.47588 


.52412 


2.1014 


.54107 


1.8482 


1.1370 


.12049 


.87951 


35 


26 


. 7613 


. 2386 


.1002 


. 4145 


.8469 


.1372 


. 2063 


. 7937 


34 


27 


. 7639 


. 2361 


.0991 


. 4183 


.8456 


.1373 


. 2077 


. 7923 


33 


28 


. 7665 


. 2335 


.0980 


. 4220 


.8443 


.1375 


. 2090 


. 7909 


32 


29 


. 7690 


. 2310 


. .0969 


. 4258 


.8430 


.1377 


. 2104 


. 7895 


31 


30 


.47716 


.52284 


2.0957 


.54295 


1.8418 


1.1379 


.12118 


.87882 


30 


m 


. 7741 


. 2258 


.0946 


. 4333 


.8405 


.1381 


. 2132 


. 7868 


29 


32 


. 7767 


. 2233 


.0935 


. 4371 


.8392 


.1382 


. 2146 


. 7854 


28 


33 


. 7792 


. 2207 


.0924 


. 4409 


.8379 


.1384 


. 2160 


. 7840 


27 


34 


. 7818 


. 2182 


.0912 


. 4446 


.8367 


.1386 


. 2174 


. 7826 


26 


35 


.47844 


.52156 


2.0901 


.54484 


1.8354 


1.1388 


.12188 


.87812 


25 


36 


. 7869 


. 2131 


.0890 


. 4522 


.8341 


.1390 


. 2202 


. 7798 


24 


37 


. 7895 


. 2105 


.0879 


. 4559 


.8329 


.1391 


. 2216 


. 7784 


23 


38 


. 7920 


. 2080 


.0868 


. 4597 


.8316 


.1393 


. 2229 


. 7770 


22 


39 


. 7946 


. 2054 


.0857 


. 4635 


.8303 


.1395 


. 2243 


. 7756 


21 


40 


.47971 


.52029 


2.0846 


.54673 


1.8291 


1.1397 


.12257 


.87742 


20 


41 


. 7997 


. 2003 


.0835 


. 4711 


.8278 


.1399 


. 2271 


. 7728 


19 


42 


. 8022 


. 1978 


.0824 


. 4748 


.8265 


.1401 


. 2285 


. 7715 


18 


43 


. 8048 


. 1952 


.0812 


. 4786 


.8253 


.1402 


. 2299 


. 7701 


17 


44 


. 8073 


. 1927 


.0801 


. 4824 


.8240 


.1404 


. 2313 


. 7687 


16 


45 


.48099 


.51901 


2.0790 


.54862 


1.8227 


1.1406 


.12327 


.87673 


15 


46 


. 8124 


. 1876 


.0779 


. 4900 


.8215 


.1408 


. 2341 


. 7659 


14 


47 


. 8150 


. 1850 


.0768 


. 4937 


.8202 


.1410 


. 2355 


. 7645 


13 


48 


. 8175 


. 1825 


.0757 


. 4975 


.8190 


.1411 


. 2369 


. 7631 


12 


49 


. 8201 


. 1799 


.0746 


. 5013 


.8177 


.1413 


. 2383 


. 7617 


11 


50 


.48226 


.51774 


2.0735 


.55051 


1.8165 


1.1415 


.12397 


.87603 


10 


51 


. 8252 


. 1748 


.0725 


. 5089 


.8152 


.1417 


. 2411 


. 7588 


9 


52 


. 8277 


. 1723 


.0714 


. 5127 


.8140 


.1419 


. 2425 


. 7574 


8 


53 


. 8:503 


. 1697 


.0703 


. 5165 


.8127 


.1421 


. 2439 


. 7560 


7 


54 


. 8328 


. 1672 


.0692 


. 5203 


.8115 


.1422 


. 2453 


. 7546 


6 


55 


.48354 


.51646 


2.0681 


.55241 


1.8102 


1.1424 


.12468 


.87532 


5 


56 


. 8379 


. 1621 


.0670 


. 5279 


.8090 


.1426 


. 2482 


. 7518 


i 


57 


. 8405 


. 1595 


.0659 


. 5317 


.8078 


.1428 


. 2496 


. 7504 


3 


58 


. 8430 


. 1570 


.0648 


. 5355 


.8065 


.1430 


. 2510 


. 7490 


2 


59 


. 8455 


. 1544 


.0637 


. 5393 


.8053 


.1432 


. 2524 


. 7476 


1 


60 


. 8481 


. 1519 


.0627 


. 5431 


.8040 


.1433 


. 2538 


. 7462 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



118° 



61° 



Natural Functions. 



163 



29 c 




Natural Trigonometrical Functions. 


150° 


M. 


Sine. 


Vrs. cos. i 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 


^o 


.48481 


.51519 


2.0627 


.55431 


1.8040 


1.1433 


.12538 


.87462 


60 


1 


. 8506 


. 1493 


.0616 


. 5469 


.8028 


.1435 


. 2552 


. 7448 


59 


2 


. 8532 


. 1468 


.0605 


. 5507 


.8016 


.1437 


. 2566 


. 7434 


58 


3 


. 8557 


. 1443 


.0594 


. 5545 


.8003 


.1439 


. 2580 


. 7420 


57 


4 


. 8583 


. 1417 


.0583 


. 5583 


.7991 


.1441 


. 2594 


. 7405 


56 


5 


.48608 


.51392 


2.0573 


.55621 


1.7979 


1.1443 


.12609 


.87391 


55 


6 


. 8633 


. 1366 


.0562 


. 5659 


.7966 


.1445 


. 2623 


. 7377 


54 


. 7 


. 8659 


. 1341 


.0551 


. 5697 


.7954 


.1446 


. 2637 


. 7363 


53 


8 


. 8684 


. 1316 


.0540 


. 5735 


.7942 


.1448 


. 2651 


. 7349 


52 


9 


. 8710 


. 1290 


.0530 


. 5774 


.7930 


.1450 


. 2665 


. 7335 


51 


10 


.48735 


.51265 


2.0519 


.55812 


1.7917 


1.1452 


.12679 


.87320 


50 


11 


. 8760 


. 1239 


.0508 


. 5850 


.7905 


.1454 


. 2694 


. 7306 


49 


12 


. 8786 


. 1214 


.0498 


. 5888 


.7893 


.1456 


. 2708 


. 7292 


48 


13 


. 8811 


. 1189 


.0487 


. 5926 


.7881 


.1458 


. 2722 


. 7278 


47 


14 


. 8837 


. 1163 


.0476 


. 5964 


.7868 


.1459 


. 2736 


. 7264 


46 


15 


.48862 


.51138 


2.0466 


.56003 


1.7856 


1.1461 


.12750 


.87250 


45 


16 


. 8887 


. 1112 


.0455 


. 6041 


.7844 


.1463 


. 2765 


. 7235 


44 


17 


. 8913 


. 1087 


.0444 


. 6079 


.7832 


.1465 


. 2779 


. 7221 


43 


18 


. 8938 


. 1062 


.0434 


. 6117 


.7820 


.1467 


. 2793 


. 7207 


42 


19 


. 8964 


. 1036 


.0423 


. 6156 


.7808 


.1469 


. 2807 


. 7193 


41 


20 


.48989 


.51011 


2.0413 


.56194 


1.7795 


1.1471 


.12821 


.87178 


40 


21 


. 9014 


. 0986 


.0402 


. 6232 


.7783 


.1473 


. 2836 


. 7164 


39 


22 


. 9040 


. 0960 


.0392 


. 6270 


.7771 


.1474 


. 2850 


. 7150 


38 


23 


. 9065 


. 0935 


.0381 


. 6309 


.7759 


.1476 


. 2864 


. 7136 


37 


24 


. 9090 


. 0910 


.0370 


. 6347 


.7747 


.1478 


. 2879 


. 7121 


36 


25 


.49116 


.50884 


2.0360 


.56385 


1.7735 


1.1480 


.12893 


.87107 


35 


26 


. 9141 


. 0859 


.0349 


. 6424 


.7723 


.1482 


. 2907 


. 7093 


34 


27 


. 9166 


. 0834 


.0339 


. 6462 


.7711 


.1484 


. 2921 


. 7078 


33 


28 


. 9192 


. 0808 


.0329 


. 6500 


.7699 


.1486 


. 2936 


. 7064 


32 


29 


. 9217 


. 0783 


.0318 


. 6539 


.7687 


.1488 


. 2950 


. 7050 


31 


30 


.49242 


.50758 


2.0308 


.56577 


1.7675 


1.1489 


.12964 


.87035 


30 


31 


. 9268 


. 0732 


.0297 


. 6616 


.7663 


.1491 


. 2979 


. 7021 


29 


32 


. 9293 


. 0707 


.0287 


. 6654 


.7651 


.1493 


. 2993 


. 7007 


28 


33 


. 9318 


. 0682 


.0276 


. 6692 


.7639 


.1495 


. 3007 


. 6992 


27 


34 


. 9343 


. 0656 


.0266 


. 6731 


.7627 


.1497 


. 3022 


. 6978 


26 


35 


.49369 


.50631 


2.0256 


.56769 


1.7615 


1.1499 


.13036 


.86964 


25 


36 


. 9394 


. 0606 


.0245 


. 6808 


.7603 


.1501 


. 3050 


. 6949 


24 


37 


. 9419 


. 0580 


.0235 


. 6846 


.7591 


.1503 


. 3065 


. 6935 


23 


38 


. 9445 


. 0555 


.0224 


. 6885 


.7579 


.1505 


. 3079 


. 6921 


22 


39 


. 9470 


. 0530 


.0214 


. 6923 


.7567 


.1507 


. 3094 


. 6906 


21 


40 


.49495 


.50505 


2.0204 


.56962 


1.7555 


1.1508 


.13108 


.86892 


20 


41 


. 9521 


. 0479 


.0194 


. 7000 


.7544 


.1510 


. 3122 


. 6877 


19 


42 


. 9546 


. 0454 


.0183 


. 7039 


.7532 


.1512 


. 3137 


. 6863 


18 


43 


. 9571 


. 0429 


.0173 


. 7077 


.7520 


.1514 


. 3151 


. 6849 


17 


44 


. 9596 


. 0404 


.0163 


. 7116 


.7508 


.1516 


. 3166 


. 6834 


16 


45 


.49622 


.50378 


2.0152 


.57155 


1.7496 


1.1518 


.13180 


.86820 


15 


46 


. 9647 


. 0353 


.0142 


. 7193 


.7484 


.1520 


. 3194 


. 6805 


14 


47 


. 9672 


. 0328 


.0132 


. 7232 


.7473 


.1522 


. 3209 


. 6791 


13 


48 


. 9697 


. 0303 


.0122 


. 7270 


.7461 


.1524 


. 3223 


. 6776 


12 


49 


. 9723 


. 0277 


.0111 


. 7309 


.7449 


.1526 


. 3238 


. 6762 


11 


50 


.49748 


.50252 


2.0101 


.57348 


1.7437 


1.1528 


.13252 


.86748 


10 


51 


. 9773 


. 0227 


.0091 


. 7386 


.7426 


.1530 


. 3267 


. 6733 


9 


52 


. 9798 


. 0202 


.0081 


. 7425 


.7414 


.1531 


. 3281 


. 6719 


8 


53 


. 9823 


. 0176 


.0071 


. 7464 


.7402 


.1533 


. 3296 


. 6704 


7 


54 


. 9849 


. 0151 


.0061 


. 7502 


.7390 


.1535 


. 3310 


. 6690 


6 


55 


.49874 


.50126 


2.0050 


.57541 


1.7379 


1.1537 


.13325 


.86675 


5 


56 


. 9899 


. 0101 


.0040 


. 7580 


.7367 


.1539 


. 3339 


. 6661 


4 


57 


. 9924 


. 0076 


.0030 


. 7619 


.7355 


.1541 


. 3354 


. 6646 


3 


58 


. 9950 


. 0050 


.0020 


. 7657 


.7344 


.1543 


. 3368 


. 6632 


2 


1 59 


. 9975 


. 0025 


.0010 


. 7696 


.7332 


.1545 


. 3383 


. 6617 


1 


60 


.50000 


. 0000 


.0000 


. 7735 


.7320 


.1547 


. 3397 


. 6602 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Sine. 


Vrs. cos. 


M. 



119° 



60° 



164 



Natural Functions. 



30° 



Natural Trigonometrical Functions. 



149° 



M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.50000 


.50000 


2.0000 


.57735 


1.7320 


1.1547 


.13397 


.86602 


60 


1 


. 0025 


.49975 


1.9990 


. 7774 


.7309 


.1549 


. 3412 


. 6588 


59 


2 


. 0050 


. 9950 


.9980 


. 7813 


.7297 


.1551 


. 3426 


. 6573 


58 


3 


. 0075 


. 9924 


.9970 


. 7851 


.7286 


.1553 


. 3441 


. 6559 


57 


4 


. 0101 


. 9899 


.9960 


. 7890 


.7274 


.1555 


. 3456 


. 6544 


56 


5 


.50126 


.49874 


1.9950 


.57929 


1.7262 


1.1557 


.13470 


.86530 


55 


6 


. 0151 


. 9849 


.9940 


. 7968 


.7251 


.1559 


. 3485 


. 6515 


54 


7 


. 0176 


. 9824 


.9930 


. 8007 


.7239 


.1561 


. 3499 


. 6500 


53 


8 


. 0201 


. 9799 


.9920 


. 8046 


.7228 


.1562 


. 3514 


. 6486 


52 


9 


. 0226 


. 9773 


.9910 


. 8085 


.7216 


.1564 


. 3529 


. 6471 


51 


10 


.50252 


.49748 


1.9900 


.58123 


1.7205 


1.1566 


.13543 


.86457 


50 


11 


. 0277 


. 9723 


.9890 


. 8162 


.7193 


.1568 


. 3558 


. 6442 


49 


12 


. 0302 


. 9698 


.9880 


. 8201 


.7182 


.1570 


. 3572 


. 6427 


48 


13 


. 0327 


. 9673 


.9870 


. 8240 


.7170 


.1572 


. 3587 


. 6413 


47 


14 


. 0352 


. 9648 


.9860 


. 8279 


.7159 


.1574 


. 3602 


. 6398 


46 


15 


.50377 


.49623 


1.9850 


.58318 


1.7147 


1.1576 


.13616 


.86383 


45 


16 


.0402 


. 9597 


.9840 


. 8357 


.7136 


.1578 


. 3631 


. 6369 


44 


17 


. 0428 


. 9572 


.9830 


. 8396 


.7124 


.1580 


. 3646 


. 6354 


43 


18 


. 0453 


. 9547 


.9820 


. 8435 


.7113 


.1582 


. 3660 


. 6339 


42 


19 


. 0478 


. 9522 


.9811 


. 8474 


.7101 


.1584 


. 3675 


. 6325 


41 


20 


.50503 


.49497 


1.9801 


.58513 


1.7090 


1.1586 


.13690 


.86310 


40 


21 


. 0528 


. 9472 


.9791 


. 8552 


.7079 


.1588 


. 3704 


. 6295 


39 


22 


. 0553 


. 9447 


.9781 


. 8591 


.7067 


.1590 


. 3719 


. 6281 


38 


23 


. 0578 


. 9422 


.9771 


. 8630 


.7056 


.1592 


. 3734 


. 6266 


37 


24 


. 0603 


. 9397 


.9761 


. 8670 


.7044 


.1594 


. 3749 


. 6251 


36 


25 


.50628 


.49371 


1.9752 


.58709 


1.7033 


1.1596 


.13763 


.86237 


35 


26 


. 0653 


. 9346 


.9742 


. 8748 


.7022 


.1598 


. 3778 


. 6222 


34 


27 


. 0679 


. 9321 


.9732 


. 8787 


.7010 


.1600 


. 3793 


. 6207 


33 


28 


. 0704 


. 9296 


.9722 


. 8826 


.6999 


.1602 


. 3807 


. 6192 


32 


29 


. 0729 


. 9271 


.9713 


. 8865 


.6988 


.1604 


. 3822 


. 6178 


31 


30 


.50754 


.49246 


1.9703 


.58904 


1.6977 


1.1606 


.13837 


.86163 


30 


31 


. 0779 


. 9221 


.9693 


. 8944 


.6965 


.1608 


. 3852 


. 6148 


29 


82 


. 0804 


. 9196 


.9683 


. 8983 


.6954 


.1610 


. 3867 


. 6133 


28 


33 


. 0829 


. 9171 


.9674 


. 9022 


.6943 


.1612 


. 3881 


. 6118 


27 


34 


. 0854 


. 9146 


.9664 


. 9061 


.6931 


.1614 


. 3896 


. 6104 


26 


35 


.50879 


.49121 


1.9654 


.59100 


1.6920 


1.1616 


.13911 


.86089 


25 


36 


. 0904 


. 9096 


.9645 


. 9140 


.6909 


.1618 


. 3926 


. 6074 


24 


37 


. 0929 


. 9071 


.9635 


. 9179 


.6898 


.1620 


. 3941 


. 6059 


23 


38 


. 0954 


. 9046 


.9625 


. 9218 


.6887 


.1622 


. 3955 


. 6044 


22 


39 


. 0979 


. 9021 


.9616 


. 9258 


.6875 


.1624 


. 3970 


. 6030 


21 


40 


.51004 


.48996 


1.9606 


.59297 


1.6864 


1.1626 


.13985 


.86015 


20 


41 


. 1029 


. 8971 


.9596 


. 9336 


.6853 


.1628 


. 4000 


. 6000 


19 


42 


. 1054 


. 8946 


.9587 


. 9376 


.6842 


.1630 


. 4015 


. 5985 


18 


43 


. 1079 


. 8921 


.9577 


. 9415 


.6831 


.1632 


. 4030 


. 5970 


17 


44 


. 1104 


. 8896 


.9568 


. 9454 


.6820 


.1634 


. 4044 


. 5955 


16 


45 


.51129 


.48871 


1.9558 


.59494 


1.6808 


1.1636 


.14059 


.85941 


15 


46 


. 1154 


. 8846 


.9549 


. 9533 


.6797 


.1638 


. 4074 


. 5926 


14 


47 


. 1179 


. 8821 


.9539 


. 9572 


.6786 


.1640 


. 4089 


. 5911 


13 


48 


. 12M 


. 8796 


.9530 


. 9612 


.6775 


.1642 


. 4104 


. 5896 


12 


49 


. 1229 


. 8771 


.9520 


. 9651 


.6764 


.1644 


. 4119 


. 5881 


11 


50 


.51254 


.48746 


1.9510 


.59691 


1.6753 


1.1646 


.14134 


.85866 


10 


51 


. 1279 


. 872] 


.9501 


. 9730 


.6742 


.1648 


. 4149 


. 5851 


9 


52 


. 1304 


. 8696 


.9491 


. 9770 


.6731 


.1650 


. 4164 


. 5836 


8 


53 


. 1329 


. 8671 


.9482 


. 9809 


.6720 


.1652 


. 4178 


. 5821 


7 


54 


. 1354 


. 8646 


.9473 


. 9849 


.6709 


.1654 


. 4193 


. 5806 


6 


55 


.51379 


.48621 


1.9463 


.59888 


1.6698 


1.1656 


.14208 


.85791 


5 


56 


. llol 


. 8596 


.9454 


. 9928 


.6687 


.1658 


. 4223 


. 5777 


4 


57 


. 1429 


. 8571 


.9444 


. 9967 


.6676 


.1660 


. 4238 


. 5762 


3 


58 


. li:. 1 


. 8546 


.9435 


.60007 


.6665 


.1662 


. 4253 


. 5747 


2 


59 


. 1479 


. 8521 


.9425 


. 0046 


.6654 


.1664 


. 4268 


. 5732 


1 


60 


. 1504 


. 8496 


.9416 


. 0086 


.6643 


.1666 


. 4283 


. 5717 


M. 


1 o-iiio. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt| 


Vrs. cos. 


Sine. |m. 



120° 



59° 



Natural Functions. 



165 



31 c 




Natural Trigonometrical Functions. 


14 


8° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 


>0 


.51504 


.48496 


1.9416 


.60086 


1.6643 


1.1666 


.14283 


.85717 


60 


1 


. 1529 


. 8471 


.9407 


. 0126 


.6632 


.1668 


. 4298 


. 5702 


59 


2 


. 1554 


. 8446 


.9397 


. 0165 


.6621 


.1670 


. 4313 


. 5687 


58 


3 


. 1578 


. 8421 


.9388 


. 0205 


.6610 


.1672 


. 4328 


. 5672 


57 


4 


. 1603 


. 8396 


.9378 


. 0244 


.6599 


.1674 


. 4343 


. 5657 


56 


5 


.51628 


.48371 


1.9369 


.60284 


1.6588 


1.1676 


.14358 


.85642 


55 


6 


. 1653 


. 8347 


.9360 


. 0324 


.6577 


.1678 


. 4373 


. 5627 


54 


7 


. 1678 


. 8322 


.9350 


. 0363 


.6566 


.1681 


. 4388 


. 5612 


53 


8 


. 1703 


. 8297 


.9341 


. 0403 


.6555 


.1683 


. 4403 


. 5597 


52 


9 


. 1728 


. 8272 


.9332 


. 0443 


.6544 


.1685 


. 4418 


. 5582 


51 


10 


.51753 


.48247 


1.9322 


.60483 


1.6534 


1.1687 


.14433 


.85566 


50 


11 


. 1778 


. 8222 


.9313 


. 0522 


.6523 


.1689 


. 4448 


. 5551 


49 


12 


. 1803 


. 8197 


.9304 


. 0562 


.6512 


.1691 


. 4463 


. 5536 


48 


13 


. 1827 


. 8172 


.9295 


. 0602 


.6501 


.1693 


. 4479 


. 5521 


47 


14 


. 1852 


. 8147 


.9285 


. 0642 


.6490 


.1695 


. 4494 


. 5506 


46 


15 


.51877 


.48123 


1.9276 


.60681 


1.6479 


1.1697 


.14509 


.85491 


45 


16 


. 1902 


. 8098 


.9267 


. 0721 


.6469 


.1699 


. 4524 


. 5476 


44 


17 


. 1927 


. 8073 


.9258 


. 0761 


.6458 


.1701 


. 4539 


. 5461 


43 


18 


. 1952 


. 8048 


.9248 


. 0801 


.6447 


.1703 


. 4554 


. 5446 


42 


19 


. 1977 


. 8023 


.9239 


. 0841 


.6436 


.1705 


. 4569 


. 5431 


41 


20 


.52002 


.47998 


1.9230 


.60881 


1.6425 


1.1707 


.14584 


.85416 


40 


21 


. 2026 


. 7973 


.9221 


. 0920 


.6415 


.1709 


. 4599 


. 5400 


39 


22 


. 2051 


. 7949 


.9212 


. 0960 


.6404 


.1712 


. 4615 


. 5385 


38 


23 


. 2076 


. 7924 


.9203 


. 1000 


.6393 


.1714 


. 4630 


. 5370 


37 


24 


. 2101 


. 7899 


.9193 


. 1040 


.6383 


.1716 


. 4645 


. 5355 


36 


25 


.52126 


.47874 


1.9184 


.61080 


1.6372 


1.1718 


.14660 


.85340 


35 


26 


. 2151 


. 7849 


.9175 


. 1120 


.6361 


.1720 


. 4675 


. 5325 


34 


27 


. 2175 


. 7824 


.9166 


. 1160 


.6350 


.1722 


. 4690 


. 5309 


33 


28 


. 2200 


. 7800 


.9157 


. 1200 


.6340 


.1724 


. 4706 


. 5294 


32 


29 


. 2225 


. 7775 


.9148 


. 1240 


.6329 


.1726 


. 4721 


. 5279 


31 


30 


.52250 


.47750 


1.9139 


.61280 


1.6318 


1.1728 


.14736 


.85264 


30 


31 


. 2275 


. 7725 


.9130 


. 1320 


.6308 


.1730 


. 4751 


. 5249 


29 


32 


. 2299 


. 7700 


.9121 


. 1360 


.6297 


.1732 


. 4766 


. 5234 


28 


33 


. 2324 


. 7676 


.9112 


. 1400 


.6286 


.1734 


. 4782 


. 5218 


27 


34 


. 2349 


. 7651 


.9102 


. 1440 


.6276 


.1737 


. 4797 


. 5203 


26 


35 


.52374 


.47626 


1.9093 


.61480 


1.6265 


1.1739 


.14812 


.85188 


25 


36 


. 2398 


. 7601 


.9084 


. 1520 


.6255 


.1741 


. 4827 


. 5173 


24 


37 


. 2423 


. 7577 


.9075 


. 1560 


.6244 


.1743 


. 4842 


. 5157 


23 


38 


. 2448 


. 7552 


.9066 


. 1601 


.6233 


.1745 


. 4858 


. 5142 


22 


39 


. 2473 


. 7527 


.9057 


. 1641 


.6223 


.1747 


. 4873 


. 5127 


21 


40 


.52498 


.47502 


1.9048 


.61681 


1.6212 


1.1749 


.14888 


.85112 


20 


41 


. 2522 


. 7477 


.9039 


. 1721 


.6202 


.1751 


. 4904 


. 5096 


19 


42 


. 2547 


. 7453 


.9030 


. 1761 


.6191 


.1753 


. 4919 


. 5081 


18 


43 


. 2572 


. 7428 


.9021 


. 1801 


.6181 


.1756 


. 4934 


. 5066 


17 


44 


. 2597 


. 7403 


.9013 


. 1842 


.6170 


.1758 


. 4949 


. 5050 


16 


45 


.52621 


.47379 


1.9004 


.61882 


1.6160 


1.1760 


.14965 


.85035 


15 


46 


. 2646 


. 7354 


.8995 


. 1922 


.6149 


.1762 


. 4980 


. 5020 


14 


47 


. 2671 


. 7329 


.8986 


. 1962 


.6139 


.1764 


. 4995 


. 5004 


13 


48 


. 2695 


. 7304 


.8977 


. 2004 


.6128 


.1766 


. 5011 


. 4989 


12 


49 


. 2720 


. 7280 


.8968 


. 2043 


.6118 


.1768 


. 5026 


. 4974 


11 


50 


.52745 


.47255 


1.8959 


.62083 


1.6107 


1.1770 


.15041 


.84959 


10 


51 


. 2770 


. 7230 


.8950 


. 2123 


.6097 


.1772 


. 5057 


. 4943 


9 


52 


. 2794 


. 7205 


.8941 


. 2164 


.6086 


.1775 


. 5072 


. 4928 


8 


53 


. 2819 


. 7181 


.8932 


. 2204 


.6076 


.1777 


. 5087 


. 4912 


7 


54 


. 2844 


. 7156 


.8924 


. 2244 


.6066 


.1779 


. 5103 


. 4897 


6 


55 


.52868 


.47131 


1.8915 


.62285 


1.6055 


1.1781 


.15118 


.84882 


5 


56 


. 2893 


. 7107 


.8906 


. 2325 


.6045 


.1783 


. 5133 


. 4866 


4 


57 


. 2918 


. 7082 


.8897 


. 2366 


.6034 


.1785 


. 5149 


. 4851 


3 


58 


. 2942 


. 7057 


.8888 


. 2406 


.6024 


.1787 


. 5164 


. 4836 


2 


59 


. 2967 


. 7033 


.8879 


. 2446 


.6014 


.1790 


. 5180 


. 4820 


1 


60 


. 2992 


. 7008 


.8871 


. 2487 


.6003 


.1792 


. 5195 


. 4805 





M. 


j Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



121° 



58° 



166 



Natural Functions. 



32° 


Natural Trigonometrical Functions. 


1414 


M Sine. 




• ' 


Twig. 


1.6003 


1.1792 


.15195 


Cosine. M. 


o .52992 


.47008 




EM87 


B B 60-** - 


1 . 3016 


8 


88 - 


2527 




.1794 


. 5211 


59 


3041 




- 


- - 


! B 


.1796 


522 


58 


3 . 3066 


. 6934 




. aeoe 


" 72 


17 - 


. 5241 


1758 57 


4 . 9090 




88 


. 2649 




.1800 


" 




^115 










a - 




54728 


6 . 3140 


- 


8818 


23 


.5941 




5288 


. 4712 


54 


7 . 


3 


88 


277 






. 5303 


. 4697 




B 


811 


.8801 


. 2S11 


.5921 


J809 


. 5319 


. 46S1 




3214 


7S 




2851 




.1>11 




. 4666 


51 


53238 


i - 




2S - 


1.5900 


1.1813 


.15350 


.84650 


50 


11 . - 


3 


B77S 


- 






. 5365 




49 


IS 


" . 


83 




588 


1818 




. 4619 


48 


13 . - 




8751 


. 3014 




.1820 




. 4604 




14 . 3337 


. 6663 


^~ 




585 




. 5412 




46 


15 .53361 


- 




.6309-5 




1.1821 


.15427 


45 


16 


. 6614 


83 1 


. 3136 


58 




. 5443 


3 44 


17 . 3411 




B72 


. 3177 




1828 




. 4542 43 


IS . 


. 656? 


.8714 


. 3217 


5818 


.1831 


. 5474 


42 


19 . 3460 


. 6540 


^~ 


258 


58 - 






. 4511 


41 


■ 


.46516 


- " 


.63299 




1JL835 




.84495 


40 


21 


. 6491 






5788 




. 5520 


. 4479 


39 


22 . 


. 6466 


- M 


v 


5778 


.1839 




. 4464 


38 


23 . 


. 


.8671 




■ - 


.1841 


5552 


. 444S 


' 


M 


. 6417 


B 


. 3462 





.1844 


" " 


. 4433 


36 


K 






" 


1 5747 






.84417 




. 


. 636S 








L8 3 


. 5598 


. 4402 


34 


3656 




.8 ■ 


184 


5723 


.1850 


. 5614 




33 


0681 


. 6319 


.8629 


. 3625 


.5717 


1852 


. 5630 


. 4370 




29 . _ " 


. 6294 


.8620 












31 


' ' 


.46270 


1.S611 


1 7 






.15661 




30 


31 . " 




B 


- 


83 


J859 


3 




29 


. 


. 6221 


BE " 


78! 


" 


.1861 


■ - 




28 


33 . 3S03 


. 6196 




a 


5667 






. 4292 


23 


34 . 828 


72 


8578 


871 


" 57 




572 


. 4276 


26 


35 .53852 


.46147 


1.8569 


.63912 


1.5646 


U 3 




.84261 


25 


36 .3877 


. 6123 


.8561 


. 3953 


" 


.1S70 


575J 


. 4245 


24 


3901 


. 6098 


8552 


. 3994 


_ 


1872 


. 5770 


. 4229 




B 


. 6074 






..5616 


J874 




. 4214 


22 




. 6049 


K 


. 4076 


.5606 


.1877 


. 5802 


. 419S 


21 




- 




.64117 


15596 


1.1879 


.15817 


.84182 


20 


41 


. 6000 


.8519 






.1881 


58 


. 4167 


19 


4024 


S 3 




. 4199 


5577 


188 




. 4151 


18 


43 




8£ - 


. 4240 






58 


. 4135 


17 


44 


' _- 




. 4281 




.1888 


588 


. 4120 


16 








.64322 






.15896 


.84104 


15 


4122 










18 - 


. 5912 




14 


4146 




5 - 


. 4404 


5527 




I 23 


. 4072 


13 


4171 






. 4446 








. 4057 


12 


49 . 








" 




. 5959 


. 4041 


11 


50 . 








1.5497 


1.1901 




.84025 10 


51 . 4244 


. 








.1903 


. 5991 


. 4009 9 








. 4610 




.1906 


. 6006 


. 3993 


53 








! 


.1908 


22 


78 7 


54 










.1910 


. 6036 


6 


55 .' 










1.1912 




5 


56 . 


. 5634 








.1915 


. 6070 


. 3930 


4 


"- 


. 5609 








.1917 




. 3914 


3 


58 . 4415 












. 6101 


. 3-99 


2- 


59 . 










.1921 


. 6117 




l m ' 


60 . 






. 4941 


.5399 


.1922 


. 6133 


. 3867 





sine. 




& Mil 


Cotang. 


Tang. 


Coeec'nt 


Vrs. cos. 


Sine. M. 



122° 



57° 



Natural Functions. 



167 



33° 



Natural Trigonometrical Functions. 



146° 



M. 


Sine. 






Tang. 


- 






e. If. 





.54464 


.45536 


u 1 1 


.04941 




1.1924 


.16133 


.83867 60 


1 






- ' - 






.19-26 


. 6149 






2 


. 4513 




.8344 




.5379 




. 616-5 


. -.5 


L't 


3 


. 4537 


. .5463 


- 




I 


.1930 


. 6180 


. 3819 


ffl 


4 


. 4561 


. 5438 


- - 




.5359 


.1933 


. 6196 




56 


5 


.54586 


.45414 


1 ■ - 


!65148 


1.5350 


1.1935 


1 212 




' 


6 


. 4610 


. 5390 


.8311 




.53,40 


.1937 


__- 


. 3772 


54 


7 


. 4634 


. 5365 




. 5231 


' 


.1939 


. 6244 


" 


'3 


8 


. 1659 


. 5341 




. 5272 


' - 


.1942 


. 6260 


- - 


9 


. 4683 


. 5317 


8287 


. 531 4 


.5311 


.1944 


. 6276 


. 3724 51 


10 


.•54708 


.45292 


1.-279 


' " 


1 5 : 


1.1946 


- 2 - 


• 2 


11 


. 4732 


'- - 


.-271 


' -~ 


-_ 1 


.1948 




- 


12 


. 4756 


. 524 4 


-- 




.5282 


.1951 


. 6323 


i 


13 


. 4781 


. 5219 


.8J55 




5872 


.1953 


. 6339 


6 J 47 


14 


. 4-05 


. 5195 




. 5521 


"- - 


.1955 


. 6355 


-A 46 


15 


.54829 


.45171 


L82 - 




1.5252 


1.1958 


.16371 


■ _9 45 


16 


. 4854 


. 5146 


-_ 


. 5604 


.5243 


.1960 


' 


. 3613 44 


17 


1878 


. 5122 


-___ 




5233 


.1962 




. 3597 43 


18 


. 4902 


. 5098 


.8214 


' -: 


.5223 


.1964 


. 6419 


. 35-1 42 


19 


. 4926 


. 5073 


K 


. 5729 


.5214 


.1967 


. 6435 


5 41 


20 


..54951 


.45049 


1 -: - 


~ _ ~1 


1.5204 


1.1969 


.16451 


1 9 40 


21 


. 4975 


. " _~ 


.8190 






.1971 


. 6467 


. 353 


22 


. 4999 


. 5000 


.8182 






.1974 




. 3517 .38 


23 


. 5024 


. 4976 


.8174 




.5175 


.1976 


. 6499 


. 3-501 37 


24 


. 5048 


. 4952 


.-: ■ 


.5938 


.5166 


.1978 


. 6515 


185 36 


25 


.55072 


.44928 




.65980 


1,51-56 


1.1980 


.16531 


3 35 


26 


. 5097 


. 4903 


.8150 


_; 


I ' 


.19-3 


. 6-547 


. 3453 34 


27 


. 5121 


. 4879 


.8142 




.5137 


.19-5 


" 


. 3437 33 


28 


. 5145 




.8134 


6105 


.5127 


' 


. 6579 


. 3421 


- 


29 


. 5169 


. 4830 


.8126 


. 6147 


.5118 


.1990 


. 6595 


. 3405 


31 


30 


.55194 


.44-06 


1.8118 


.66188 


1.5108 


1.1992 


.16611 




30 


31 


. 5218 


. 47-2 


.8110 


. 6230 


,5099 


.1994 


. 6627 


. 3372 


a 


32 


. 5242 


4758 


.8102 


. 6272 


..5089 


.1997 


. 6643 


r 


33 


. 5266 


. 4733 


.8094 


. 6314 


,5080 


.1999 


. 6660 


. 3:340 27 


34 


. 5291 


. 4709 


■ ■ 


. 6>;56 


,5070 


.2001 


' 


. 3324 26 


35 


.55315 


.4468-5 


1.8078 


.66398 


1,5061 


1.2004 


.16692 


• 


25 


36 


. 5339 


. 4661 


.8070 


. 6440 


.5051 


.2006 


. 6708 


. 3292 


24 


37 


. 5363 


. 4637 


H - 


. 64-2 


,5042 


.2008 


. 6724 


. 3276 


23 


38 




. 4612 




. 6524 


,5032 


.2010 


. 6740 


. 3260 


-i 


39 


. 5412 


. 4588 


.-047 


. 6566 


,5023 


.2013 


" 


. 3244 


21 


40 


.55436 


.44-564 


1.8039 


.66608 


1,5013 


1.2015 


.16772 


- 228 


20 


41 


. .5460 


. 4-540 


.8031 


. 66-50 


,5004 


.2017 




. 3211 


19 


42 


. 5484 


. 4515 


- _ 


. 6692 


.4994 


.2020 




. 3195 


18 


43 


. 5-509 


. 4491 


.8015 


. 6734 


.4985 


.2022 


- 


. 3179 


17 


44 


. 5533 


. 4467 


.8007 


. 6776 


.4975 


.2024 


' 


. 3163 


16 


45 


.55557 


.44443 


1.7999 


.66818 


1.4966 


1.2027 


.16-53 


.-3147 


15 


46 


. 5581 


. 4419 


.7992 


. 6-60 


.4957 


.2029 




. 3131 


14 


47 


. 5605 


. 4395 


.7984 


. 6902 


.4947 


.2031 


■' 


. 3115 


13 


48 


. 5629 


. 4370 


.7976 


. 6944 




.2034 


. 6901 


. 3098 


12 


49 


. 5654 


. 4346 


.7968 


■ 


.4928 


.2036 


. 6918 


■_ 


11 


50 


.5507- 


.44322 


1.7960 


.670-28 


1.4919 


1.2039 


.16934 




10 


51 


. 5702 


. 4298 


.7953 


. 7071 


.4910 


.2041 




. 3050 


9 


52 


. 5726 


. 4274 


.7945 


. 7113 


.4900 


.2043 


. 6966 


. 30:34 


8 


53 


. 5750 


. 4250 


.7937 


. 7155 


.4891 


.2046 


. 69-2 


. 3017 7 


54 


. 5774 


. 4225 


.7929 


. 7197 




.2048 


. 6999 


. 3001 6 


55 


.55799 


.44201 


1.7921 


.67239 


1.4872 


1.20-50 


.17015 


52985 5 


56 


. 5-23 


. 4177 


.7914 


7282 


.4-63 


.2053 


. 7031 


. 2969 


4 


57 


. 5-17 


. 4153 


.79,6 


" 24 


.4853 


.2055 


. 7047 


. 2952 


3 


58 


. 5871 


. 4129 


' ■ 


" 


.4844 


.2057 


. 7064 


. 2936 


2 


59 


. 5895 


. 4105 


.7891 


" • 


.4835 


.2060 


. 7080 


. 2920 1 


60 


i . 5919 


. 4081 




. 7451 


.4826 


.2062 


" 


. 2904 


M. 


Cosine. 


Vrs. gin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. M. 



123° 



56° 



168 



Natural Functions. 



34° 


Natural Trigonometrical Functions. 


145° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.55919 


.44081 


1.7883 


.67451 


1.4826 


1.2062 


.17096 


.82904 


00 


1 


. 5943 


. 4057 


.7875 


. 7193 


.4816 


.2064 


. 7112 


. 2887 


59 


2 


. 5967 


. 4032 


.7867 


. 7535 


.4807 


.2067 


. 7129 


. 2871 


58 


3 


. 5992 


. 4008 


.7860 


. 7578 


.4798 


.2069 


. 7145 


. 2855 


57 


4 


. 6016 


. 3984 


.7852 


. 7620 


.4788 


.2072 


. 7101 


. 2839 


50 


5 


.56040 


.43960 


1.7844 


.07063 


1.4779 


1.2074 


.17178 


.82822 


55 


6 


. 6064 


. 3936 


.7837 


. 7705 


.4770 


.2076 


. 7194 


. 2806 


54 


7 


. 6088 


. 3912 


.7829 


. 7747 


.4761 


.2079 


. 7210 


. 2790 


53 


8 


. 6112 


. 3888 


.7821 


. 7790 


.4751 


.2081 


. 7227 


. 2773 


52 


9 


. 6136 


. 3864 


.7814 


. 7832 


.4742 


.2083 


. 7243 


. 2757 


51 


10 


.56160 


.43840 


1.7806 


.67875 


1.4733 


1.2086 


.17259 


.82741 


50 


11 


. 6184 


. 3816 


.7798 


. 7917 


.4724 


.2088 


. 7276 


. 2724 


49 


12 


. 6208 


. 3792 


.7791 


. 7960 


.4714 


.2091 


. 7292 


. 2708 


48 


13 


. 6232 


. 3768 


.7783 


. 8002 


.4705 


.2093 


. 7308 


. 2692 


47 


11 


. 6256 


. 3743 


.7776 


. 8045 


.4696 


.2095 


. 7325 


. 2675 


40 


15 


.56280 


.43719 


1.7768 


.68087 


1.4087 


1.2098 


.17341 


.82659 


45 


16 


. 6304 


. 3695 


.7760 


. 8130 


.4678 


.2100 


. 7357 


. 2643 


44 


17 


. 6328 


. 3671 


.7753 


. 8173 


.4009 


.2103 


. 7374 


. 2626 


43 


18 


. 6353 


. 3647 


.7745 


. 8215 


.4659 


.2105 


. 7390 


. 2610 


42 


19 


. 6377 


. 3623 


.7738 


. 8258 


.4650 


.2107 


. 7406 


. 2593 


41 


20 


.56401 


.43599 


1.7730 


.68301 


1.4641 


1.2110 


.17423 


.82577 


40 


21 


. 6425 


. 3575 


.7723 


. 8343 


.4632 


.2112 


. 7439 


. 2561 


39 


22 


. 6449 


. 3551 


.7715 


. 8386 


.4623 


.2115 


. 7456 


. 2544 


38 


23 


. 6473 


. 3527 


.7708 


. 8429 


.4614 


.2117 


. 7472 


. 2528 


37 


24 


. 6497 


. 3503 


.7700 


. 8471 


.4605 


.2119 


. 7489 


. 2511 


30 


25 


.56521 


.43479 


1.7693 


.68514 


1.4595 


1.2122 


.17505 


.82495 


35 


26 


. 6545 


. 3455 


.7085 


. 8557 


.4580 


.2124 


. 7521 


. 2478 


34 


27 


. 6569 


. 3431 


.7078 


. 8600 


.4577 


.2127 


. 753,8 


. 2462 


33 


28 


. 6593 


. 3407 


.7670 


. 8642 


.4568 


.2129 


. 7554 


. 2445 


32 


29 


. 6617 


. 3383 


.7663 


. 8685 


.4559 


.2132 


. 7571 


. 2429 


31 


30 


.56641 


.43359 


1.7655 


.68728 


1.4550 


1.2134 


.17587 


.82413 


30 


a 


. 6664 


. 3335 


.7648 


. 8771 


.4541 


.2136 


. 7604 


. 2396 


29 


32 


. 6688 


. 3311 


.7640 


. 8814 


.4532 


.2139 


. 7020 


. 22,80 


28 


33 


. 6712 


. 3287 


.7633 


. 8857 


.4523 


.2141 


. 7637 


. 22,03 


27 


34 


. 6736 


. 3/203 


.7025 


. 8899 


.1511 


.2144 


. 705:', 


. 2347 


20 


35 


.56760 


.43239 


1.7018 


.68942 


1.4505 


1.21 10 


.17070 


.822,30 


25 


36 


. 6784 


. 3216 


.7610 


. 8985 


.4496 


.2149 


. 7686 


. 2314 


24 


37 


. 6808 


. 3192 


.7603 


. 9028 


.4487 


.2151 


. 7703 


. 2297 


22, 


38 


. 6832 


. 3168 


.7596 


. 9071 


.4478 


.2153 


. 7719 


. 2280 


22 


39 


. 6856 


. 3144 


.7588 


. 9114 


.4469 


.2156 


. 77: '.0 


. 2264 


21 


40 


.56880 


.43120 


1.75S1 


.09157 


1.4400 


1.2158 


.17752 


.82247 


20 


41 


. 6904 


. 3096 


.7573 


. 9200 


.4451 


.2161 


. 7709 


. 2231 


19 


42 


. 6928 


. 3072 


.7566 


. 9243 


.4442 


.2103 


. 7786 


. 2214 


18 


43 


. 6952 


. 3048 


.7559 


. 9286 


.4433 


.2166 


. 7802 


. 2198 


17 


41 


. 6976 


. 3024 


.7551 


. 9329 


.4424 


.2168 


. 7819 


. 2181 


10 


45 


.57000 


.43000 


1.7544 


.69372 


1.4415 


1.2171 


.17835 


.82105 


15 


40 


. 7023 


. 2976 


.7537 


. 9415 


.4406 


.2173 


. 7852 


. 2148 


14 


47 


. 7047 


. 2952 


.7520 


. 9459 


.4397 


.2175 


. 7868 


. 2131 


13 


48 


. 7071 


. 2929 


.7522 


. 9502 


.4388 


.2178 


. 7885 


. 2115 


12 


49 


. 7095 


. 2905 


.7514 


. 9545 


.4379 


.2 ISO 


. 7902 


. 2098 


11 


50 


.57119 


.42881 


1.7507 


.69588 


1.4370 


1.2183 


.17918 


.82082 


10 


51 


. 71 13 


. 2857 


.7.500 


. 9631 


.4361 


.2185 


. 793,5 


. 2065 


9 


52 


. 7167 




.71'.):', 


. 9674 


.4352 


.2188 


. 7951 


. 2018 


8 


53 


. 7191 


. 2809 


.7485 


. 9718 


.4343 


.2190 


. 7968 


. 202,2 


7 


54 


. 7214 


. 2785 


.7478 


. 9761 


,1 :;:;:, 


.2193 


. 7985 


. 2015 





55 


.57238 


.127.;] 


1.7171 


.69804 


1.4326 


1.2195 


.18001 


.81998 


5 


56 


. 7262 


. 2738 


.7463 


. 9847 


.4317 


.2198 


. 8018 


. 1982 


4 


57 




. 2714 


.7150 


. 9891 


.4308 


.2200 


. 8035 


. 1905 


3 


58 


. 7310 


. 2090 


.7449 


. 9934 


.4299 


.2203 


. 8051 


. 1948 


2 


59 




. 2666 


.7112 


. 9977 


.4290 


.2205 


. 8008 


. 193,2 


1 


GU 




. 2642 


.7434 


.70021 


.4 '281 


.2208 


. 8085 


. 1015 





M. 


Cosine. 


Vrs. Kin. 


Secant. 


Co tang. 


Tang. 


Cosec'nt] 


Vrs. cos. 


Sine. 


M. 



124° 



55° 



Natural Functions. 



169 



35 


■> 


Natural Trigonometrical Functions. 


144° 


M. 


Sine. 


Vrs. cos. 


'Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.57358 


.42642 


1.7434 


.70021 


1.4281 


1.2208 


.18085 


.81915 


60 


1 


. 7381 


. 2618 


.7427 


. 0064 


.4273 


.2210 


. 8101 


. 1898 


59 


2 


. 7405 


. 2595 


.7420 


. 0107 


.4261 


.2213 


. 8118 


. 1882 


58 


3 


. 7429 


. 2571 


.7413 


. 0151 


.4255 


.2215 


. 8135 


. 1865 


57 


4 


. 7453 


. 2547 


.7405 


. 0194 


.4246 


.2218 


. 8151 


. 1848 


56 


5 


.57477 


.42523 


1.7398 


.70238 


1.4237 


1.2220 


.18168 


.81832 


55 


6 


. 7500 


. 2499 


.7391 


. 0281 


.4228 


.2223 


. 8185 


. 1815 


54 


7 


. 7524 


. 2476 


.7384 


. 0325 


.4220 


.2225 


. 8202 


. 1798 


53 


8 


. 7548 


. 2452 


.7377 


. 0368 


.4211 


.2228 


. 8218 


. 1781 


52 


9 


. 7572 


. 2428 


.7369 


. 0412 


.4202 


.2230 


. 8235 


. 1765 


51 


10 


.57591) 


.42404 


1.7362 


.70455 


1.4193 


1.2233 


.18252 


.81748 


bO 


11 


. 7619 


. 2380 


.7355 


. 0499 


.4185 


.2235 


. 8269 


. 1731 


49 


12 


. 7643 


. 2357 


.7348 


. 0542 


.4176 


.2238 


. 8285 


. 1714 


48 


13 


. 7667 


. 2333 


.7341 


. 0586 


.4167 


.2240 


. 8302 


. 1698 


47 


14 


. 7691 


. 2309 


.7334 


. 0629 


.4158 


.2243 


. 8319 


. 1681 


46 


15 


.57714 


.42285 


1.7327 


.70673 


1.4150 


1.2245 


.18336 


.81664 


45 


16 


' . 7738 


. 2262 


.7319 


. 0717 


.4141 


.2248 


. 8353 


. 1647 


44 


17 


. 7762 


. 2238 


.7312 


. 0760 


.4132 


.2250 


. 8369 


. 1630 


43 


18 


. 7786 


. 2214 


.7305 


. 0804 


.4123 


.2253 


. 8386 


. 1614 


42 


19 


. 7809 


. 2190 


.729S 


. 0848 


.4115 


.2255 


. 8403 


. 1597 


41 


20 


.57833 


.42167 


1.7291 


.70891 


1.4106 


1.2258 


.18420 


.81580 


40 


21 


. 7857 


. 2143 


.7284 


. 0935 


.4097 


.2260 


. 8437 


. 1563 


39 


22 


. 7881 


. 2119 


.7277 


. 0979 


.4089 


.2263 


. 8453 


. 1546 


38 


23 


. 7904 


. 2096 


.7270 


. 1022 


.4080 


.2265 


. 8470 


. 1530 


37 


24 


. 7928 


. 2072 


.7263 


. 1066 


.4071 


.2268 


. 8487 


. 1513 


36 


25 


.57952 


.42048 


1.7256 


.71110 


1.4063 


1.2270 


.18504 


.81496 


35 


26 


. 7975 


. 2024 


.7249 


. 1154 


.4054 


.2273 


. 8521 


. 1479 


34 


27 


. 7999 


. 2001 


.7242 


. 1198 


.4045 


.2276 


. 8538 


. 1462 


33 


28 


. 8023 


. 1977 


.7234 


. 1241 


.4037 


.2278 


. 8555 


. 1445 


32 


29 


. 8017 


. 1953 


.7227 


. 1285 


.4028 


.2281 


. 8571 


. 1428 


31 


30 


.58070 


.41930 


1.7220 


.71329 


1.4019 


1.2283 


.18588 


.81411 


30 


31 


. 8094 


. 1906 


.7213 


. 1373 


.4011 


.2286 


. 8605 


. 1395 


29 


32 


. 8118 


. 1882 


.7206 


. 1417 


.4002 


.2288 


. 8622 


. 1378 


28 


33 


. 8141 


. 1859 


.7199 


. 1461 


.3994 


.2291 


. 8639 


. 1361 


27 


34 


. 8165 


. 1835 


.7192 


. 1505 


.3985 


.2293 


. 8656 


. 1344 


26 


35 


.58189 


.41811 


1.7185 


.71549 


1.3976 


1.2296 


.18673 


.81327 


25 


36 


• 8212 


. 1788 


.7178 


. 1593 


.3968 


.2298 


. 8690 


. 1310 


24 


37 


. 8236 


. 1764 


.7171 


. 1637 


.3959 


.2301 


. 8707 


. 1293 


23 


38 


. 8259 


. 1740 


.7164 


. 1681 


.3951 


.2304 


. 8724 


. 1276 


22 


39 


. 8283 


. 1717 


.7157 


. 1725 


.3942 


.2306 


. 8741 


. 1259 


21 


40 


.58307 


.41693 


1.7151 


.71769 


1.3933 


1.2309 


.18758 


.81242 


20 


41 


. 8330 


. 1669 


.7144 


. 1813 


.3925 


.2311 


. 8775 


. 1225 


19 


42 


. 8354 


. 16-16 


.7137 


. 1857 


.3916 


.2314 


. 8792 


. 1208 


18 


43 


. 8378 


. 1622 


.7130 


. 1901 


.3908 


.2316 


. 8809 


. 1191 


17 


44 


. 8401 


. 1599 


.7123 


. 1945 


.3899 


.2319 


. 8826 


. 1174 


16 


45 


.58425 


.41575 


1.7116 


.71990 


1.3891 


1.2322 


.18843 


.81157 


15 


46 


. 8448 


. 1551 


.7109 


. 2034 


.3882 


.2324 


. 8860 


. 1140 


14 


47 


. 8472 


. 1528 


.7102 


. 2078 


.3874 


.2327 


. 8877 


. 1123 


13 


48 


. 8496 


. 1504 


.7095 


. 2122 


.3865 


.2329 


. 8894 


. 1106 


12 


49 


. 8519 


. 1481 


.7088 


. 2166 


.3857 


.2332 


. 8911 


. 1089 


11 


50 


.58543 


.41457 


1.7081 


.72211 


1.3848 


1.2335 


.18928 


.81072 


10 


51 


. 8566 


. 1433 


.7075 


. 2255 


.3840 


.2337 


. S945 


. 1055 


9 


52 


. 8590 


. 1410 


.7068 


. 2299 


.3831 


.2340 


. 8962 


. 1038 


8 


53 


. 8614 


. 1386 


.7061 


. 2344 


.3823 


.2342 


. 8979 


. 1021 


7 


54 


. 8637 


. 1363 


.7054 


. 2388 


.3814 


.2345 


. 8996 


. 1004 


6 


55 


.58661 


.41339 


1.7047 


.72132 


1.3806 


1.2348 


.19013 


.80987 


5 


56 


. 8684 


. 1316 


.7040 


. 2177 


.3797 


.2350 


. 9030 


. 0970 


4 


57 


. 8708 


. 1292 


.7033 


. 2521 


.3789 


.2353 


. 9047 


. 0953 


3 


58 


. 8731 


. 1268 


.7027 


. 2565 


.3781 


.2355 


. 9064 


. 0936 


2 


59 


. 8755 


. 1245 


.7020 


. 2610 


.3772 


.2358 


. 9081 


. 0919 


1 


60 


. 8778 


. 1221 


.7013 


. 2654 


.3764 


.2361 


. 9098 


. 0902 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



125° 



54° 



170 



Natukal Functions. 



36° 



Natural Trigonometrical Functions. 



143° 



M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.58778 


.41221 


1.7013 


.72654 


1.3764 


1.2361 


.19098 


.80902 


60 


1 


. 8802 


. 1198 


.7006 


. 2699 


.3755 


.2363 


. 9115 


. 0885 


59 


2 


. 8825 


. 1174 


.6999 


. 2743 


.3747 


.2366 


. 9132 


. 0867 


58 


3 


. 8849 


. 1151 


.6993 


. 2788 


.3738 


.2368 


. 9150 


. 0850 


57 


4 


. 8873 


. 1127 


.6986 


. 2832 


.3730 


.2371 


. 9167 


. 0833 


56 


5 


.58896 


.41104 


1.6979 


.72877 


1.3722 


1.2374 


.19184 


.80816 


55 


6 


. 8920 


. 1080 


.6972 


. 2921 


.3713 


.2376 


. 9201 


. 0799 


54 


7 


. 8943 


. 1057 


.6965 


. 2966 


.3705 


.2379 


. 9218 


. 0782 


53 


8 


. 8967 


. 1033 


.6959 


. 3010 


.3697 


.2382 


. 9235 


. 0765 


52 


9 


. 8990 


. 1010 


.6952 


. 3055 


.3688 


.2384 


. 9252 


. 0747 


51 


10 


.59014 


.40986 


1.6945 


.73100 


1.3680 


1.2387 


.19270 


.80730 


50 


11 


. 9037 


. 0963 


.6938 


. 3144 


.3672 


.2389 


. 9287 


. 0713 


49 


12 


. 9060 


. 0939 


.6932 


. 3189 


.3663 


.2392 


. 9304 


. 0696 


48 


13 


. 9084 


. 0916 


.6925 


. 3234 


.3655 


.2395 


. 9321 


. 0679 


47 


14 


. 9107 


. 0892 


.6918 


. 3278 


.3647 


.2397 


. 9338 


. 0662 


46 


15 


.59131 


.40869 


1.6912 


.73323 


1.3638 


1.2400 


.19355 


.80644 


45 


16 


. 9154 


. 0845 


.6905 


. 3368 


.3630 


.2403 


. 9373 


. 0627 


44 


17 


. 9178 


. 0822 


.6898 


. 3412 


.3622 


.2405 


. 9390 


. 0610 


43 


18 


. 9201 


. 0799 


.6891 


. 3457 


.3613 


.2408 


. 9407 


. 0593 


42 


19 


. 9225 


. 0775 


.6885 


. 3502 


.3605 


.2411 


. 9424 


. 0576 


41 


20 


.59248 


.40752 


1.6878 


.73547 


1.3597 


1.2413 


.19442 


.80558 


40 


21 


. 9272 


. 0728 


.6871 


. 3592 


.3588 


.2416 


. 9459 


. 0541 


39 


22 


. 9295 


. 0705 


.6865 


. 3637 


.3580 


.2419 


. 9476 


. 0524 


38 


23 


. 9318 


. 0681 


.6858 


. 3681 


.3572 


.2421 


. 9493 


. 0507 


37 


24 


. 9342 


. 0658 


.6851 


. 3726 


.3564 


.2424 


. 9511 


. 0489 


36 


25 


.59365 


.40635 


1.6845 


.73771 


1.3555 


1.2427 


.19528 


.80472 


35 


26 


. 9389 


. 0611 


.6838 


. 3816 


.3547 


.2429 


. 9545 


. 0455 


34 


27 


. 9412 


. 0588 


.6831 


. 3861 


.3539 


.2432 


. 9562 


. 0437 


33 


28 


. 9435 


. 0564 


.6825 


. 3906 


.3531 


.2435 


. 9580 


. 0420 


32 


29 


. 9459 


. 0541 


.6818 


. 3951 


.3522 


.2437 


. 9597 


. 0403 


31 


30 


.59482 


.40518 


1.6812 


.73996 


1.3514 


1.2440 


.19614 


.80386 


30 


31 


. 9506 


. 0494 


.6805 


. 4041 


.3506 


.2443 


. 9632 


. 0368 


29 


32 


. 9529 


. 0471 


.6798 


. 4086 


.3498 


.2445 


. 9649 


. 0351 


28 


33 


. 9552 


. 0447 


.6792 


. 4131 


.3489 


.2448 


. 9666 


. 0334 


27 


34 


. 9576 


. 0424 


.6785 


. 4176 


.3481 


.2451 


. 9683 


. 0316 


26 


35 


.59599 


.40101 


1.6779 


.74221 


1.3473 


1.2453 


.19701 


.80299 


25 


36 


. 9622 


. 0377 


.6772 


. 4266 


.3465 


.2456 


. 9718 


. 0282 


24 


37 


. 9646 


. 0354 


.6766 


. 4312 


.3457 


.2459 


. 9736 


. 0264 


23 


38 


. 9669 


. 0331 


.6759 


. 4357 


.3449 


.2461 


. 9753 


. 0247 


22 


39 


. 9692 


. 0307 


.6752 


. 4402 


.3440 


.2464 


. 9770 


. 0230 


21 


40 


.59716 


.40284 


1.6746 


.74447 


1.3432 


1.2467 


.19788 


.80212 


20 


41 


. 9739 


. 0261 


.6739 


. 4492 


.3424 


.2470 


. 9805 


. 0195 


19 


42 


. 9762 


. 0237 


.6733 


. 4538 


.3416 


.2472 


. 9822 


. 0177 


18 


43 


. 9786 


. 0214 


.6726 


. 4583 


.3408 


.2475 


. 9840 


. 0160 


17 


44 


. 9809 


. 0191 


.6720 


. 4628 


.3400 


.2478 


. 9857 


. 0143 


16 


45 


.59832 


.40167 


1.0713 


.74673 


1.3392 


1.2480 


.19875 


.80125 


15 


46 


. 9856 


. 0144 


.6707 


. 4719 


.3383 


.2483 


. 9892 


. 0108 


14 


47 


. 9879 


. 0121 


.6700 


. 4764 


.3375 


.2486 


. 9909 


. 0090 


13 


48 


. 9902 


. 0098 


.6694 


. 4809 


.3367 


.2488 


. 9927 


. 0073 


12 


49 


. 9926 


. 0074 


.6687 


. 4855 


.3359 


.2491 


. 9944 


. 0056 


11 


50 


.59949 


.40051 


1.6681 


.74900 


1.3351 


1.2494 


.19962 


.80038 


10 


51 


. 9972 


. 0028 


.6674 


. 4946 


.3343 


.2497 


. 9979 


. 0021 


9 


52 


. 9995 


. 0001 


.6668 


. 4991 


.3335 


.2499 


. 9997 


. 0003 


8 


53 


.60019 


.39981 


.6661 


. 5037 


.3327 


.2502 


.20014 


.79986 


7 


54 


. 0042 


. 9958 


.6655 


. 5082 


.3319 


.2505 


. 0031 


. 9968 


6 


55 


.60065 


.39935 


1.6648 


.75128 


1.3311 


1.2508 


.20049 


.79951 


5 


56 


. 0088 


. 9911 


.6642 


. 5173 


.3303 


.2510 


. 0066 


. 9933 


4 


57 


. 0112 


. 9888 


.6636 


. 5219 


.3294 


.2513 


. 0084 


. 9916 


3 


58 


. 0135 


. 9865 


.6629 


. 5264 


.3286 


.2516 


. 0101 


. 9898 


2 


59 


. 0158 


. 9842 


.6623 


. 5: no 


.3278 


.2519 


. 0119 


. 9881 


1 


60 


. 0181 


. 9818 


.6616 


. 5355 


.3270 


.2521 


. 0136 


. 9863 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


Iff. 



126° 



53° 



Natural Functions. 



171 



37 


3 


Natural Trigonometrical 


Functions. 


142° 


M. Sine. 


Yrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.60181 


.39818 


1.6616 


.75355 


1.3270 


1.2521 


.20136 


.79863 


60 


1 


. 0205 


. 9795 


.6610 


. 5401 


.3262 


.2524 


. 0154 


. 9846 


59 


2 


. 0228 


. 9772 


.6603 


. 5447 


.3254 


.2527 


. 0171 


. 9828 


58 


3 


. 0251 


. 9749 


.6597 


. 5492 


.3246 


.2530 


. 0189 


. 9811 


57 


4 


. 0274 


. 9726 


.6591 


. 5538 


.3238 


.2532 


. 0206 


. 9793 


56 


5 


.60298 


.39702 


1.6584 


.75584 


1.3230 


1.2535 


.20224 


.79776 


55 


6 


. 0320 


. 9679 


.6578 


. 5629 


.3222 


.2538 


. 0242 


. 9758 


54 


7 


. 0344 


. 9656 


.6572 


. 5675 


.3214 


.2541 


. 0259 


. 9741 


53 


8 


. 0367 


. 9633 


.6565 


. 5721 


.3206 


.2543 


. 0277 


. 9723 


52 


9 


. 0390 


. 9610 


.6559 


. 5767 


.3198 


.2546 


. 0294 


. 9706 


51 


10 


.60413 


.39586 


1.6552 


.75812 


1.3190 


1.2549 


.20312 


.79688 


50 


11 


. 0437 


. 9563 


.6546 


. 5858 


.3182 


.2552 


. 0329 


. 9670 


49 


12 


. 0460 


. 9540 


.6540 


. 5904 


.3174 


.2554 


. 0347 


. 9653 


48 


13 


. 0483 


. 9517 


.6533 


. 5950 


.3166 


.2557 


. 0365 


. 9635 


47 


14 


. 0506 


. 9494 


1 .6527 


. 5996 


.3159 


.2560 


. 0382 


. 9618 


46 


15' 


.60529 


.39471 


1.6521 


.76042 


1.3151 


1.2563 


.20400 


.79600 


45 


16 


. 0552 


. 9447 


.6514 


. 6088 


.3143 


.2565 


. 0417 


. 9582 


44 


17 


. 0576 


. 9424 


.6508 


. 6134 


.3135 


.2568 


. 0435 


. 9565 


43 


18 


. 0599 


. 9401 


.6502 


. 6179 


.3127 


.2571 


. 0453 


. 9547 


42 


19 


. 0622 


. 9378 


.6496 


. 6225 


.3119 


.2574 


. 0470 


. 9530 


41 


20 


.60645 


.39355 


1.6489 


.76271 


1.3111 


1.2577 


.20488 


.79512 


40 


21 


. 0668 


. 9332 


.6483 


. 6317 


.3103 


.2579 


. 0505 


. 9494 


39 


22 


. 0691 


. 9309 


.6477 


. 6364 


.3095 


.2582 


. 0523 


. 9477 


38 


23 


. 0714 


. 9285 


.6470 


. 6410 


.3087 


.2585 


. 0541 


. 9459 


37 


24 


. 0737 


. 9262 


.6464 


. 6456 


.3079 


.2588 


. 0558 


. 9441 


36 


25 


.60761 


.39239 


1.6458 


.76502 


1.3071 


1.2591 


.20576 


.79424 


35 


26 


. 0784 


. 9216 


.6452 


. 6548 


.3064 


.2593 


. 0594 


. 9406 


34 


27 


. 0807 


. 9193 


.6445 


. 6594 


.3056 


.2596 


. 0611 


. 9388 


33 


28 


. 0830 


. 9170 


.6439 


. 6640 


.3048 


.2599 


. 0629 


. 9371 


32 


29 


. 0853 


. 9147 


.6433 


. 6686 


.3040 


.2602 


. 0647 


. 9353 


31 


30 


.60876 


.39124 


1.6427 


.76733 


1.3032 


1.2605 


.20665 


.79335 


30 


31 


. 0899 


. 9101 


.6420 


. 6779 


.3024 


.2607 


. 0682 


. 9318 


29 


32 


. 0922 


. 9078 


.6414 


. 6825 


.3016 


.2610 


. 0700 


. 9300 


28 


33 


. 0945 


. 9055 


.6408 


. 6871 


.3009 


.2613 


. 0718 


. 9282 


27 


34 


. 0963 


. 9031 


.6402 


. 6918 


.3001 


.2616 


. 0735 


. 9264 


26 


35 


.60991 


.39008 


1.6396 


.76964 


1.2993 


1.2619 


.20753 


.79247 


25 


36 


. 1014 


. 8985 


.6389 


. 7010 


.2985 


.2622 


. 0771 


. 9229 


24 


37 


. 1037 


. 8962 


.6383 


. 7057 


.2977 


.2624 


. 0789 


. 9211 


23 


38 


. 1061 


. 8939 


.6377 


. 7103 


.2970 


.2627 


. 0806 


. 9193 


22 


39 


. 1084 


. 8916 


.6371 


. 7149 


.2962 


.2630 


. 0824 


. 9176 


21 


40 


.61107 


.38893 


1.6365 


.77196 


1.2954 


1.2633 


.20842 


.79158 


20 


41 


. 1130 


. 8870 


.6359 


. 7242 


.2946 


.2636 


. 0860 


. 9140 


19 


42 


. 1153 


. 8847 


.6352 


. 7289 


.2938 


.2639 


. 0878 


. 9122 


18 


43 


. 1176 


. 8824 


.6346 


. 7335 


.2931 


.2641 


. 0895 


. 9104 


17 


44 


. 1199 


. 8801 


.6340 


. 7382 


.2923 


.2644 


. 0913 


. 9087 


16 


45 


.61222 


.38778 


1.6334 


.77428 


1.2915 


1.2647 


.20931 


.79069 


15 


46 


. 1245 


. 8755 


.6328 


. 7475 


.2907 


.2650 


. 0949 


. 9051 


14 


47 


. 1268 


. 8732 


.6322 


. 7521 


.2900 


.2653 


. 0967 


. 9033 


13 


48 


. 1290 


. 8709 


.6316 


. 7568 


.2892 


.2656 


. 0984 


. 9015 


12 


49 


. 1314 


. 8686 


.6309 


. 7614 


.2884 


.2659 


. 1002 


. 8998 


11 


50 


.61337 


.38663 


1.6303 


.77661 


1.2876 


1.2661 


.21020 


.78980 


10 


51 


. 1360 


. 8640 


.6297 


. 7708 


.2869 


.2664 


. 1038 


. 8962 


9 


52 


. 1383 


. 8617 


.6291 


. 7754 


.2861 


.2667 


. 1056 


. 8944 


8 


53 


. 1405 


. 8594 


.6285 


. 7801 


.2853 


.2670 


. 1074 


. 8926 


7 


54 


. 1428 


. 8571 


.6279 


. 7848 


.2845 


.2673 


. 1091 


. 8908 


6 


55 


.61451 


.38548 


1.6273 


.77895 


1.2838 


1.2676 


.21109 


.78890 


5 


56 


. 1474 


. 8525 


.6267 


. 7941 


.2830 


.2679 


. 1127 


. 8873 


4 


57 


. 1497 


. 8503 


.6261 


. 7988 


.2822 


.2681 


. 1145 


. 8855 


3 


58 


. 1520 


. 8480 


.6255 


. 8035 


.2815 


.2684 


. 1163 


. 8837 


2 


59 


. 1543 


. 8457 


.6249 


. 8082 


.2807 


.2687 


. 1181 


. 8819 


1 


60 


. 1566 


. 8434 


.6243 


. 8128 


.2799 


.2690 


. 1199 


. 8801 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt) 


Vrs. cos. 


Sine. 


M. 



127° 



52° 



172 



Natural Functions. 



38° 


Natural Trigonometrical Functions. 


141° 


M. 


Sine. 


Vrs. cos. 


Oosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.61566 


.38434 


1.6243 


.78128 


1.2799 


1.2690 


.21199 


.78801 


60 


1 


. 1589 


. 8411 


.6237 


. 8175 


.2792 


.2693 


. 1217 


. 8783 


59 


2 


. 1612 


. 8388 


.6231 


. 8222 


.2784 


.2696 


. 1235 


. 8765 


58 


3 


. 1635 


. 8365 


.6224 


. 8269 


.2776 


.2699 


. 1253 


. 8747 


57 


4 


. 1658 


. 8342 


.6218 


. 8316 


.2769 


.2702 


. 1271 


. 8729 


56 


5 


.61681 


.38319 


1.6212 


.78363 


1.2761 


1.2705 


.21288 


.78711 


55 


6 


. 1703 


. 8296 


.6206 


. 8410 


.2753 


.2707 


. 1306 


. 8693 


54 


7 


. 1726 


. 8273 


.6200 


. 8457 


.2746 


.2710 


. 1324 


. 8675 


53 


8 


. 1749 


. 8251 


.6194 


. 8504 


.2738 


.2713 


. 1342 


. 8657 


52 


9 


. 1772 


. 8228 


.6188 


. 8551 


.2730 


.2716 


. 1360 


. 8640 


51 


10 


.61795 


.38205 


1.6182 


.78598 


1.2723 


1.2719 


.21378 


.78622 


50 


11 


. 1818 


. 8182 


.6176 


. 8645 


.2715 


.2722 


. 1396 


. 8604 


49 


12 


. 1841 


. 8159 


.6170 


. 8692 


.2708 


.2725 


. 1414 


. 8586 


48 


13 


. 1864 


. 8136 


.6164 


. 8739 


.2700 


.2728 


. 1432 


. 8568 


47 


14 


. 1886 


. 8113 


.6159 


. 8786 


.2692 


.2731 


. 1450 


. 8550 


46 


15 


.61909 


.38091 


1.6153 


.78834 


1.2685 


1.2734 


.21468 


.78532 


45 


16 


. 1932 


. 8068 


.6147 


. 8881 


.2677 


.2737 


. 1486 


. 8514 


44 


17 


. 1955 


. 8045 


.6141 


. 8928 


.2670 


.2739 


. 1504 


. 8496 


43 


18 


. 1978 


. 8022 


.6135 


. 8975 


.2662 


.2742 


. 1522 


. 8478 


42 


19 


. 2001 


. 7999 


.6129 


. 9022 


.2655 


.2745 


. 1540 


. 8460 


41 


20 


.62023 


.37976 


1.6123 


.79070 


1.2647 


1.2748 


.21558 


.78441 


40 


21 


. 2046 


. 7954 


.6117 


. 9117 


.2639 


.2751 


. 1576 


. 8423 


39 


22 


. 2069 


. 7931 


.6111 


. 9164 


.2632 


.2754 


. 1594 


. 8405 


38 


23 


. 2092 


. 7908 


.6105 


. 9212 


.2624 


.2757 


. 1612 


. 8387 


37 


24 


. 2115 


. 7885 


.6099 


. 9259 


.2617 


.2760 


. 1631 


. 8369 


36 


25 


.62137 


.37862 


1.6093 


.79306 


1.2609 


1.2763 


.21649 


.78351 


35 


26 


. 2160 


. 7840 


.6087 


. 9354 


.2602 


.2766 


. 1667 


. 8333 


34 


27 


. 2183 


. 7817 


.6081 


. 9401 


.2594 


.2769 


. 1685 


. 8315 


33 


28 


. 2206 


. 7794 


.6077 


. 9449 


.2587 


.2772 


. 1703 


. 8297 


32 


29 


. 2229 


. 7771 


.6070 


. 9496 


.2579 


.2775 


. 1721 


. 8279 


31 


30 


.62251 


.37748 


1.6064 


.79543 


1.2572 


1.2778 


.21739 


.78261 


30 


& 


. 2274 


. 7726 


.6058 


. 9591 


.2564 


.2781 


. 1757 


. 8243 


29 


32 


. 2297 


. 7703 


.6052 


. 9639 


.2557 


.2784 


. 1775 


. 8224 


28 


33 


. 2320 


. 7680 


.6046 


. 9686 


.2549 


.2787 


. 1793 


. 8206 


27 


34 


. 2342 


. 7657 


.6040 


. 9734 


.2542 


.2790 


. 1812 


. 8188 


26 


35 


.62365 


.37635 


1.6034 


.79781 


1.2534 


1.2793 


.21830 


.78170 


25 


36 


. 2388 


. 7612 


.6029 


. 9829 


.2527 


.2795 


. 1848 


. 8152 


24 


37 


. 2411 


. 7589 


.6023 


. 9876 


.2519 


.2798 


. 1866 


. 8134 


23 


38 


. 2433 


. 7566 


.6017 


. 9924 


.2512 


.2801 


. 1884 


. 8116 


22 


39 


. 2456 


. 7544 


.6011 


. 9972 


.2504 


.2804 


. 1902 


. 8097 


21 


40 


.62479 


.37521 


1.6005 


.80020 


1.2497 


1.2807 


.21921 


.78079 


20 


41 


. 2501 


. 7498 


.6000 


. 0067 


.2489 


.2810 


. 1939 


. 8061 


19 


42 


. 2524 


. 7476 


.5994 


. 0115 


.2482 


.2813 


. 1957 


. 8043 


18 


43 


. 2547 


. 7453 


.5988 


. 0163 


.2475 


.2816 


. 1975 


. 8025 


17 


44 


. 2570 


. 7430 


.5982 


. 0211 


.2467 


.2819 


. 1993 


. 8007 


16 


45 


.62592 


.37408 


1.5976 


.80258 


1.2460 


1.2822 


.22011 


.77988 


15 


46 


. 2615 


. 7385 


.5971 


. 0306 


.2452 


.2825 


. 2030 


. 7970 


14 


47 


. 2638 


. 7362 


.5965 


. 0354 


.2445 


.2828 


. 2048 


. 7952 


13 


48 


. 2660 


. 7340 


.5959 


. 0402 


.2437 


.2831 


. 2066 


. 7934 


12 


49 . 2683 


. 7317 


.5953 


. 0450 


.2430 


.2834 


. 2084 


. 7915 


11 


50 .62706 


.37294 


1.5947 


.80498 


1.2123 


1.2837 


.22103 


.77897 


10 


51 


. 2728 


. 7272 


.5942 


. 0546 


.2415 


.2840 


. 2121 


. 7879 


9 


52 


. 2751 


. 7249 


.5936 


. 0594 


.2408 


.2843 


. 2139 


. 7861 


8 


53 


. 2774 


. 7226 


.5930 


. 0642 


.2400 


.2846 


. 2157 


. 7842 


7 


54 


. 2796 


. 7204 


.5924 


. 0690 


.2393 


.2849 


. 2176 


. 7824 


6 


55 


.62819 
. 2841 


.37181 


1.5919 


.80738 


1.2386 


1.2852 


.22194 


.77806 


5 


56 


. 7158 


.5913 


. 0786 


.2378 


.2855 


. 2212 


. 7788 


4 


57 


. 2864 


. 7186 


.5907 


. 0834 


.2371 


.2858 


. 2230 


. 7769 


3 


58 


. 2887 


. 7113 


.5901 


. 0882 


.2364 


.2861 


. 2249 


. 7751 


2 


59 


. 2909 


. 7090 


.6896 


. 09: X) 


.2366 


.2864 


. 2267 


. 7733 


1 


60 


. 2932 


. 7068 


.5890 


. 0978 


.2349 


.2867 


. 2285 


. 7715 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


at 



128° 



51° 



Natural Functions. 



173 



39° 


Natural Trigonom 


etrical Functions. 


140° 


M. 


Sine. 


Yrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.62932 


.37068 


1.5890 


.80978 


1.2349 


1.2867 


.22285 


.77715 


6C 


1 


. 2955 


. 7045 


.5884 


. 1026 


.2342 


.2871 


. 2304 


. 7696 


59 


2 


. 2977 


. 7023 


.5879 


. 1075 


.2334 


.2874 


. 2322 


. 7678 


58 


3 


. 3000 


. 7000 


.5873 


. 1123 


.2327 


.2877 


. 2340 


. 7660 


57 


4 


. 3022 


. 6977 


.5867 


. 1171 


.2320 


.2880 


. 2359 


. 7641 


56 


5 


.63045 


.36955 


1.5862 


.81219 


1.2312 


1.2883 


.22377 


.77623 


55 


6 


. 3067 


.6932 


.5856 


. 1268 


.2305 


.2886 


. 2395 


. 7605 


54 


7 


. 3090 


. 6910 


.5850 


. 1316 


.2297 


.2889 


. 2414 


. 7586 


53 


8 


. 3113 


. 6887 


.5845 


. 1364 


.2290 


.2892 


. 2432 


. 7568 


52 


9 


. 3135 


. 6865 


.5839 


. 1413 


.22S3 


.2895 


. 2450 


. 7549 


51 


10 


.63158 


.36842 


1.5833 


.81461 


1.2276 


1.2898 


.22469 


.77531 


50 


11 


. 3180 


. 6820 


.5828 


. 1509 


.2268 


.2901 


. 2487 


. 7513 


49 


12 


. 3203 


. 6797 


.5822 


. 1558 


.2261 


.2904 


. 2505 


. 7494 


48 


13 


. 3225 


. 6774 


.5816 


. 1606 


.2254 


.2907 


. 2524 


. 7476 


47 


14 


. 3248 


. 6752 


.5811 


. 1655 


.2247 


.2910 


. 2542 


. 7458 


46 


15 


.63270 


.36729 


1.5805 


.81703 


1.2239 


1.2913 


.22561 


.77439 


45 


16 


. 3293 


. 6707 


.5799 


. 1752 


.2232 


.2916 


. 2579 


. 7421 


44 


17 


. 3315 


. 6684 


.5794 


. 1800 


.2225 


.2919 


. 2597 


. 7402 


43 


18 


. 3338 


. 6662 


.5788 


. 1849 


.2218 


.2922 


. 2616 


. 7384 


42 


19 


. 3360 


. 6639 


.5783 


. 1898 


.2210 


.2926 


. 2634 


. 7365 


41 


20 


.63383 


.36617 


1.5777 


.81946 


1.2203 


1.2929 


.22653 


.77347 


40 


21 


. 3405 


. 6594 


.5771 


. 1995 


.2196 


.2932 


. 2671 


. 7329 


39 


22 


. 3428 


. 6572 


.5766 


. 2043 


.2189 


.2935 


. 2690 


. 7310 


38 


23 


. 3450 


. 6549 


.5760 


. 2092 


.2181 


.2938 


. 2708 


. 7292 


37 


24 


. 3473 


. 6527 


.5755 


. 2141 


.2174 


.2941 


. 2727 


. 7273 


36 


25 


.63495 


.36504 


1.5749 


.82190 


1.2167 


1.2944 


.22745 


.77255 


35 


26 


. 3518 


. 6482 


.5743 


. 2238 


.2160 


.2947 


. 2763 


. 7236 


34 


27 


. 3540 


. 6459 


.5738 


. 2287 


.2152 


.2950 


. 2782 


. 7218 


33 


28 


. 3563 


. 6437 


.5732 


. 2336 


.2145 


.2953 


. 2800 


. 7199 


32 


29 


. 3585 


. 6415 


.5727 


. 2385 


.2138 


.2956 


. 2819 


. 7181 


31 


30 


.63608 


.36392 


1.5721 


.82434 


1.2131 


1.2960 


.22837 ' 


.77162 


30 


31 


. 3630 


. 6370 


.5716 


. 2482 


.2124 


.2963 


. 2856 


. 7144 


29 


32 


. 3653 


. 6347 


.5710 


. 2531 


.2117 


.2966 


. 2874 


. 7125 


28 


33 


. 3675 


. 6325 


.5705 


. 2580 


.2109 


.2969 


. 2893 


. 7107 


27 


34 


. 3697 


. 6302 


.5699 


. 2629 


.2102 


.2972 


. 2912 


. 7088 


26 


35 


.63720 


.36280 


1.5694 


.82678 


1.2095 


1.2975 


.22930 


.77070 


25 


36 


. 3742 


. 6258 


.5688 


. 2727 


.2088 


.2978 


. 2949 


. 7051 


24 


37 


. 3765 


. 6235 


.5683 


. 2776 


.2081 


.2981 


. 2967 


. 7033 


23 


38 


. 3787 


. 6213 


.5677 


. 2825 


.2074 


.2985 


. 2986 


. 7014 


22 


39 


. 3810 


. 6190 


.5672 


. 2874 


.2066 


.2988 


. 3004 


. 6996 


21 


40 


.63832 


.36168 


1.5666 


.82923 


1.2059 


1.2991 


.23023 


.76977 


20 


41 


. 3854 


. 6146 


.5661 


. 2972 


.2052 


.2994 


. 3041 


. 6958 


19 


42 


. 3877 


. 6123 


.5655 


. 3022 


.2045 


.2997 


. 3060 


. 6940 


18 


43 


. 3899 


. 6101 


.5650 


. 3071 


.2038 


.3000 


. 3079 


. 6921 


17 


44 


. 3921 


. 6078 


.5644 


. 3120 


.2031 


.3003 


. 3097 


. 6903 


16 


45 


.63944 


.36056 


1.5639 


.83169 


1.2024 


1.3006 


.23116 


.76884 


15 


46 


. 3966 


. 6034 


.5633 


. 3218 


.2016 


.3010 


. 3134 


. 6865 


14 


47 


. 3989 


. 6011 


.5628 


. 3267 


.2009 


.3013 


. 3153 


. 6847 


13 


48 


. 4011 


. 5989 


.5622 


. 3317 


.2002 


.3016 


. 3172 


. G828 


12 


49 


. 4033 


. 5967 


.5617 


. 3366 


.1995 


.3019 


. 3190 


. 6810 


11 


50 


.64056 


.35944 


1.5611 


.83415 


1.1988 


1.3022 


.23209 


.76791 


10 


51 


. 4078 


. 5922 


.5606 


. 3465 


.1981 


.3025 


. 3227 


. 6772 


9 


52 


. 4100 


. 5900 


.5600 


. 3514 


.1974 


.3029 


. 3246 


. 6754 


8 


53 


. 4123 


. 5877 


.5595 


. 3563 


.1967 


.3032 


. 3265 


. 6735 


7 


54 


. 4145 


.. 5855 


.5590 


. 3613 


.1960 


.3035 


. 3283 


. 6716 


6 


55 


.64167 


.35833 


1.5584 


.83662 


1.1953 


1.3038 


.23302 


.76698 


5 


56 


. 4189 


. 5810 


.5579 


. 3712 


.1946 


.3041 


. 3321 


. 6679 


4 


57 


. 4212 


. 5788 


.5573 


. 3761 


.1939 


.3044 


. 3339 


. 6660 


3 


5S 


. 4234 


. 5766 


.5568 


. 3811 


.1932 


.3048 


. 3358 


. 6642 


2 


59 


. 4256 


. 5743 


.5563 


. 3860 


.1924 


.3051 


. 3377 


. 6623 


1 


60 


. 4279 


. 5721 


.5557 


. 3910 


.1917 


.3054 


. 3395 


. 6604 





JK. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



129° 



50° 



174 



Natural Functions. 



40° 


Natural Trigonometrical Functions. 


139° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Yrs. sin. 


Cosine. 


M. 





.64279 


.35721 


1.5557 


.83910 


1.1917 


1.3054 


.23395 


.76604 


60 


1 


. 4301 


. 5699 


.5552 


. 3959 


.1910 


.3057 


. 3414 


. 6586 


59 


2 


. 4323 


. 5677 


.5546 


. 4009 


.1903 


.3060 


. 3433 


. 6567 


58 


3 


. 4345 


. 5654 


.5541 


. 4059 


.1896 


.3064 


. 3452 


. 6548 


57 


4 


. 4368 


. 5632 


- .5536 


. 4108 


.1889 


.3067 


. 3470 


. 6530 


56 


5 


.64390 


.35610 


1.5530 


.84158 


1.1882 


1.3070 


.23489 


.76511 


55 


6 


. 4412 


. 5588 


.5525 


. 4208 


.1875 


.3073 


. 3508 


. 6492 


54 


7 


. 4435 


. 5565 


.5520 


. 4257 


.1868 


.3076 


. 3527 


. 6473 


53 


8 


. 4457 


. 5543 


.5514 


. 4307 


.1861 


.3080 


. 3545 


. 6455 


52 


9 


. 4479 


. 5521 


.5509 


. 4357 


.1854 


.3083 


. 3564 


. 6436 


51 


10 


.64501 


.35499 


1.5503 


.84407 


1.1847 


1.3086 


.23583 


.76417 


50 


11 


. 4523 


. 5476 


.5498 


. 4457 


.1840 


.3089 


. 3602 


. 6398 


49 


12 


. 4546 


. 5454 


.5493 


. 4506 


.1833 


.3092 


. 3620 


. 6380 


48 


13 


. 4568 


. 5432 


.5487 


. 4556 


.1826 


.3096 


. 3639 


. 6361 


47 


14 


. 4590 


. 5410 


.5482 


. 4606 


.1819 


.3099 


. 3658 


. 6342 


46 


15 


.64612 


.35388 


1.5477 


.84656 


1.1812 


1.3102 


.23677 


.76323 


45 


16 


. 4635 


. 5365 


.5471 


. 4706 


.1805 


.3105 


. 3695 


. 6304 


44 


17 


. 4657 


. 5343 


.5466 


. 4756 


.1798 


.3109 


. 3714 


. 6286 


43 


18 


. 4679 


. 5321 


.5461 


. 4806 


.1791 


.3112 


. 3733 


. 6267 


42 


19 


. 4701 


. 5299 


.5456 


. 4856 


.1785 


.3115 


. 3752 


. 6248 


41 


20 


.64723 


.35277 


1.5450 


.84906 


1.1778 


1.3118 


.23771 


.76229 


40 


21 


. 4745 


. 5254 


.5445 


. 4956 


.1771 


.3121 


. 3790 


. 6210 


39 


22 


. 4768 


. 5232 


.5440 


. 5006 


.1764 


.3125 


. 3808 


. 6191 


38 


23 


. 4790 


. 5210 


.5434 


. 5056 


.1757 


.3128 


. 3827 


. 6173 


37 


24 . 4812 


. 5188 


.5429 


. 5107 


.1750 


.3131 


. 3846 


. 6154 


36 


25 ! .64834 


.35166 


1.5424 


.85157 


1.1743 


1.3134 


.23865 


.76135 


35 


26 


. 4856 


. 5144 


.5419 


. 5207 


.1736 


.3138 


. 3884 


. 6116 


34 


27 


. 4878 


. 5121 


.5413 


. 5257 


.1729 


.3141 


. 3903 


. 6097 


33 


28 


. 4900 


. 5099 


.5408 


. 5307 


.1722 


.3144 


. 3922 


. 6078 


32 


29 . 4923 


. 5077 


.5403 


. 5358 


.1715 


.3148 


. 3940 


. 6059 


31 


30 | .64945 


.35055 


1.5398 


.85408 


1.1708 


1.3151 


.23959 


.76041 


30 


31 . 4967 


. 5033 


.5392 


. 5458 


.1702 


.3154 


. 3978 


. 6022 


29 


32 .4989 


. 5011 


.5387 


. 5509 


.1695 


.3157 


. 3997 


. 6003 


28 


33 . 5011 


. 4989 


.5382 


. 5559 


.1688 


.3161 


. 4016 


. 5984 


27 


34 . 5033 


. 4967 


.5377 


. 5609 


.1681 


.3164 


. 4035 


. 5965 


26 


35 .65055 


.34945 


1.5371 


.85660 


1.1674 


1.3167 


.24054 


.75946 


25 


36 . 5077 


. 4922 


.5366 


. 5710 


.1667 


.3170 


. 4073 


. 5927 


24 


37 . 5099 


. 4900 


.5361 


. 5761 


.1660 


.3174 


. 4092 


. 5908 


23 


38 . 5121 


. 4878 


.5356 


. 5811 


.1653 


.3177 


. 4111 


. 5889 


22 


39 . 5144 


. 4856 


.5351 


. 5862 


.1647 


.3180 


. 4130 


. 5870 


21 


40 


.65166 


.34834 


1.5345 


.85912 


1.1640 


1.3184 


.24149 


.75851 


20 


41 


. 5188 


. 4812 


.5340 


. 5963 


.1633 


.3187 


. 4168 


. 5832 


19 


42 


. 5210 


. 4790 


.5335 


. 6013 


.1626 


.3190 


. 4186 


. 5813 


18 


43 


. 5232 


. 4768 


.5330 


. 6064 


.1619 


.3193 


. 4205 


. 5794 


17 


44 


. 5254 


. 4746 


.5325 


. 6115 


.1612 


.3197 


. 4224 


. 5775 


16 


45 


.65276 


.34724 


1.5319 


.86165 


1.1605 


1.3200 


.24243 


.75756 


15 


46 


. 5298 


. 4702 


.5314 


. 6216 


.1599 


.3203 


. 4262 


. 5737 


14 


47 


. 5320 


. 4680 


.5309 


. 6267 


.1592 


.3207 


. 4281 


. 5718 


13 


48 


. 5342 


. 4658 


.5304 


. 6318 


.1585 


.3210 


. 4300 


. 5699 


12 


49 


. 5364 


. 4636 


.5299 


. 6368 


.1578 


.3213 


. 4319 


. 5680 


11 


50 


.65386 


.34614 


1.5294 


.86419 


1.1571 


1.3217 


.24338 


.75661 


10 


51 


. 5408 


. 4592 


.5289 


. 6470 


.1565 


.3220 


. 4357 


. 5642 


9 


52 


. 5430 


. 4570 


.5283 


. 6521 


.1558 


.3223 


. 4376 


. 5623 


8 


53 


. 5452 


. 4548 


.5278 


. 6572 


.1551 


.3227 


. 4396 


. 5604 


7 


54 


. 5474 


. 4526 


.5273 


. 6623 


.1544 


.3230 


. 4415 


. 5585 


6 


55 


.65496 


.34501 


1.5268 


.86674 


1.1537 


1.3233 


.24434 


.75566 


5 


56 


. 5518 


. 4482 


.5263 


. 6725 


.1531 


.3237 


. 4453 


. 5547 


4 


57 


. 5510 


. 4460 


.5258 


. 6775 


.1524 


.3240 


. 4472 


. 5528 


3 


58 


. 6562 


. 4438 


.5253 


. 6826 


.1517 


.3243 


. 4491 


. 5509 


2 


59 


. 5584 


. 4416 


.5248 


. 6878 


.1510 


.3247 


. 4510 


. 5490 


1 


60 


. 5606 


. 4394 


.5242 


. 6929 


.1504 


.3250 


. 4529 


. 5471 





M. 


Cosine. 


Vrs. sin. 


Secant. 


CotaDg. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



130° 



49° 



Natural Functions. 



175 



41° 


Natural Trigonometrical Functions. 


138° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Co tang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.65606 


.34394 


1.5242 


.86929 


1.1504 


1.3250 


.24529 


.75471 


60 


1 


. 5628 


. 4372 


.5237 


. 6980 


.1497 


.3253 


. 4548 


. 5452 


59 


2 


. 5650 


. 4350 


.5232 


. 7031 


.1490 


.3257 


. 4567 


. 5433 


58 


3 


•. 5672 


. 4328 


.5227 


. 7082 


.1483 


.3260 


. 4586 


. 5414 


57 


4 


. 5694 


. 4306 


.5222 


. 7133 


.1477 


.3263 


. 4605 


. 5394 


56 


5 


.65716 


.34284 


1.5217 


.87184 


1.1470 


1.3267 


.24624 


.75375 


55 


6 


. 5737 


. 4262 


.5212 


. 7235 


.1463 


.3270 


. 4644 


. 5356 


54 


7 


. 5759 


. 4240 


.5207 


. 7287 


.1456 


.3274 


. 4663 


. 5337 


53 


8 


. 5781 


. 4219 


.5202 


. 7338 


.1450 


.3277 


. 4682 


. 5318 


52 


9 


. 5803 


. 4197 


.5197 


. 7389 


.1443 


.3280 


. 4701 


. 5299 


51 


10 


.65825 


.34175 


1.5192 


.87441 


1.1436 


1.3284 


.24720 


.75280 


50 


11 


. 5847 


. 4153 


.5187 


. 7492 


.1430 


.3287 


. 4739 


. 5261 


49 


12 


. 5869 


. 4131 


.5182 


. 7543 


.1423 


.3290 


. 4758 


. 5241 


48 


13 


. 5891 


. 4109 


.5177 


. 7595 


.1416 


.3294 


. 4778 


. 5222 


47 


14 


. 5913 


. 4087 


.5171 


. 7646 


.1409 


.3297 


. 4797 


. 5203 


46 


15 


.65934 


.34065 


1.5166 


.87698 


1.1403 


1.3301 


.24816 


.75184 


45 


16 


. 5956 


. 4043 


.5161 


. 7749 


.1396 


.3304 


. 4835 


. 5165 


44 


17 


. 5978 


. 4022 


.5156 


. 7801 


.1389 


.3307 


. 4854 


. 5146 


43 


18 


. 6000 


. 4000 


.5151 


. 7852 


.1383 


.3311 


. 4873 


. 5126 


42 


19 


. 6022 


. 3978 


.5146 


. 7904 


.1376 


.3314 


. 4893 


. 5107 


41 


20 


.66044 


.33956 


1.5141 


.87955 


1.1369 


1.3318 


.24912 


.75088 


40 


21 


. 6066 


. 3934 


.5136 


. 8007 


.1363 


.3321 


. 4931 


. 5069 


39 


22 


. 6087 


. 3912 


.5131 


. 8058 


.1356 


.3324 


. 4950 


. 5049 


38 


23 


. 6109 


. 3891 


.5126 


. 8110 


.1349 


.3328 


. 4970 


. 5030 


37 


24 


. 6131 


. 3869 


.5121 


. 8162 


.1343 


.3331 


. 4989 


. 5011 


36 


25 


.66153 


.33847 


1.5116 


.88213 


1.1336 


1.3335 


.25008 


.74992 


35 


26 


. 6175 


. 3825 


.5111 


. 8265 


.1329 


.3338 


. 5027 


. 4973 


34 


27 


. 6197 


. 3803 


.5106 


. 8317 


.1323 


.3342 


. 5047 


. 4953 


33 


28 


. 6218 


. 3781 


.5101 


. 8369 


.1316 


.3345 


. 5066 


. 4934 


32 


29 


. 6240 


. 3760 


.5096 


. 8421 


.1309 


.3348 


. 5085 


. 4915 


31 


30 


.66262 


.33738 


1.5092 


.88472 


1.1303 


1.3352 


.25104 


.74895 


30 


31 


. 6284 


. 3716 


.5087 


. 8524 


.1296 


.3355 


. 5124 


. 4876 


29 


32 


. 6305 


. 3694 


.5082 


. 8576 


.1290 


.3359 


. 5143 


. 4857 


28 


33 


. 6327 


. 3673 


.5077 


. 8628 


.1283 


.3362 


. 5162 


. 4838 


27 


34 


. 6349 


. 3651 


.5072 


. 8680 


.1276 


.3366 


. 5181 


. 4818 


26 


35 


.66371 


.33629 


1.5067 


.88732 


1.1270 


1.3369 


.25201 


,74799 


25 


36 


. 6393 


. 3607 


.5062 


. 8784 


.1263 


.3372 


. 5220 


. 4780 


24 


37 


. 6414 


. 3586 


.5057 


. 8836 


.1257 


.3376 


. 5239 


. 4760 


23 


38 


. 6436 


. 3564 


.5052 


. 8888 


.1250 


.3379 


. 5259 


. 4741 


22 


39 


. 6458 


. 3542 


.5047 


. 8940 


.1243 


.3383 


. 5278 


. 4722 


21 


40 


.66479 


.33520 


1.5042 


.88992 


1.1237 


1.3386 


.25297 


.74702 


20 


41 


. 6501 


. 3499 


.5037 


. 9044 


.1230 


.3390 


. 5317 


. 4683 


19 


42 


. 6523 


. 3477 


.5032 


. 9097 


.1224 


.3393 


. 5336 


. 4664 


18 


43 


. 6545 


. 3455 


.5027 


. 9149 


.1217 


.3397 


. 5355 


. 4644 


17 


44 


. 6566 


. 3433 


.5022 


. 9201 


.1211 


.3400 


. 5375 


. 4625 


16 


45 


.66588 


.33412 


1.5018 


.89253 


1.1204 


1.3404 


.25394 


.74606 


15 


46 


. 6610 


. 3390 


.5013 


. 9306 


.1197 


.3407 


. 5414 


. 4586 


14 


47 


. 6631 


. 3368 


.5008 


. 9358 


.1191 


.3411 


. 5433 


. 4567 


13 


48 


. 6653 


. 3347 


.5003 


. 9410 


.1184 


.3414 


. 5452 


. 4548 


12 


49 


. 6675 


. 3325 


.4998 


. 9463 


.1178 


.3418 


. 5472 


. 4528 


11 


50 


.66697 


.33303 


1.4993 


.89515 


1.1171 


1.3421 


.25491 


.74509 


10 


51 


. 6718 


. 3282 


.4988 


. 9567 


.1165 


.3425 


. 5510 


. 4489 


9 


52 


. 6740 


. 3260 


.4983 


. 9620 


.1158 


.3428 


. 5530 


. 4470 


8 


53 


. 6762 


. 3238 


.4979 


. 9672 


.1152 


.3432 


. 5549 


. 4450 


7 


54 


. 6783 


. 3217 


.4974 


. 9725 


.1145 


.3435 


. 5569 


. 4431 


6 


55 


.66805 


.33195 


1.4969 


.89777 


1.1139 


1.3439 


.25588 


.74412 


5 


56 


. 6826 


. 3173 


.4964 


. 9830 


.1132 


.3442 


. 5608 


. 4392 


4 


57 


. 6848 


. 3152 


.4959 


. 9882 


.1126 


.3446 


. 5627 


. 4373 


3 


58 


. 6870 


. 3130 


.4954 


. 9935 


.1119 


.3449 


. 5647 


. 4353 


2 


59 


. 6891 


. 3108 


.4949 


. 9988 


.1113 


.3453 


. 5666 


. 4334 


1 


60 


. 6913 


. 3087 


.4945 


.90040 


.1106 


.3456 


. 5685 


. 4314 





M. 


Cosine. 


Yrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



I3i c 



48° 



176 



Natural Functions. 



42 c 


> 


Natural Trigonometrical Functions. 


137° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.66913 


.33087 


1.4945 


.90040 


1.1106 


1.3456 


.25685 


.74314 


60 


1 


. 6935 


. 3065 


.4940 


. 0093 


.1100 


.3460 


. 5705 


. 4295 


59 


2 


. 6956 


. 3044 


.4935 


. 0146 


.1093 


.3463 


. 5724 


. 4275 


58 


3 


. 6978 


. 3022 


.4930 


. 0198 


.1086 


.3467 


. 5744 


. 4256 


57 


4 


. 6999 


. 3000 


.4925 


. 0251 


.1080 


.3470 


. 5763 


. 4236 


56 


5 


.67021 


.32979 


1.4921 


.90304 


1.1074 


1.3474 


.25783 


.74217 


55 


6 


. 7043 


. 2957 


.4916 


. 0357 


.1067 


.3477 


. 5802 


. 4197 


54 


7 


. 7064 


. 2936 


.4911 


. 0410 


.1061 


.3481 


. 5822 


. 4178 


53 


8 


. 7086 


. 2914 


.4906 


. 0463 


.1054 


.3485 


. 5841 


. 4158 


52 


9 


. 7107 


. 2893 


.4901 


. 0515 


.1048 


.3488 


. 5861 


. 4139 


51 


10 


.67129 


.32871 


1.4897 


.90568 


1.1041 


1.3492 


.25880 


.74119 


50 


11 


. 7150 


. 2849 


.4892 


. 0621 


.1035 


.3495 


. 5900 


. 4100 


49 


12 


. 7172 


. 2828 


.4887 


. 0674 


.1028 


.3499 


. 5919 


. 4080 


48 


13 


. 7194 


. 2806 


.4882 


. 0727 


.1022 


.3502 


. 5939 


. 4061 


47 


14 


. 7215 


. 2785 


.4877 


. 0780 


.1015 


.3506 


. 5959 


. 4041 


46 


15 


.67237 


.32763 


1.4873 


.90834 


1.1009 


1.3509 


.25978 


.74022 


45 


16 


. 7258 


. 2742 


.4868 


. 0887 


.1003 


.3513 


. 5998 


. 4002 


44 


17 


. 7280 


. 2720 


.4863 


. 0940 


.0996 


.3517 


. 6017 


. 3983 


43 


18 


. 7301 


. 2699 


.4858 


. 0993 


.0990 


.3520 


. 6037 


. 3963 


42 


19 


. 7323 


. 2677 


.4854 


. 1046 


.0983 


.3524 


. 6056 


. 3943 


41 


20 


.67344 


.32656 


1.4849 


.91099 


1.0977 


1.3527 


.26076 


.73924 


40 


21 


. 7366 


. 2634 


.4844 


. 1153 


.0971 


.3531 


. 6096 


. 3904 


39 


22 


. 7387 


. 2613 


.4839 


. 1206 


.0964 


.3534 


. 6115 


. 3885 


38 


23 


. 7409 


. 2591 


.4835 


. 1259 


.0958 


.3538 


. 6135 


. 3865 


37 


24 


. 7430 


. 2570 


.4830 


. 1312 


.0951 


.3542 


. 6154 


. 3845 


36 


25 


.67452 


.32548 


1.4825 


.91366 


1.0945 


1.3545 


.26174 


.73826 


35 


26 


. 7473 


. 2527 


.4821 


. 1419 


.0939 


.3549 


. 6194 


. 3806 


34 


27 


. 7495 


. 2505 


.4816 


. 1473 


.0932 


.3552 


. 6213 


. 3787 


33 


28 


. 7516 


. 2484 


.4811 


. 1526 


.0926 


.3556 


. 6233 


. 3767 


32 


29 


. 7537 


. 2462 


.4806 


. 1580 


.0919 


.3560 


. 6253 


. 3747 


31 


30 


.67559 


.32441 


1.4802 


.91633 


1.0913 


1.3563 


.26272 


.73728 


30 


31 


. 7580 


. 2419 


.4797 


. 1687 


.0907 


.3567 


. 6292 


. 3708 


29 


32 


. 7602 


. 2398 


.4792 


. 1740 


.0900 


.3571 


. 6311 


. 3688 


28 


33 


. 7623 


. 2377 


.4788 


. 1794 


.0894 


.3574 


. 6331 


. 3669 


'27 


34 


. 7645 


. 2355 


.4783 


. 1847 


.0888 


.3578 


. 6351 


. 3649 


26 


35 


.67666 


.32334 


1.4778 


.91901 


1.0881 


1.3581 


.26371 


.73629 


25 


36 


. 7688 


. 2312 


.4774 


. 1955 


.0875 


.3585 


. 6390 


. 3610 


24 


37 


. 7709 


. 2291 


.4769 


. 2008 


.0868 


.3589 


. 6410 


. 3590 


23 


38 


. 7730 


. 2269 


.4764 


. 2062 


.0862 


.3592 


. 6430 


. 3570 


22 


39 


. 7752 


. 2248 


.4760 


. 2116 


.0856 


.3596 


. 6449 


. 3551 


21 


40 


.67773 


.32227 


1.4755 


.92170 


1.0849 


1.3600 


.26469 


.73531 


20 


41 


. 7794 


. 2205 


.4750 


. 2223 


.0843 


.3603 


. 6489 


. 3511 


19 


42 


. 7816 


. 2184 


.4746 


. 2277 


.0837 


.3607 


. 6508 


. 3491 


18 


43 


. 7837 


. 2163 


.4741 


. 2331 


.0830 


.3611 


. 6528 


. 3472 


17 


44 


. 7859 


. 2141 


.4736 


. 2385 


.0824 


.3614 


. 6548 


. 3452 


16 


45 


.67880 


.32120 


1.4732 


.92439 


1.0818 


1.3618 


.26568 


.73432 


15 


46 


. 7901 


. 2098 


.4727 


. 2493 


.0812 


.3622 


. 6587 


. 3412 


14 


47 


. 7923 


. 2077 


.4723 


. 2547 


.0805 


.3625 


. 6607 


. 3393 


13 


48 


. 7944 


. 2056 


.4718 


. 2601 


.0799 


.3629 


. 6627 


. 3373 


12 


49 


. 7965 


. 2034 


.4713 


. 2655 


.0793 


.3633 


. 6647 


. 3353 


11 


50 


.67987 


.32013 


1.4709 


.92709 


1.07S6 


1.3636 


.26666 


.73333 


10 


51 


. 8008 


. 1992 


.4704 


. 2763 


.0780 


.3640 


. 6686 


. 3314 


9 


52 


. 8029 


. 1970 


.4699 


. 2817 


.0774 


.3644 


. 6706 


. 3294 


8 


53 


. 8051 


. 1949 


.4695 


. 2871 


.0767 


.3647 


. 6726 


. 3274 


7 


54 


. 8072 


. 1928 


.4690 


. 2926 


.0761 


.3651 


. 6746 


. 3254 


6 


55 


.68093 


.31907 


1.4686 


.92980 


1.0755 


1.3655 


.26765 


.73234 


5 


56 


. 8115 


. 1885 


1 .4681 


. 3034 


.0749 


.3658 


. 6785 


. 3215 


4 


57 


. 8136 


. 1864 


1 .4676 


. 3088 


.0742 


.3662 


. 6805 


. 3195 


3 


58 


. 8157 


. 1843 


.4672 


. 3143 


.0736 


.3666 


. 6825 


. 3175 


2 


59 


. sits 


. 1821 


.4667 


3197 


.0730 


.3669 


. 6845 


. 3155 


1 


60 


. 8200 


. 1800 


.4663 


. 3251 


.0724 


.3673 


. 6865 


. 3135 





M. 


Cosine. 


Vrs. sin. 


i Secant. 


Cotang. 


, Tang. 


Cosec'nt 


iVrs. cos. 


Sine. 


M. 



132° 



47° 



Natukal Functions. 



177 



43° 


Natural Trigonometrical Functions. 


136° 


M. 


JSine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.68200 


.31800 


1.4663 


.93251 


1.0724 


1.3673 


.26865 


.73135 


60 


1 


. 8221 


. 1779 


.4658 


. 3306 


.0717 


.3677 


. 6884 


. 3115 


59 


2 


. 8242 


. 1758 


.4654 


. 3360 


.0711 


.3681 


. 6904 


. 3096 


58 


3 


. 8264 


. 1736 


.4649 


. 3415 


.0705 


.3684 


. 6924 


. 3076 


57 


4 


. 8285 


. 1715 


.4644 


. 3469 


.0699 


.3688 


. 6944 


. 3056 


56 


5 


.68306 


.31694 


1.4640 


.93524 


1.0692 


1.3692 


.26964 


.73036 


55 


6 


. 8327 


. 1673 


.4635 


. 3578 


.0686 


.3695 


. 6984 


. 3016 


54 


7 


. 8349 


. 1651 


.4631 


. 3633 


.0680 


.3699 


. 7004 


. 2996 


53 


8 


. 8370 


. 1630 


.4626 


. 3687 


.0674 


.3703 


. 7023 


. 2976 


52 


9 


. 8391 


. 1609 


.4622 


. 3742 


.0667 


.3707 


. 7043 


. 2956 


51 


10 


=68412 


.31588 


1.4617 


.93797 


1.0661 


1.3710 


.27063 


.72937 


50 


11 


. 8433 


. 1566 


.4613 


. 3851 


.0655 


.3714 


. 7083 


. 2917 


49 


12 


. 8455 


. 1545 


.4608 


. 3906 


.0649 


.3718 


. 7103 


. 2897 


48 


13 


. 8476 


. 1524 


.4604 


. 3961 


.0643 


.3722 


. 7123 


. 2877 


47 


14 


. 8497 


. 1503 


.4599 


. 4016 


.0636 


.3725 


. 7143 


. 2857 


46 


15 


.68518 


.31482 


1.4595 


.94071 


1.0630 


1.3729 


.27163 


.72837 


45 


16 


. 8539 


. 1460 


.4590 


. 4125 


.0624 


.3733 


. 7183 


. 2817 


44 


17 


. 8561 


. 1439 


.4586 


. 4180 


.0618 


.3737 


. 7203 


. 2797 


43 


18 


. 8582 


. 1418 


.4581 


. 4235 


.0612 


.3740 


. 7223 


. 2777 


42 


19 


. 8603 


. 1397 


.4577 


. 4290 


.0605 


.3744 


. 7243 


. 2757 


41 


20 


.68624 


.31376 


1.4572 


.94345 


1.0599 


1.3748 


.27263 


.72737 


40 


21 


. 8645 


. 1355 


.4568 


. 4400 


.0593 


.3752 


. 7283 


. 2717 


39 


22 


. 8666 


. 1333 


.4563 


. 4455 


.0587 


.3756 


. 7302 


. 2697 


38 


23 


. 8688 


. 1312 


.4559 


. 4510 


.0581 


.3759 


. 7322 


. 2677 


37 


24 


. 8709 


. 1291 


.4554 


. 4565 


.0575 


.3763 


. 7342 


. 2657 


36 


25 


.68730 


.31270 


1.4550 


.94620 


1.0568 


1.3767 


.27362 


.72637 


35 


26 


. 8751 


. 1249 


.4545 


. 4675 


.0562 


.3771 


. 7382 


. 2617 


34 


27 


. 8772 


. 1228 


.4541 


. 4731 


.0556 


.3774 


. 7402 


. 2597 


33 


28 


. 8793 


. 1207 


.4536 


. 4786 


.0550 


.3778 


. 7422 


. 2577 


32 


29 


. 8814 


. 1186 


.4532 


. 4841 


.0544 


.3782 


. 7442 


. 2557 


31 


30 


.68835 


.31164 


1.4527 


.94896 


1.0538 


1.3786 


.27462 


.72537 


30 


31 


. 8856 


. 1143 


.4523 


. 4952 


.0532 


.3790 


. 7482 


. 2517 


29 


32 


. 8878 


. 1122 


.4518 


. 5007 


.0525 


.3794 


. 7503 


. 2497 


28 


33 


. 8899 


. 1101 


.4514 


. 5062 


.0519 


.3797 


. 7523 


. 2477 


27 


34 


. 8920 


. 1080 


.4510 


. 5118 


.0513 


.3801 


. 7543 


. 2457 


26 


35 


.68941 


.31059 


1.4505 


.95173 


1.0507 


1.3805 


.27563 


.72437 


25 


36 


. 8962 


. 1038 


.4501 


. 5229 


.0501 


.3809 


. 7583 


. 2417 


24 


37 


. 8983 


. 1017 


.4496 


. 5284 


.0495 


.3813 


. 7603 


. 2397 


23 


38 


. 9004 


. 0996 


.4492 


. 5340 


.0489 


.3816 


. 7623 


. 2377 


22 


39 


. 9025 


. 0975 


.4487 


. 5395 


.0483 


.3820 


. 7643 


. 2357 


21 


40 


.69046 


.30954 


1.4483 


.95451 


1.0476 


1.3824 


.27663 


.72337 


20 


41 


. 9067 


. 0933 


.4479 


. 5506 


.0470 


.3828 


. 7683 


. 2317 


19 


42 


. 9088 


. 0912 


.4474 


. 5562 


.0464 


.3832 


. 7703 


. 2297 


18 


43 


. 9109 


. 0891 


.4470 


. 5618 


.0458 


.3836 


. 7723 


. 2277 


17 


44 


. 9130 


. 0870 


.4465 


. 5673 


.0452 


.3839 


. 7743 


. 2256 


16 


45 


.69151 


.30849 


1.4461 


.95729 


1.0446 


1.3843 


.27764 


.72236 


15 


46 


. 9172 


. 0828 


.4457 


. 5785 


.0440 


.3847 


. 7784 


. 2216 


14 


47 


. 9193 


. 0807 


.4452 


. 5841 


.0434 


.3851 


. 7804 


. 2196 


13 


48 


. 9214 


. 0786 


.4448 


. 5896 


.0428 


.3855 


. 7824 


. 2176 


12 


49 


. 9235 


. 0765 


.4443 


. 5952 


.0422 


.3859 


. 7844 


. 2156 


11 


50 


.69256 


.30744 


1.4439 


.96008 


1.0416 


1.3863 


.27864 


.72136 


10 


51 


. 9277 


. 0723 


.4435 


. 6064 


.0410 


.3867 


. 7884 


. 2115 


9 


52 


. 9298 


. 0702 


.4430 


. 6120 


.0404 


.3870 


. 7904 


. 2095 


8 


53 


. 9319 


. 0681 


.4426 


. 6176 


.0397 


.3874 


. 7925 


. 2075 


7 


54 


. 9340 


. 0660 


.4422 


. 6232 


.0391 


.3878 


. 7945 


. 2055 


6 


55 


.69361 


.30639 


1.4417 


.96288 


1.0385 


1.3882 


.27965 


.72035 


5 


56 


. 9382 


. 0618 


.4413 


. 6344 


.0379 


.3886 


. 7985 


. 2015 


4 


57 


. 9403 


0597 


.4408 


. 6400 


.0373 


.3890 


. 8005 


. 1994 


3 


58 


. 9424 


. 05^6 


.4404 


. 6456 


.0367 


.3894 


. 8026 


. 1974 


2 


59 


. 9445 


. 0555 


.4400 


. 6513 


.0361 


.3898 


. 8046 


. 1954 


1 


60 


. 9466 


. 0534 


.4395 


. 6569 


.0355 


.3902 


. 8066 


. 1934 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec 'nt 


Vrs. cos. 


Sine. 


M. 



133° 



46° 



12 



178 



Natural Functions. 



44° 


Natural Trigonometrical Functions. 


135° 


M. 


Sine. 


Vrs. cos. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Vrs. sin. 


Cosine. 


M. 





.69466 


.30534 


1.4395 


.96569 


1.0355 


1.3902 


.28066 


.71934 


60 


1 


. 9487 


. 0513 


.4391 


. 6625 


.0349 


.3905 


. 8086 


. 1914 


59 


2 


. 9508 


. 0492 


.4387 


. 6681 


.0343 


.3909 


. 8106 


. 1893 


58 


3 


. 9528 


. 0471 


.4382 


. 6738 


.0337 


.3913 


. 8127 


. 1873 


57 


4 


. 9549 


. 0450 


.4378 


. 6794 


.0331 


.3917 


. 8147 


. 1853 


56 


5 


.69570 


.30430 


1.4374 


.96850 


1.0325 


1.3921 


.28167 


.71833 


55 


6 


. 9591 


. 0409 


.4370 


. 6907 


.0319 


.3925 


. 8187 


. 1813 


54 


7 


. 9612 


. 0388 


.4365 


. 6963 


.0313 


.3929 


. 8208 


. 1792 


53 


8 


. 9633 


. 0367 


.4361 


. 7020 


.0307 


.3933 


. 8228 


. 1772 


52 


9 


. 9654 


. 0346 


.4357 


. 7076 


.0301 


.3937 


. 8248 


. 1752 


51 


10 


.69675 


.30325 


1.4352 


.97133 


1.0295 


1.3941 


.28268 


.71732 


50 


11 


. 9696 


. 0304 


.4348 


. 7189 


.0289 


.3945 


. 8289 


. 1711 


49 


12 


. 9716 


. 0283 


.4344 


. 7246 


.0283 


.3949 


. 8309 


. 1691 


48 


13 


. 9737 


. 0263 


.4339 


. 7302 


.0277 


.3953 


. 8329 


. 1671 


47 


14 


. 9758 


. 0242 


.4335 


. 7359 


.0271 


.3957 


. 8349 


. 1650 


46 


15 


.69779 


.30221 


1.4331 


.97416 


1.0265 


1.3960 


.28370 


.71630 


45 


16 


. 9800 


. 0200 


.4327 


. 7472 


.0259 


.3964 


. 8390 


. 1610 


44 


17 


. 9821 


. 0179 


.4322 


. 7529 


.0253 


.3968 


. 8410 


. 1589 


43 


18 


. 9841 


. 0158 


.4318 


. 7586 


.0247 


.3972 


. 8431 


. 1569 


42 


19 


. 9862 


. 0138 


.4314 


. 7643 


.0241 


.3976 


. 8451 


. 1549 


41 


20 


.69883 


.30117 


1.4310 


.97699 


1.0235 


1.3980 


.28471 


.71529 


40 


21 


. 9904 


. 0096 


.4305 


. 7756 


.0229 


.3984 


. 8492 


. 1508 


39 


22 


. 9925 


. 0075 


.4301 


. 7813 


.0223 


.3988 


. 8512 


. 1488 


38 


23 


. 9945 


. 0054 


.4297 


. 7870 


.0218 


.3992 


. 8532 


. 1468 


37 


24 


. 9966 


. 0034 


.4292 


. 7927 


.0212 


.3996 


. 8553 


. 1447 


36 


25 


.69987 


.30013 


1.4288 


.97984 


1.0206 


1.4000 


.28573 


.71427 


35 


26 


.70008 


.29992 


.4284 


. 8041 


.0200 


.4004 


. 8593 


. 1406 


34 


27 


. 0029 


. 9971 


.4280 


. 8098 


.0194 


.4008 


. 8614 


. 1386 


33 


28 


. 0049 


. 9950 


.4276 


. 8155 


.0188 


.4012 


. 8634 


. 1366 


32 


29 


. 0070 


. 9930 


.4271 


. 8212 


.0182 


.4016 


. 8654 


. 1345 


31 


30 


.70091 


.29909 


1.4267 


.98270 


1.0176 


1.4020 


.28675 


.71325 


30 


81 


. 0112 


. 9888 


.4263 


. 8327 


.0170 


.4024 


. 8695 


. 1305 


29 


32 


. 0132 


. 9867 


.4259 


. 8384 


.0164 


.4028 


. 8716 


. 1284 


28 


33 


. 0153 


. 9847 


.4254 


. 8441 


.0158 


.4032 


. 8736 


. 1264 


27 


34 


. 0174 


. 9826 


.4250 


. 8499 


.0152 


.4036 


. 8756 


. 1243 


26 


35 


.70194 


.29805 


1.4246 


.98556 


1.0146 


1.4040 


.28777 


.71223 


25 


36 


. 0215 


. 9785 


.4242 


. 8613 


.0141 


.4044 


. 8797 


. 1203 


24 


37 


. 0236 


. 9764 


.4238 


. 8671 


.0135 


.4048 


. 8818 


. 1182 


23 


38 


. 0257 


. 9743 


.4233 


. 8728 


.0129 


.4052 


. 8838 


. 1162 


22 


39 


. 0277 


. 9722 


.4229 


. 8786 


.0123 


.4056 


. 8859 


. 1141 


21 


40 


.70298 


.29702 


1.4225 


.98843 


1.0117 


1.4060 


.28879 


.71121 


20 


41 


. 0319 


. 9681 


.4221 


. 8901 


.0111 


.4065 


. 8899 


. 1100 


19 


42 


. 0339 


. 9660 


.4217 


. 8958 


.0105 


.4069 


. 8920 


. 1080 


18 


43 


. 0360 


. 9640 


.4212 


. 9016 


.0099 


.4073 


. 8940 


. 1059 


17 


44 


. 0381 


. 9619 


.4208 


. 9073 


.0093 


.4077 


. 8961 


. 1039 


16 


45 


.70101 


.29598 


1.4204 


.99131 


1.0088 


1.4081 


.28981 


.71018 


15 


46 


. 0122 


. 9578 


.4200 


. 9189 


.0082 


.4085 


. 9002 


. 0998 


14 


47 


. 0113 


. 9557 


.4196 


. 9246 


.0076 


.4089 


. 9022 


. 0977 


13 


48 


. 0463 


. 9536 


.4192 


. 9304 


.0070 


.4093 


. 9043 


. 0957 


12 


49 


. 0484 


. 9516 


.4188 


. 9362 


.0064 


.4097 


. 9063 


. 0936 


11 


50 


.70505 


.29495 


1.4183 


.99420 


1.0058 


1.4101 


.29084 


.70916 


10 


51 


. 0525 


. 9475 


.117'.) 


. 9478 


.0052 


.4105 


. 9104 


. 0895 


9 


52 


. 0546 


. 9454 


.4175 


. 9536 


.0047 


.4109 


. 9125 


. 0875 


8 


53 


. 0566 


. 9433 


.4171 


. 9593 


.0041 


.4113 


. 9145 


. 0854 


7 


54 


. 0587 


. 9413 


.4167 


. 9651 


.0035 


.4117 


. 9166 


. 0834 


6 


55 


.70608 


.29392 


1.4163 


.99709 


1.0029 


1.4122 


.29186 


.70813 


5 


56 


. 0628 


. 9372 


.4159 


. 9767 


.0023 


.4126 


. 9207 


. 0793 


4 


57 


. 0649 


. 9351 


.4154 


. 9826 


.0017 


.4130 


. 9228 


. 0772 


3 


56 


. 0669 


. 9330 


,1150 


. 9884 


.0012 


.4134 


. 9248 


. 0752 


2 


59 


. 0690 


. 9310 


.4146 


. 9942 


.0006 


.4138 


. 9269 


. 0731 


1 


60 


. 0711 


. 9289 


.4142 


1.0000 


.0000 


.4112 


. 9289 


. 0711 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Co tang. 


Tang. 


Cosec'nt 


Vrs. cos. 


Sine. 


M. 



134° 



45° 



Logarithmic Angular Functions. 



179 



0° 



Logarithms. 



179° 



M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





Inf. Neg. 


Infinite. 


Inf. Neg. 


Infinite. 


10.00000 


10.00000 


60 


1 


6.46373- 


13.53627 


6.46373 


13.53627 


00000 


00000 


59 


2 


76476 


23524 


76476 


23524 


00000 


00000 


58 


3 


94085 


05915 


94085 


05915 


00000 


00000 


57 


4 


7.06579 


12.93421 


7.06579 


12.93421 


00000 


00000 


56 


5 


7.16270 


12.83730 


7.16270 


12.83730 


10.00000 


10.00000 


55 


6 


24188 


75812 


24188 . 


75812 


00000 


OuOOO 


54 


7 


30882 


69118 


30882 


69118 


00000 


00000 


53 


8 


36682 


63318 


36682 


63318 


00000 


00000 


52 


9 


41797 


58203 


41797 


58203 


00000 


00000 


51 


10 


7.46373 


12.53627 


7.46373 


12.53627 


10.00000 


10.00000 


50 


11 


50512 


49488 


50512 


49488 


00000 


00000 


49 


12 


54291 


45709 


54291 


45709 


00000 


00000 


48 


13 


57767 


42233 


57767 


42233 


00000 


00000 


47 


14 


60985 


39015 


60986 


39014 


00000 


00000 


46 


15 


7.63982 


12.36018 


7.63982 


12.36018 


10.00000 


10.00000 


45 


16 


66784 


33216 


66785 


33215 


00000 


00000 


44 


17 


69417 


30583 


69418 


30582 


00001 


9.99999 


43 


18 


71900 


28100 


71900 


28100 


00001 


99999 


42 


19 


74248 


25752 


74248 


25752 


00001 


99999 


41 


20 


7.76475 


12.23525 


7.76476 


12.23524 


10.00001 


9.99999 


40 


21 


78594 


21406 


78595 


21405 


00001 


99999 


39 


22 


80615 


19385 


80615 


19385 


00001 


99999 


38 


23 


82545 


17455 


82546 


17454 


00001 


99999 


37 


24 


84393 


15607 


84394 


15606 


00001 


99999 


36 


25 


7.86166 


12.13834 


7.86167 


12.13833 


10.00001 


9.99999 


35 


26 


87870 


12130 


87871 


12129 


00001 


99999 


34 


27 


89509 


10491 


89510 


10490 


00001 


99999 


33 


28 


91088 


08912 


91089 


08911 


00001 


99999 


32 


29 


92612 


07388 


92613 


07387 


00002 


99998 


31 


30 


7.94084 
95508 


12.05916 
04492 


7.94086 


12.05914 


10.00002 
00002 


9.99998 
99998 


30 
29 


31 


95510 


04490 


32 


96887 


03113 


96889 


03111 


00002 


99998 


28 


33 


98223 


01777 


98225 


01775 


00002 


99998 


27 


34 


99520 


00480 


99522 


00478 


00002 


99998 


26 


35 


8.00779 


11.99221 


8.00781 


11.99219 


10.00002 


9.99998 


25 


36 


02002 


97998 


02004 


97996 


00002 


99998 


24 


37 


03192 


96808 


03194 


96806 


00003 


99997 


23 


38 


04350 


95650 


04353 


95647 


00003 


99997 


22 


39 


05478 


94522 


05481 


94519 


00003 


99997 


21 


40 


8.06578 


11.93422 


8.06581 


11.93419 


10.00003 


9.99997 


20 


41 


07650 


92350 


07653 


92347 


00003 


99997 


19 


42 


08696 


91304 


08700 


91300 


00003 


99997 


18 


43 


09718 


90282 


09722 


90278 


00003 


99997 


17 


44 


10717 


89283 


10720 


89280 


00004 


99996 


16 


45 


8.11693 


11.88307 


8.11696 


11.88304 


10.00004 


9.99996 


15 


46 


12647 


87353 


12651 


87349 


00004 


99996 


14 


47 


13581 


86419 


13585 


86415 


00004 


99996 


13 


48 


14495 


85505 


14500 


85500 


00004 


99996 


12 


49 


15391 


84609 


15395 


84605 


00004 


99996 


11 


50 


8.16268 


11.83732 


8.16273 


11.83727 


10.00005 


9.99995 


10 


51 


17128 


82872 


17133 


82867 


00005 


99995 


9 


52 


17971 


82029 


17976 


82024 


00005 


99995 


8 


53 


18798 


81202 


18804 


81196 


00005 


99995 


7 


54 


19610 


80390 


19616 


80384 


00005 


99995 


6 


55 


8.20407 


11.79593 


8.20413 


11.79587 


10.00006 


9.99994 


5 


56 


21189 


78811 


21195 


78805 


00006 


99994 


4 


57 


21958 


78042 


21964 


78036 


00006 


99994 


3 


58 


22713 


77287 


22720 


77280 


00006 


99994 


2 


59 


23456 


76544 


23462 


76538 


00006 


99994 


1 


60 


24186 


75814 


24192 


75808 


00007 


99993 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



90° 



89° 



180 



Logarithmic Angular Functions. 



1° 






Logarithms. 




178° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


| Secant. 


Cosine. 


M. 





8.24186 


11.75814 


8.24192 


11.75808 


10.00007 


9.99993 


60 


1 


24903 


75097 


24910 


75090 


00007 


99993 


59 


2 


25609 


74391 


25616 


74384 


00007 


99993 


58 


3 


26304 


73696 


26312 


73688 


00007 


99993 


57 


4 


26988 


73012 


26996 


73004 


00008 


99992 


56 


5 


8.27661 


11.72339 


8.27669 


11.72331 


10.00008 


9.99992 


55 


6 


28324 


71676 


28332 


71668 


00008 


99992 


54 


7 


28977 


71023 


28986 


71014 


00008 


99992 


53 


8 


29621 


70379 


29629 


70371 


00008 


99992 


52 


9 


30255 


69745 


30263 


69737 


00009 


99991 


51 


10 


8.30879 


11.69121 


8.30888 


11.69112 


10.00009 


9.99991 


50 


11 


31495 


68505 


31505 


68495 


00009 


99991 


49 


12 


32103 


67897 


32112 


67888 


00010 


99990 


48 


13 


32702 


67298 


32711 


67289 


00010 


99990 


47 


14 


33292 


66708 


33302 


66698 


00010 


99990 


46 


15 


8.33875 


11.66125 


8.33886 


11.66114 


10.00010 


9.99990 


45 


16 


34450 


65550 


34461 


65539 


00011 


99989 


44 


17 


35018 


64982 


35029 


64971 


00011 


99989 


43 


18 


35578 


64422 


35590 


64410 


00011 


99989 


42 


19 


36131 


63869 


36143 


63857 


00011 


99989 


41 


20 


8.36678 


11.63322 


8.36689 


11.63311 


10.00012 


9.99988 


40 


21 


37217 


62783 


37229 


62771 


00012 


99988 


39 


22 


37750 


62250 


37762 


62238 


00012 


99988 


38 


23 


38276 


61724 


38289 


61711 


00013 


99987 


37 


24 


38796 


61204 


38809 


61191 


00013 


99987 


36 


25 


8.39310 


11.60690 


8.39323 


11.60677 


10.00013 


9.99987 


35 


26 


39818 


60182 


39832 


60168 


00014 


99986 


34 


27 


40320 


59680 


40334 


59666 


00014 


99986 


33 


28 


40816 


59184 


40830 


59170 


00014 


99986 


32 


29 


41307 


58693 


41321 


58679 


00015 


99985 


31 


30 


8.41792 


11.58208 


8.41807 


11.58193 


10.00015 


9.99985 


30 


31 


42272 


57728 


42287 


57713 


00015 


99985 


29 


32 


42746 


57254 


42762 


57238 


00016 


99984 


28 


33 


43216 


56784 


43232 


56768 


00016 


99984 


27 


34 


43680 


56320 


43696 


56304 


00016 


99984 


26 


35 


8.44139 


11.55861 


8.44156 


11.55844 


10.00017 


9.99983 


25 


36 


44594 


55406 


44611 


55389 


00017 


99983 


24 


37 


45044 


54956 


45061 


54939 


00017 


99983 


23 


38 


45489 


54511 


45507 


54493 


00018 


99982 


22 


39 


45930 


54070 


45948 


54052 


00018 


99982 


21 


40 


8.46366 


11.53634 


8.46385 


11.53615 


10.00018 


9.99982 


20 


41 


46799 


53201 


46817 


53183 


00019 


99981 


19 


42 


47226 


52774 


47215 


52755 


00019 


99981 


18 


43 


47650 


52350 


47669 


52331 


00019 


99981 


17 


44 


48069 


51931 


48089 


51911 


00020 


99980 


16 


45 


8.48485 


11.51515 


8.48505 


11.51495 


10.00020 


9.99980 


15 


46 


48896 


51104 


48917 


51083 


00021 


99979 


14 


47 


49304 


50696 


49325 


50675 


00021 


99979 


13 


48 


49708 


50292 


49729 


50271 


00021 


99979 


12 


49 


50108 


49892 


50130 


49870 


00022 


99978 


11 


50 


8.50504 


11.49496 


8.50527 


11.49473 


10.00022 


9.99978 


10 


51 


50897 


49103 


50920 


49080 


00023 


99977 


9 


52 


51287 


48713 


61310 


48690 


00023 


99977 


8 


53 


51673 


48327 


51696 


48304 


00023 


99977 


7 


54 


52055 


47945 


52079 


47921 


00024 


99976 


6 


56 


8.52434 


11.47566 


8.52459 


11.47541 


10.00024 


9.99976 


5 


56 


52810 


47190 


52s:i5 


47165 


00025 


99975 


4 


57 


53183 


46817 


53208 


46792 


00025 


99975 


3 


58 


53552 


46448 


58578 


46422 


00026 


99974 


2 


59 


53919 


16081 


53945 


46055 


00026 


99974 


1 


60 


54282 


45718 


54308 


45692 


00026 


99974 





M. 


Cosine. 


;mt. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



88° 



Logarithmic Angular Functions. 



181 



2° 






Logarithms. 




177° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





8.54282 


11.45718 


8.54308 


11.45692 


10.00026 


9.99974 


60 


1 


54642 


45358 


54669 


45331 


00027 


99973 


59 


2 


54999 


45001 


55027 


44973 


00027 


99973 


58 


3 


55354 


44646 


55382 


44618 


00028 


99972 


57 


4 


55705 


44295 


55734 


44266 


00028 


99972 


56 


5 


8.56054 


11.43946 


8.56083 


11.43917 


10.00029 


9.99971 


55 


6 


56400 


43600 


56429 


43571 


00029 


99971 


54 


7 


56743 


43257 


56773 


43227 


00030 


99970 


53 


8 


57084 


42916 


57114 


42886 


00030 


99970 


52 


9 


57421 


42579 


57452 


42548 


00031 


99969 


51 


10 


8.57757 


11.42243 


8.57788 


11.42212 


10.00031 


9.99969 


50 


11 


58089 


41911 


58121 


41879 


00032 


99968 


49 


12 


58419 


41581 


58451 


41549 


00032 


99968 


48 


13 


58747 


41253 


58779 


41221 


00033 


99967 


47 


14 


59072 


40928 


59105 


40895 


00033 


99967 


46 


15 


8.59395 


11.40605 


8.59428 


11.40572 


10.00033 


9.99967 


45 


16 


59715 


40285 


59749 


40251 


00034 


99966 


44 


17 


60033 


39967 


60068 


39932 


00034 


99966 


43 


18 


60349 


39651 


60384 


39616 


00035 


99965 


42 


19 


60662 


39338 


60698 


39302 


00036 


99964 


41 


20 


8.60973 


11.39027 


8.61009 


11.38991 


10.00036 


9.99964 


40 


21 


61282 


38718 


61319 


38681 


00037 


99963 


39 


22 


61589 


38411 


61626 


38374 


00037 


99963 


38 


23 


61894 


38106 


61931 


38069 


00038 


99962 


37 


24 


62196 


37804 


62234 


37766 


00038 


99962 


36 


25 


8.62497 


11.37503 


8.62535 


11.37465 


10.00039 


9.99961 


35 


26 


62795 


37205 


62834 


37166 


00039 


99961 


34 


27 


63091 


36909 


63131 


36869 


00040 


99960 


33 


28 


63385 


36615 


63426 


36574 


00040 


99960 


32 


29 


63678 


36322 


63718 


36282 


00041 


99959 


31 


30 


8.63968 


11.36032 


8.64009 


11.35991 


10.00041 


9.99959 


30 


31 


64256 


35744 


64298 


35702 


00042 


99958 


29 


32 


64543 


35457 


64585 


35415 


00042 


99958 


28 


33 


64827 


35173 


64870 


35130 


00043 


99957 


27 


34 


65110 


34890 


65154 


34846 


00044 


99956 


26 


35 


8.65391 


11.34609 


8.65435 


11.34565 


10.00044 


9.99956 


25 


36 


65670 


34330 


65715 


34285 


00045 


99955 


24 


37 


65947 


34053 


65993 


34007 


00045 


99955 


23 


38 


66223 


33777 


66269 


33731 


00046 


99954 


22 


39 


66497 


33503 


66543 


33457 


00046 


99954 


21 


40 


8.66769 


11.33231 


8.66816 


11.33184 


10.00047 


9.99953 


20 


41 


67039 


32961 


67087 


32913 


00048 


99952 


19 


42 


67308 


32692 


67356 


32644 


00048 


99952 


18 


43 


67575 


32425 


67624 


32376 


00049 


99951 


17 


44 


67841 


32159 


67890 


32110 


00049 


99951 


16 


45 


8.68104 


11.31896 


8.68154 


11.31846 


10.00050 


9.99950 


15 


46 


68367 


31633 


68417 


31583 


00051 


99949 


14 


47 


68627 


31373 


68678 


31322 


00051 


99949 


13 


48 


68886 


31114 


68938 


31062 


00052 


99948 


12 


49 


69144 


30856 


69196 


30804 


00052 


99948 


11 


50 


8.69400 


11.30600 


8.69453 


11.30547 


10.00053 


9.99947 


10 


51 


69654 


30346 


69708 


30292 


00054 


99946 


9 


52 


69907 


30093 


69962 


30038 


00054 


99946 


8 


53 


70159 


29841 


70214 


29786 


00055 


99945 


7 


54 


70409 


29591 


70465 


29535 


00056 


99944 


6 


55 


8.70658 


11.29342 


8.70714 


11.29286 


10.00056 


9.99944 


5 


56 


70905 


29095 


70962 


29038 


00057 


99943 


4 


57 


71151 


28849 


71208 


28792 


00058 


99942 


3 


58 


71395 


28605 


71453 


28547 


00058 


99942 


2 


59 


71638 


28362 


71697 


28303 


00059 


99941 


1 


60 


71880 


28120 


71940 


28060 


00060 


99940 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



92° 



87° 



182 



Logarithmic Angular Functions. 



3° 






Logarithms. 






176° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent, 


Secant. 


Cosine. 


M. 





8.71880 


11.28120 


8.71940 


11.28060 


10.00060 


9.99940 


60 


1 


72120 


27880 


72181 


27819 


00060 


99940 


59 


2 


72359 


27641 


72420 


27580 


00061 


99939 


58 


3 


72597 


27403 


72659 


27341 


00062 


99938 


57 


4 


72834 


27166 


72896 


27104 


00062 


99938 


56 


5 


8.73069 


11.26931 


8.73132 


11.26868 


10.00063 


9.99937 


55 


6 


73303 


26697 


73366 


26634 


00064 


99936 


54 


7 


73535 


26465 


73600 


26400 


00064 


99936 


53 


8 


73767 


26233 


73832 


26168 


00065 


99935 


52 


9 


73997 


26003 


74063 


25937 


00066 


99934 


51 


10 


8.74226 


11.25774 


8.74292 


11.25708 


10.00066 


9.99934 


50 


11 


74454 


25546 


74521 


25479 


00067 


99933 


49 


12 


74680 


25320 


74748 


25252 


00068 


99932 


48 


13 


74906 


25094 


74974 


25026 


00068 


99932 


47 


14 


75130 


24870 


75199 


24801 


00069 


99931 


46 


15 


8.75353 


11.24647 


8.75423 


11.24577 


10.00070 


9.99930 


45 


16 


75575 


24425 


75645 


24355 


00071 


99929 


44 


17 


75795 


24205 


75867 


24133 


00071 


99929 


43 


18 


76015 


23985 


76087 


23913 


00072 


99928 


42 


19 


76234 


23766 


76306 


23694 


00073 


99927 


41 


20 


8.76451 


11.23549 


8.76525 


11.23475 


10.00074 


9.99926 


40 


21 


76667 


23333 


76742 


23258 


00074 


99926 


39 


22 


76883 


23117 


76958 


23042 


00075 


99925 


38 


23 


77097 


22903 


77173 


22827 


00076 


99924 


37 


24 


77310 


22690 


77387 


22613 


00077 


99923 


36 


25 


8.77522 


11.22478 


8.77600 


11.22400 


10.00077 


9.99923 


35 


26 


77733 


22267 


77811 


22189 


00078 


99922 


34 


27 


77943 


22057 


78022 


21978 


00079 


99921 


33 


28 


78152 


21848 


78232 


21768 


00080 


99920 


32 


29 


78360 


21640 


78441 


21559 


00080 


99920 


31 


30 


8.78568 


11.21432 


8.78649 


11.21351 


10.00081 


9.99919 


30 


31 


78774 


21226 


78855 


21145 


00082 


99918 


29 


32 


78979 


21021 


79061 


20939 


00083 


99917 


28 


33 


79183 


20817 


79266 


20734 


00083 


99917 


27 


34 


79386 


20614 


79470 


20530 


00084 


99916 


26 


35 


8.79588 


11.20412 


8.79673 


11.20327 


10.00085 


9.99915 


25 


36 


79789 


20211 


79875 


20125 


00086 


99914 


24 


37 


79990 


20010 


80076 


19924 


00087 


99913 


23 


38 


80189 


19811 


80277 


19723 


00087 


99913 


22 


39 


80388 


19612 


80476 


19524 


00088 


99912 


21 


40 


8.80585 


11.19415 


8.80674 


11.19326 


10.00089 


9.99911 


20 


41 


80782 


19218 


80872 


19128 


00090 


99910 


19 


42 


80978 


19022 


81068 


18932 


00091 


99909 


18 


43 


81173 


18827 


81264 


18736 


00091 


99909 


17 


44 


81367 


18633 


81459 


18541 


00092 


99908 


16 


45 


8.81560 


11.18440 


8.81653 


11.18347 


10.00093 


9.99907 


15 


46 


81752 


18248 


81846 


18154 


00094 


99906 


14 


47 


81944 


18056 


82038 


17962 


00095 


99905 


13 


48 


82134 


17866 


82230 


17770 


00096 


99904 


12 


49 


82324 


17676 


82420 


17580 


00096 


99904 


11 


50 


8.82513 


11.17487 


8.82610 


11.17390 


10.00097 


9.99903 


10 


51 


82701 


17299 


82799 


17201 


00098 


99902 


9 


52 


82888 


17112 


82987 


17013 


00099 


99901 


8 


53 


83075 


16925 


83175 


16825 


00100 


99900 


7 


54 


83261 


16739 


83361 


16639 


00101 


99899 


6 


55 


8.83446 


11.16554 


8.83547 


11.16453 


10.00102 


9.99898 


5 


56 


83630 


16370 


83732 


16268 


00102 


99898 


4 


57 


83813 


16187 


83916 


16084 


00103 


99897 


3 


58 


83996 


16004 


84100 


15900 


00104 


99896 


2 


59 


84177 


15823 


84282 


15718 


00105 


99895 


1 


60 


84358 


15642 


84464 


15536 


00106 


99894 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



86° 



LOGARITHMIC ANGULAR FUNCTIONS. 



183 



40 






Logar 


ithms. 




175° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





8.84358 


11.15642 


8.84464 


11.15536 


10.00106 


9.99894 


60 


1 


84539 


15461 


84646 


15354 


00107 


99893 


59 


2 


84718 


15282 


84826 


15174 


00108 


99892 


58 


3 


84897 


15103 


85006 


14994 


00109 


99891 


57 


4 


85075 


14925 


85185 


14815 


00109 


99891 


56 


5 


8.85252 


11.14748 


8.85363 


11.14637 


10.00110 


9.99890 


55 


6 


85429 


14571 


85540 


14460 


00111 


99889 


54 


7 


85605 


14395 


85717 


14283 


00112 


99888 


53 


8 


85780 


14220 


85893 


14107 


00113 


99887 


52 


9 


85955 


14045 


86069 


13931 


00114 


99886 


51 


10 


8.86128 


11.13872 


8.86243 


11.13757 


10.00115 


9.99885 


50 


11 


86301 


13699 


86417 


13583 


00116 


99884 


49 


12 


86474 


13526 


86591 


13409 


00117 


99883 


48 


13 


86645 


13355 


86763 


13237 


00118 


99882 


47 


14 


86816 


13184 


86935 


13065 


00119 


99881 


46 


15 


8.86987 


11.13013 


8.87106 


11.12894 


10.00120 


9.99880 


45 


16 


87156 


12844 


87277 


12723 


00121 


99879 


44 


17 


87325 


12675 


87447 


12553 


00121 


99879 


43 


18 


87494 


12506 


87616 


12384 


00122 


99878 


42 


19 


87661 


12339 


87785 


12215 


00123 


99877 


41 


20 


8.87829 


ll.i2171 


8.87953 


11.12047 


10.00124 


9.99876 


40 


21 


87995 


12005 


88120 


11880 


00125 


99875 


39 


22 


88161 


11839 


88287 


11713 


00126 


99874 


38 


23 


88326 


11674 


88453 


11547 


00127 


99873 


37 


24 


88490 


11510 


88618 


11382 


00128 


99872 


36 


25 


8.88654 


11.11346 


8.88783 


11.11217 


10.00129 


9.99871 


35 


• 26 


88817 


11183 


88948 


11052 


00130 


99870 


34 


27 


88980 


11020 


89111 


10889 


00131 


99869 


33 


28 


89142 


10858 


89274 


10726 


00132 


99868 


32 


29 


89304 


10696 


89437 


10563 


00133 


99867 


31 


30 


8.89464 


11.10536 


8.89598 


11.10402 


10.00134 


9.99866 


30 


31 


89625 


10375 


89760 


10240 


00135 


99865 


29 


32 


89784 


10216 


89920 


10080 


00136 


99864 


28 


33 


89943 


10057 


90080 


09920 


00137 


99863 


27 


34 


90102 


09898 


90240 


09760 


00138 


99862 


26 


35 


8.90260 


11.09740 


8.90399 


11.09601 


10.00139 


9.99861 


25 


36 


90417 


09583 


90557 


09443 


00140 


99860 


24 


37 


90574 


09426 


90715 


09285 


00141 


99859 


23 


38 


90730 


09270 


90872 


09128 


00142 


99858 


22 


39 


90885 


09115 


91029 


08971 


00143 


99857 


21 


40 


8.91040 


11.08960 


8.91185 


11.08815 


10.00144 


9.99856 


20 


41 


91195 


08805 


91340 


08660 


00145 


99855 


19 


42 


91349 


08651 


91495 


08505 


00146 


99854 


18 


43 


91502 


08498 


91650 


08350 


00147 


99853 


17 


44 


91655 


08345 


91803 


08197 


00148 


99852 


16 


45 


8.91807 


11.08193 


8.91957 


11.08043 


10.00149 


9.99851 


15 


46 


91959 


08041 


92110 


07890 


00150 


99850 


14 


47 


92110 


07890 


92262 


07738 


00152 


99848 


13 


48 


92261 


07739 


92414 


07586 


00153 


99847 


12 


49 


92411 


07589 


92565 


07435 


00154 


99846 


11 


50 


8.92561 


11.07439 


8.92716 


11.07284 


10.00155 


9.99845 


10 


51 


92710 


07290 


92866 


07134 


00156 


99844 


9 


52 


92859 


07141 


93016 


06984 


00157 


99843 


8 


53 


93007 


06993 


93165 


06835 


00158 


99842 


7 


54 


93154 


06846 


93313 


06687 


00159 


99841 


6 


55 


8.93301 


11.06699 


8.93462 


11.06538 


10.00160 


9.99840 


5 


56 


93448 


06552 


93609 


06391 


00161 


99839 


4 


57 


93594 


06406 


93756 


06244 


00162 


99838 


3 


58 


93740 


06260 


93903 


06097 


00163 


99837 


2 


59 


93885 


06115 


94049 


05951 


00164 


99836 


1 


60 


94030 


05970 


94195 


05805 


00166 


99834 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. | 


Cosecant. 


Sine. 


M. 



184 



Logarithmic Angular Functions. 



5° 






Logarithms. 




174° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





8.94030 


11.05970 


8.94195 


11.05805 


10.00166 


9.99834 


60 


1 


94174 


05826 


94340 


05660 


00167 


99833 


59 


2 


94317 


05683 


94485 


05515 


00168 


99832 


58 


3 


94461 


05539 


94630 


05370 


00169 


99831 


57 


4 


94603 


05397 


94773 


05227 


00170 


99830 


56 


5 


8.94746 


11.05254 


8.94917 


11.05083 


10.00171 


9.99829 


55 


6 


94887 


05113 


95060 


04940 


00172 


99828 


54 


7 


95029 


04971 


95202 


04798 


00173 


99827 


53 


8 


95170 


04830 


95344 


04656 


00175 


99825 


52 


9 


95310 


04690 


95486 


04514 


00176 


99824 


51 


10 


8.95450 


11.04550 


8.95627 


11.04373 


10.00177 


9.99823 


50 


11 


95589 


04411 


95767 


04233 


00178 


99822 


49 


12 


95728 


04272 


95908 


04092 


00179 


99821 


48 


13 


95867 


04133 


96047 


03953 


00180 


99820 


47 


14 


96005 


03995 


96187 


03813 


00181 


99819 


46 


15 


8.96143 


11.03857 


8.96325 


11.03675 


10.00183 


9.99817 


45 


16 


96280 


03720 


96464 


03536 


00184 


99816 


44 


17 


96417 


03583 


96602 


03398 


00185 


99815 


43 


18 


96553 


03447 


96739 


03261 


00186 


99814 


42 


19 


96689 


03311 


96877 


03123 


00187 


99813 


41 


20 


8.96825 


11.03175 


8.97013 


11.02987 


10.00188 


9.99812 


40 


21 


96960 


03040 


97150 


02850 


00190 


99810 


39 


22 


97095 


02905 


97285 


02715 


00191 


99809 


38 


23 


97229 


02771 


97421 


02579 


00192 


99808 


37 


24 


97363 


02637 


97556 


02444 


00193 


99807 


36 


25 


8.97496 


11.02504 


8.97691 


11.02309 


10.00194 


9.99806 


35 


26 


97629 


02371 


97825 


02175 


00196 


99804 


34 


27 


97762 


02238 


97959 


02041 


00197 


99803 


33 


28 


97894 


02106 


98092 


01908 


00198 


99802 


32 


29 


98026 


01974 


98225 


01775 


00199 


99801 


31 


30 


8.98157 


11.01843 


8.98358 


11.01642 


10.00200 


9.99800 


30 


31 


98288 


01712 


98490 


01510 


00202 


99798 


29 


32 


98419 


01581 


98622 


01378 


00203 


99797 


28 


33 


98549 


01451 


98753 


01247 


00204 


99796 


27 


34 


98679 


01321 


98884 


01116 


00205 


99795 


26 


35 


8.98808 


11.01192 


8.99015 


11.00985 


10.00207 


9.99793 


25 


36 


98937 


01063 


99145 


00855 


00208 


99792 


24 


37 


99066 


00934 


99275 


00725 


00209 


99791 


23 


38 


99194 


00806 


99405 


00595 


00210 


99790 


22 


39 


99322 


00678 


99534 


00466 


00212 


99788 


21 


40 


8.99450 


11.00550 


8.99662 


11.00338 


10.00213 


9.99787 


20 


41 


99577 


00423 


99791 


00209 


00214 


99786 


19 


42 


99704 


00296 


99919 


00081 


00215 


99785 


18 


43 


99830 


00170 


9.00046 


10.99954 


00217 


99783 


17 


44 


99956 


00044 


00174 


99826 


00218 


99782 


16 


45 


9.00082 


10.99918 


9.00301 


10.99699 


10.00219 


9.99781 


15 


46 


00207 


99793 


00427 


99573 


00220 


99780 


14 


47 


00332 


99668 


00553 


99447 


00222 


99778 


13 


48 


00456 


99544 


00679 


99321 


00223 


99777 


12 


49 


00581 


99419 


00805 


99195 


00224 


99776 


11 


50 


9.00704 


10.99296 


9.00930 


10.99070 


10.00225 


9.99775 


10 


51 


00828 


99172 


01055 


98945 


00227 


99773 


9 


52 


00951 


99049 


01179 


98821 


00228 


99772 


8 


53 


01074 


98926 


01303 


98697 


00229 


99771 


7 


54 


01196 


98804 


01427 


98573 


00231 


99769 


6 


55 


9.01318 


10.98682 


9.01550 


10.98450 


10.00232 


9.99768 


5 


56 


01440 


98560 


01673 


98327 


00233 


99767 


4 


57 


01561 


98439 


01796 


98204 


00235 


99765 


3 


58 


01682 


98318 


01918 


98082 


00236 


99764 


2 


59 


01803 


98197 


02040 


97960 


00237 


99763 


1 


60 


01923 


98077 


02162 


97838 


00239 


99761 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



84° 



LOGARITHMIC ANGULAR FUNCTIONS. 



185 



6° 






Logarithms. 




173° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.01923 


10.98077 


9.02162 


10.97838 


10.00239 


9.99761 


60 


1 


02043 


97957 


02283 


97717 


00240 


99760 


59 


2 


02163 


97837 


02404 


97596 


00241 


99759 


58 


3 


02283 


97717 


02525 


97475 


00243 


99757 


57 


4 


02402 


97598 


02645 


97355 


00244 


99756 


56 


5 


9.02520 


10.97480 


9.02766 


10.97234 


10.00245 


9.99755 


55 


6 


02639 


97361 


02885 


97115 


00247 


99753 


54 


7 


02757 


97243 


03005 


96995 


00248 


99752 


53 


8 


02874 


97126 


03124 


96876 


00249 


99751 


52 


9 


02992 


97008 


03242 


96758 


00251 


99749 


51 


10 


9.03109 


10.96891 


9.03361 


10.96639 


10.00252 


9.99748 


50 


11 


03226 


96774 


03479 


96521 


00253 


99747 


49 


12 


03342 


96658 


03597 


96403 


00255 


99745 


48 


13 


03458 


96542 


03714 


96286 


00256 


99744 


47 


14 


03574 


96426 


03832 


96168 


00258 


99742 


46 


15 


9.03690 


10.96310 


9.03948 


10.96052 


10.00259 


9.99741 


45 


16 


03805 


96195 


04065 


95935 


00260 


99740 


44 


17 


03920 


96080 


04181 


95819 


00262 


99738 


43 


18 


04034 


95966 


04297 


95703 


00263 


99737 


42 


19 


04149 


95851 


04413 


95587 


00264 


99736 


41 


20 


9.04262 


10.95738 


9.04528 


10.95472 


10.00266 


9.99734 


40 


21 


04376 


95624 


04643 


95357 


00267 


99733 


39 


22 


04490 


95510 


04758 


95242 


00269 


99731 


38 


23 


04603 


95397 


04873 


95127 


00270 


99730 


37 


24 


04715 


95285 


04987 


95013 


00272 


99728 


36 


25 


9.04828 


10.95172 


9.05101 


10.94899 


10.00273 


9.99727 


35 


26 


04940 


95060 


05214 


94786 


00274 


99726 


34 


27 


05052 


94948 


05328 


94672 


0)276 


99724 


33 


28 


05164 


94836 


05441 


94559 


00277 


99723 


32 


29 


05275 


94725 


05553 


94447 


00279 


99721 


31 


30 


9.05386 


10.94614 


9.05666 


10.94334 


10.00280 


9.99720 


30 


31 


05497 


94503 


05778 


94222 


00282 


99718 


29 


32 


05607 


94393 


05890 


94110 


00283 


99717 


28 


33 


05717 


94283 


06002 


93998 


00284 


99716 


27 


34 


05827 


94173 


06113 


93887 


00286 


99714 


26 


35 


9„05937 


10.94063 


9.06224 


10.93776 


10.00287 


9.99713 


25 


36 


06046 


93954 


06335 


93665 


00289 


99711 


24 


37 


06155 


93845 


06445 


93555 


00290 


99710 


23 


38 


06264 


93736 


06556 


93444 


00292 


99708 


22 


39 


06372 


93628 


06666 


93334 


00293 


99707 


21 


40 


9.06481 


10.93519 


9.06775 


10.93225 


10.00295 


9.99705 


20 


41 


06589 


93411 


06885 


93115 


00296 


99704 


19 


42 


06696 


93304 


06994 


93006 


00298 


99702 


18 


43 


06804 


93196 


07103 


92897 


00299 


99701 


17 


44 


06911 


93089 


07211 


92789 


00301 


99699 


16 


45 


9.07018 


10.92982 


9.07320 


10.92680 


10.00302 


9.99698 


15 


46 


07124 


92876 


07428 


92572 


00304 


99696 


14 


47 


07231 


92769 


07536 


92464 


00305 


99695 


13 


48 


07337 


92663 


07643 


92357 


00307 


99693 


12 


49 


07442 


92558 


07751 


92249 


00308 


99692 


11 


50 


9.07548 


10.92452 


9.07858 


10.92142 


10.00310 


9.99690 


10 


51 


07653 


92347 


07964 


92036 


00311 


99689 


9 


52 


07758 


92242 


08071 


91929 


00313 


99687 


8 


53 


07863 


92137 


08177 


91823 


00314 


99686 


7 


54 


07968 


92032 


08283 


91717 


00316 


99684 


6 


55 


9.08072 


10.91928 


9.08389 


10.91611 


10.00317 


9.99683 


5 


56 


08176 


91824 


08495 


91505 


00319 


99681 


4 


57 


08280 


91720 


08600 


91400 


00320 


99680 


3 


58 


08383 


91617 


08705 


91295 


00322 


99678 


2 


59 


08486 


91514 


08810 


91190 


00323 


99677 


1 


60 


08589 


91411 


08914 


91086 


00325 


99675 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



83° 



186 



Logarithmic Angular Functions. 



7° 






Logarithms. 






172° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.08589 


10.91411 


9.08914 


10.91086 


10.00325 


9.99675 


60 


1 


08692 


91308 


09019 


90981 


00326 


99674 


59 


2 


08795 


91205 


09123 


90877 


00328 


99672 


58 


3 


08897 


91103 


09227 


90773 


00330 


99670 


57 


4 


08999 


91001 


09330 


90670 


00331 


99669 


56 


5 


9.09101 


10.90899 


9.09434 


10.90566 


10.00333 


9.99667 


55 


6 


09202 


90798 


09537 


90463 


00334 


99666 


54 


7 


09304 


90696 


09640 


90360 


00336 


99664 


53 


8 


09405 


90595 


09742 


90258 


00337 


99663 


52 


9 


09506 


90494 


09845 


90155 


00339 


99661 


51 


10 


9.09606 


10.90394 


9.09947 


10.90053 


10.00341 


9.99659 


50 


11 


09707 


90293 


10049 


89951 


00342 


99658 


49 


12 


09807 


90193 


10150 


89850 


00344 


99656 


48 


13 


09907 


90093 


10252 


89748 


00345 


99655 


47 


14 


10006 


89994 


10353 


89647 


00347 


99653 


46 


15 


9.10106 


10.89894 


9.10454 


10.89546 


10.00349 


9.99651 


45 


16 


10205 


89795 


10555 


89445 


00350 


99650 


44 


17 


10304 


89696 


10656 


89344 


00352 


99648 


43 


18 


10402 


89598 


10756 


89244 


00353 


99647 


42 


19 


10501 


89499 


10856 


89144 


00355 


99645 


41 


20 


9.10599 


10.89401 


9.10956 


10.89044 


10.00357 


9.99643 


40 


21 


10697 


89303 


11056 


88944 


00358 


99642 


39 


22 


10795 


89205 


11155 


88845 


00360 


99640 


38 


23 


10893 


89107 


11254 


88746 


00362 


99638 


37 


24 


10990 


89010 


11353 


88647 


00363 


99637 


36 


25 


9.11087 


10.88913 


9.11452 


10.88548 


10.00365 


9.99635 


35 


26 


11184 


88816 


11551 


88449 


00367 


99633 


34 


27 


11281 


88719 


11649 


88351 


00368 


99632 


33 


28 


11377 


88623 


11747 


88253 


00370 


99630 


32 


29 


11474 


88526 


11845 


88155 


00371 


99629 


31 


30 


9.11570 


10.88430 


9.11943 


10.88057 


10.00373 


9.99627 


30 


SI 


11666 


88334 


12040 


87960 


00375 


99625 


29 


32 


11761 


88239 


12138 


87862 


00376 


99624 


28 


33 


11857 


88143 


12235 


87765 


00378 


99622 


27 


34 


11952 


88048 


12332 


87668 


00380 


99620 


26 


35 


9.12047 


10.87953 


9.12428 


10.87572 


10.00382 


9.99618 


25 


36 


12142 


87858 


12525 


87475 


00383 


99617 


24 


37 


12236 


87764 


12621 


87379 


00385 


99615 


23 


38 


12331 


87669 


12717 


87283 


00387 


99613 


22 


39 


12425 


87575 


12813 


87187 


00388 


99612 


21 


40 


9.12519 


10.87481 


9.12909 


10.87091 


10.00390 


9.99610 


20 


41 


12612 


87388 


13004 


86996 


00392 


99608 


19 


42 


12706 


87294 


13099 


86901 


00393 


99607 


18 


43 


12799 


87201 


13194 


86806 


00395 


99605 


17 


44 


12892 


87108 


13289 


86711 


00397 


99603 


16 


45 


9.12985 


10.87015 


9.13384 


10.86616 


10.00399 


9.99601 


15 


46 


13078 


86922 


13478 


86522 


00400 


99600 


14 


47 


13171 


86829 


13573 


86427 


00402 


99598 


13 


48 


13263 


86737 


13667 


86333 


00404 


99596 


12 


49 


13355 


86645 


13761 


86239 


00405 


99595 


11 


50 


9.13447 


10.86553 


9.13854 


10.86146 


10.00407 


9.99593 


10 


51 


13539 


86461 


13948 


86052 


00409 


99591 


9 


52 


13630 


86370 


14011 


85959 


00411 


99589 


8 


53 


13722 


86278 


14134 


85866 


00412 


99588 


7 


54 


13813 


86187 


14227 


85773 


00414 


99586 


6 


55 


9.13904 


10.86096 


9.14320 


10.85680 


10.00416 


9.99584 


5 


56 


13994 


86006 


14412 


85588 


00418 


99582 


4 


57 


14085 


85915 


14504 


85496 


00419 


99581 


3 


58 


14175 


85825 


14597 


85403 


00421 


99579 


2 


59 


1 1266 


85734 


14688 


85312 


00423 


99577 


1 


60 


14356 


85644 


14780 


85220 


00425 


99575 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



Logarithmic Angular Functions. 



187 



8° 






Logarithms. 






171° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.14356 


10.85644 


9.14780 


10.85220 


10.00425 


9.99575 


60 


1 


14445 


85555 


14872 


85128 


00426 


99574 


59 


2 


14535 


85465 


14963 


85037 


00428 


99572 


58 


3 


14624 


85376 


15054 


84946 


00430 


99570 


57 


4 


14714 


85286 


15145 


84855 


00432 


99568 


56 


5 


9.14803 


10.85197 


9.15236 


10.84764 


10.00434 


9.99566 


55 


6 


14891 


85109 


15327 


84673 


00435 


99565 


54 


7 


14980 


85020 


15417 


84583 


00437 


99563 


53 


8 


15069 


84931 


15508 


84492 


00439 


99561 


52 


9 


15157 


84843 


15598 


84402 


00441 


99559 


51 


10 


9.15245 


10.84755 


9.15688 


10.84312 


10.00443 


9.99557 


50 


11 


15333 


84667 


15777 


84223 


00444 


99556 


49 


12 


15421 


84579 


15867 


84133 


00446 


99554 


48 


13 


15508 


84492 


15956 


84044 


00448 


99552 


47 


14 


15596 


84404 


16046 


83954 


00450 


99550 


46 


15 


9.15683 


10.84317 


9.16135 


10.83865 


10.00452 


9.99548 


45 


16 


15770 


84230 


16224 


83776 


00454 


99546 


44 


17 


15857 


84143 


16312 


83688 


00455 


99545 


43 


18 


15944 


84056 


16401 


83599 


00457 


99543 


42 


19 


16030 


83970 


16489 


83511 


00459 


99541 


41 


20 


9.16116 


10.83884 


9.16577 


10.83423 


10.00461 


9.99539 


40 


21 


16203 


83797 


16665 


83335 


00463 


99537 


39 


22 


16289 


83711 


16753 


83247 


00465 


99535 


38 


23 


16374 


83626 


16841 


83159 


00467 


99533 


37 


24 


16460 


83540 


16928 


83072 


00468 


99532 


36 


25 


9.16545 


10.83455 


9.17016 


10.82984 


10.00470 


9.99530 


35 


26 


16631 


83369 


17103 


82897 


00472 


99528 


34 


27 


16716 


83284 


17190 


82810 


00474 


99526 


33 


28 


16801 


83199 


17277 


82723 


00476 


99524 


32 


29 


16886 


83114 


17363 


82637 


00478 


99522 


31 


30 


9.16970 


10.83030 


9.17450 


10.82550 


10.00480 


9.99520 


30 


31 


17055 


82945 


17536 


82464 


00482 


99518 


29 


32 


17139 


82861 


17622 


82378 


00483 


99517 


28 


33 


17223 


82777 


17708 


82292 


00485 


99515 


27 


34 


17307 


82693 


17794 


82206 


00487 


99513 


26 


35 


9.17391 


10.82609 


9.17880 


10.82120 


10.00489 


9.99511 


25 


36 


17474 


82526 


" 17965 


82035 


00491 


99509 


24 


37 


17558 


82442 


18051 


81949 


00493 


99507 


23 


38 


17641 


82359 


18136 


81864 


00495 


99505 


22 


39 


17724 


82276 


18221 


81779 


00497 


99503 


21 


40 


9.17807 


10.82193 


9.18306 


10.81694 


10.00499 


9.99501 


20 


41 


17890 


82110 


18391 


81609 


00501 


99499 


19 


42 


17973 


82027 


18475 


81525 


00503 


99497 


18 


43 


18055 


81945 


18560 


81440 


00505 


99495 


17 


44 


18137 


81863 


18644 


81356 


00506 


99494 


16 


45 


9.18220 


10.81780 


9.18728 


10.81272 


10.00508 


9.99492 


15 


46 


18302 


81698 


18812 


81188 


00510 


99490 


14 


47 


18383 


81617 


18896 


81104 


00512 


99488 


13 


48 


18465 


81535 


18979 


81021 


00514 


99486 


12 


49 


18547 


81453 


19063 


80937 


00516 


99484 


11 


50 


9.18628 


10.81372 


9.19146 


10.80854 


10.00518 


9.99482 


10 


51 


18709 


81291 


19229 


80771 


00520 


99480 


9 


52 


18790 


81210 


19312 


80688 


00522 


99478 


8 


53 


18871 


81129 


19395 


80605 


00524 


99476 


7 


54 


18952 


81048 


19478 


80522 


00526 


99474 


6 


55 


9.19033 


10.80967 


9.19561 


10.80439 


10.00528 


9.99472 


5 


56 


19113 


80887 


19643 


80357 


00530 


99470 


4 


57 


19193 


80807 


19725 


80275 


00532 


99468 


3 


58 


19273 


80727 


19807 


80193 


00534 


99466 


2 


59 


19353 


80647 


19889 


80111 


00536 


99464 


1 


60 


19433 


80567 


19971 


80029 


00538 


99462 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



188 



Logarithmic Angulab Functions. 



90 






Logarithms. 






170° 


M. 


Sine. 


Cosecant 


Tangent, 


Cotangent 


Secant, 


| Codne, 


!M. 





9.19488 


10.80667 


0.10071 


10.S0029 


10.00638 


0.00102 


60 


1 


L9518 


MUST 


20058 


79947 


00.. 10 


00100 




2 


19692 


80408 


20134 


70S00 


00642 


99468 


68 


8 


19672 


80328 


20210 


79784 


00644 


99456 


07 


4 


L9761 


S0210 


2029 1 ; 


79708 


oo,» 10 


99464 


56 


5 


9.19880 


10.80170 


9 20378 


10.70022 


10.00.. IS 


9.99462 


•6 


G 


19909 


80091 


20459 


79641 


00.00 


99460 


64 


7 


L9988 


80012 


20010 


70100 


00662 


99448 


58 


8 


20067 


79938 


20021 


70:570 


00564 


00 110 




9 


20145 


79865 


20701 


70200 


000,. 


w 1 1 1 


•1 


10 


9.20228 


10.70777 


0.207S2 


L0.79218 


10.00668 


9.99442 


,0 


11 


20302 


7000S 


20S02 


79188 


00660 


00 no 


19 


12 


20880 


70020 


20942 


7000S 


00002 


0010s 


IS 


18 


20458 


79542 


21022 


7S07S 


00664 


99486 


17 


11 


20585 


7010,") 


21102 


7SS0S 


00000 


00 101 


10 


L5 


9.2001:? 


10.79887 


9.21182 


L0.78818 


10.00668 


9.99482 


15 


16 


20691 


70: 500 


21201 


78739 


0061 1 


99429 


11 


17 


20768 


79282 


21841 


78659 


00678 


0012, 


18 


L8 


20845 


79155 


21120 


78580 


00675 


99426 


12 


19 


20022 


79078 


•.'l 199 


Vs. .01 


0067*3 


00120 


11 


20 


0.20000 


10.70001 


9 21678 


10.78422 


10.00679 


9.99423 


40 


21 


21(170 


78924 


21657 


78348 


00581 


99 1 1 9 


89 


22 


21153 


78847 


21736 


78264 


000s:; 


00 117 


::s 


28 


21220 


7877 1 


2isi 1 


78186 


00685 


99415 


07 


24 


21806 


7S00 1 


21898 


7S107 


00587 


99413 


86 


26 


9.21882 


10.7M. is 


0.21071 


10.7S020 


10.00689 


0.00111 


85 


26 


21 158 


78642 


22010 


77961 


00691 


00100 


84 


27 


21534 


78466 


22127 


77878 


0000:; 


00IO7 


88 


28 


21610 


78890 


22205 


77795 


00696 


00 101 


82 


29 


21685 


78815 


22283 


77717 


00698 


00102 


01 


30 


9.21761 


10.78289 


9.22861 


10.77639 


10.00000 


0.00100 


80 


SI 


21836 


7S10I 


22438 




00002 


oooos 


29 


82 


21012 


78088 


22010 


77484 


00001 


00200 


28 


88 


210S7 


78013 


22593 


77107 


00000 


99394 


27 


84 


22002 


77938 


22070 


77880 


00808 


99392 


26 


85 


9.22187 


10.77868 


9.22747 


10.7720:1 


10.00010 


9.99890 


25 




22211 


77789 


22824 


77170 


00012 


99888 


'.'i 




22286 


7771 1 


22001 


77000 


0001,, 


99385 


28 


88 


22361 




22077 


77028 


00017 


99383 


22 








23054 


70010 


00010 


99381 


21 


40 


9.22509 


10.77101 


1180 


10.76870 


10.00621 


9.99 


20 


41 




77117 


23206 


70701 


00628 




19 


42 




77:: 10 


23283 


70717 


0002,. 


99375 


IS 


48 


227:; 1 


77200 


23359 


76641 


0002s 


00072 


17 


44 








7656 • 


00680 


99370 


16 


45 


9.22878 


10.77122 


. .in 


10.76490 


10.00002 


9.99868 


L5 


46 


22952 


77048 


23686 


76414 


00634 


99366 


11 


•17 




7007") 


23661 




00636 


99364 


18 


48 


'.".(IMS 


70002 




76263 


00638 


99862 


12 


19 






23812 


76188 


000 II 


99359 


11 


r>o 


'II 


10.76' 


9 2 


10.76118 


10.00643 


9.9935' 


10 


61 


2:;:: 17 






76038 


00645 


9985 1 


9 


62 


23390 


76610 


24037 


75963 


00647 


99353 


s 


5:* 


23462 


76538 


24112 


70sss 


01)010 


99351 


7 


54 




76465 


21 ISO 


7581 1 


00652 


00:: is 


6 




.r,()7 


In 71 


'.' '1201 


in ! 


10.01 to,., 1 


9.99346 


6 


66 


23679 






766(5 ' 


00656 


99344 


•1 


67 






24410 


' ,.'■,( 


oooos 


99342 


8 






70177 


24484 


75516 


00000 


00010 


2 






7610 1 


■i .. 


75442 


00663 


00007 


1 


60 


28967 




Co tan 


nt. 


0(100., 







M 


1 


nil 


ant. 


Sine. 


Bt 



Logarithmic Angular Functions. 



189 



10° 






Logarithms. 






169° 


M. 


Sine. 


Cosecant. 


1 Tangent. 
9.24632 


Cotangent. 
10.75368 


Secant. 


Cosine. 


M. 





9.23967 


10.760:53 


10.00665 


9.99335 


60 


1 


24039 


75961 


24706 


75294 


00667 


99333 


59 


2 


24110 


75890 


24779 


75221 


00669 


99331 


58 


3 


24181 


75819 


24853 


75147 


00672 


99328 


57 


4 


24253 


75747 


24926 


75074 


00674 


99326 


56 


5 


9.24324 


10.75676 


9.25000 


10.75000 


10.00676 


9.99324 


55 


6 


24395 


75605 


25073 


74927 


00678 


99322 


54 


7 


24466 


75534 


25146 


74854 


00681 


99319 


53 


8 


24536 


75464 


25219 


74781 


00683 


99317 


52 


9 


24607 


75393 


25292 


74708 


00685 


99315 


51 


10 


9.24677 


10.75323 


9.25365 


10.74635 


10.00687 


9.99313 


50 


11 


24748 


75252 


25437 


74563 


00690 


99310 


49 


12 


24818 


75182 


25510 


74490 


00692 


99308 


48 


13 


24888 


75112 


25582 


74418 


00694 


99306 


47 


14 


24958 


75042 


25655 


74345 


00696 


99304 


46 


15 


9.25028 


10.74972 


9.25727 


10.74273 


10.00699 


9.99301 


45 


16 


25098 


74902 


25799 


74201 


00701 


99299 


44 


17 


25168 


74832 


25871 


74129 


00703 


99297 


43 


18 


25237 


74763 


25943 


74057 


00706 


99294 


42 


19 


25307 


74693 


26015 


73985 


00708 


99292 


41 


20 


9.25376 


10.74624 


9.26086 


10.73914 


10.00710 


9.99290 


40 


21 


25445 


74555 


26158 


73842 


00712 


99288 


39 


22 


25514 


74486 


26229 


73771 


00715 


99285 


38 


23 


25583 


71417 


26301 


73699 


00717 


99283 


37 


24 


25652 


74348 


26372 


73628 


00719 


99281 


36 


25 


9.25721 


10.74279 


9.26443 


10.73557 


v 10.00722 


9.99278 


35 


26 


25790 


74210 


26514 


73486 


00724 


99276 


34 


27 


25858 


74142 


26585 


73415 


00726 


99274 


33 


28 


25927 


74073 


26655 


.73345 


00729 


99271 


32 


29 


25995 


74005 


26726 


73274 


00731 


99269 


31 


30 


9.26063 


10.73937 


9.26797 


10.73203 


10.00733 


9.99267 


30 


31 


26131 


73869 


26867 


73133 


00736 


99264 


29 


32 


26199 


73801 


26937 


73063 


00738 


99262 


28 


33 


26267 


73733 


27008 


72992 


00740 


99260 


27 


34 


26335 


73665 


27078 


72922 


00743 


99257 


26 


35 


9.26403 


10.73597 


9.27148 


10.72852 


10.00745 


9.99255 


25 


36 


26470 


73530 


2721, S 


72782 


00748 


99252 


24 


37 


26538 


73462 


27288 


72712 


00750 


99250 


23 


38 


26605 


73395 


27357 


72643 


00752 


99248 


22 


39 


26672 


73328 


27427 


72573 


00755 


99245 


21 


40 


9.26739 


10.73261 


9.27496 


10.72504 


10.00757 


9.99243 


20 


41 


26806 


73194 


27566 


72434 


00759 


99241 


19 


42 


26873 


73127 


27635 


72365 


00762 


99238 


18 


43 


26940 


73060 


27704 


72296 


00764 


99236 


17 


44 


27007 


72993 


27773 


72227 


00767 


99233 


16 


45 


9.27073 


10.72927 


9.27842 


10.72158 


10.00769 


9.99231 


15 


46 


27140 


72860 


27911 


72089 


00771 


99229 


14 


47 


27206 


72794 


27980 


72020 


00774 


99226 


13 


48 


27273 


72727 


28049 


71951 


00776 


99224 


12 


49 


27339 


72661 


28117 


71883 


00779 


99221 


11 


50 


9.27405 


10.72595 


9.28186 


10.71814 


10.00781 


9.99219 


10 


51 


27471 


72529 


28254 


71746 


00783 


99217 


9 


52 


27537 


72463 


28323 


71677 


00786 


99214 


8 


53 


27602 


72398 


28391 


71609' 


00788 


99212 


7 


54 


27668 


72332 


28459 


71541 


00791 


99209 


6 


55 


9.27734 


10.72266 


9.28527 


10.71473 


10.00793 


9.99207 


5 


56 


27799 


72201 


28595 


71405 


00796 


99204 


4 


57 


27864 


72136 


28662 


71338 


00798 


99202 


3 


58 


27930 


72070 


28730 


71270 


00800 


99200 


2 


59 


27995 


72005 


28798 


71202 


00803 


99197 


1 


60 


28060 


71940 


28865 


71135 


008.05 


99195 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



100° 



79° 



190 



Logarithmic Angular Functions. 



11° 






Logarithms. 






168° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.28060 


10.71940 


9.28865 


10.71135 


10.00805 


9.99195 


60 


1 


28125 


71875 


28933 


71067 


00808 


99192 


59 


2 


28190 


71810 


29000 


71000 


00810 


99190 


58 


3 


28254 


71746 


29067 


70933 


00813 


99187 


57 


4 


28319 


71681 


29134 


70866 


00815 


99185 


56 


5 


9.28384 


10.71616 


9.29201 


10.70799 


10.00818 


9.99182 


55 


6 


28448 


71552 


29268 


70732 


00820 


99180 


54 


7 


28512 


71488 


29335 


70665 


00823 


99177 


53 


8 


28577 


71423 


29402 


70598 


00825 


99175 


52 


9 


28641 


71359 


29468 


70532 


00828 


99172 


51 


10 


9.28705 


10.71295 


9.29535 


10.70465 


10.00830 


9.99170 


50 


11 


28769 


71231 


29601 


70399 


00833 


99167 


49 


12 


28833 


71167 


29668 


70332 


00835 


99165 


48 


13 


28896 


71104 


29734 


70266 


00838 


99162 


47 


14 


28960 


71040 


29800 


70200 


00840 


99160 


46 


15 


9.29024 


10.70976 


9.29866 


10.70134 


10.00843 


9.99157 


45 


16 


29087 


70913 


29932 


70068 


00845 


99155 


44 


17 


29150 


70850 


29998 


70002 


00848 


99152 


43 


18 


29214 


70786 


30064 


69936 


00850 


99150 


42 


19 


29277 


70723 


30130 


69870 


00853 


99147 


41 


20 


9.29340 


10.70660 


9.30195 


10.69805 


10.00855 


9.99145 


40 


21 


29403 


70597 


30261 


69739 


00858 


99142 


39 


22 


29466 


70534 


30326 


69674 


00860 


99140 


38 


23 


29529 


70471 


30391 


69609 


00863 


99137 


37 


24 


29591 


70409 


30457 


- 69543 


00865 


99135 


36 


25 


9.29654 


10.70346 


9.30522 


10.69478 


10.00868 


9.99132 


35 


26 


29716 


70284 


30587 


69413 


00870 


99130 


34 


27 


29779 


70221 


30652 


69348 


00873 


99127 


33 


28 


29841 


70159 


30717 


69283 


00876 


99124 


32 


29 


29903 


70097 


30782 


69218 


00878 


99122 


31 


30 


9.29966 


10.70034 


9.30846 


10.69154 


10.00881 


9.99119 


30 


31 


30028 


69972 


30911 


69089 


00883 


99117 


29 


32 


30090 


69910 


30975 


69025 


00886 


99114 


28 


33 


30151 


69849 


31040 


68960 


00888 


99112 


27 


34 


30213 


69787 


31104 


68896 


00891 


99109 


26 


35 


9.30275 


10.69725 


9.31168 


10.68832 


10.00894 


9.99106 


25 


36 


30336 


69664 


31233 


68767 


00896 


99104 


24 


37 


30398 


69602 


31297 


68703 


00899 


99101 


23 


38 


30459 


69541 


31361 


68639 


00901 


99099 


22 


39 


30521 


69479 


31425 


68575 


00904 


99096 


21 


40 


9.30582 


10.69418 


9.31489 


10.68511 


10.00907 


9.99093 


20 


41 


30643 


69357 


31552 


68448 


00909 


99091 


19 


42 


30704 


69296 


31616 


68384 


00912 


99088 


18 


43 


30765 


69235 


31679 


68321 


00914 


99086 


17 


44 


30826 


69174 


31743 


68257 


00917 


99083 


16 


45 


9.30887 


10.69113 


9.31806 


10.68194 


10.00920 


9.99080 


15 


46 


30947 


69053 


31870 


68130 


00922 


99078 


14 


47 


31008 


68992 


31933 


68067 


00925 


99075 


13 


48 


31068 


68932 


31996 


68004 


00928 


99072 


12 


49 


31129 


68871 


32059 


67941 


00930 


99070 


11 


50 


9.31189 


10.68811 


9.32122 


10.67878 


10.00933 


9.99067 


10 


51 


31250 


68750 


32185 


67815 


00936 


99064 


9 


52 


31310 


68690 


32248 


67752 


00938 


99062 


8 


53 


31370 


68630 


32311 


67689 


00941 


99059 


7 


54 


31430 


68570 


32373 


67627 


00944 


99056 


6 


55 


9.31490 


10.68510 


9.32436 


10.67564 


10.00946 


9.99054 


5 


56 


31549 


68451 


32498 


67502 


00949 


99051 


4 


57 


31609 


68391 


32561 


67439 


00952 


99048 


8 


58 


31669 


68331 


32623 


67377 


00954 


99046 


2 


59 


31728 


68272 


32685 


67315 


00957 


99043 


1 


60 


31788 


68212 


32747 


67253 


00960 


99040 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



101° 



Logarithmic Angular Functions. 



191 



12° 






Logarithms. 






167° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent 


Secant. 


Cosine. 


M. 





9.31788 


10.68212 


9.32747 


10.67253 


10.00960 


9.99040 


60 


i 


31847 


68153 


32810 


67190 


00962 


99038 


59 


2 


31907 


68093 


32872 


67128 


00965 


99035 


58 


3 


31966 


68034 


32933 


67067 


00968 


99032 


57 


4 


32025 


67975 


32995 


67005 


00970 


99030 


56 


5 


9.32084 


10.67916 


9.33057 


10.66943 


10.00973 


9.99027 


55 


6 


32143 


67857 


33119 


66881 


00976 


99024 


54 


7 


32202 


67798 


33180 


66820 


00978 


99022 


53 


8 


32261 


67739 


33242 


66758 


00981 


99019 


52 


9 


32319 


67681 


33303 


66697 


00984 


99016 


51 


10 


9.32378 


10.67622 


9.33365 


10.66635 


10.00987 


9.99013 


50 


11 


32437 


67563 


33426 


66574 


00989 


99011 


49 


12 


32495 


67505 


33487 


66513 


00992 


99008 


48 


13 


32553 


67447 


33548 


66452 


00995 


99005 


47 


14 


32612 


67388 


33609 


66391 


00998 


99002 


46 


15 


9.32670 


10.67330 


9.33670 


10.66330 


10.01000 


9.99000 


45 


16 


32728 


67272 


33731 


66269 


01003 


98997 


44 


17 


32786 


67214 


33792 


66208 


01006 


98994 


43 


18 


32844 


67156 


33853 


66147 


01009 


98991 


42 


19 


32902 


67098 


33913 


66087 


01011 


98989 


41 


20 


9.32960 


10.67040 


9.33974 


10.66026 


10.01014 


9.98986 


40 


21 


33018 


66982 


34034 


65966 


01017 


98983 


39 


22 


33075 


66925 


34095 


65905 


01020 


98980 


38 


23 


33133 


66867 


34155 


65845 


01022 


98978 


37 


24 


33190 


66810 


34215 


65785 


01025 


98975 


36 


25 


9.33248 


10.66752 


9.34276 


10.65724 


10.01028 


9.98972 


35 


26 


33305 


66695 


34336 


65664 


01031 


98969 


34 


27 


33362 


66638 


34396 


65604 


01033 


98967 


33 


28 


33420 


66580 


34456 


65544 


01036 


98964 


32 


29 


33477 


"66523 


34516 


65484 


01039 


98961 


31 


30 


9.33534 


10.66466 


9.34576 


10.65424 


10.01042 


9.98958 


30 


31 


33591 


66409 


34635 


65365 


01045 


98955 


29 


32 


33647 


66353 


34695 


65305 


01047 


98953 


28 


33 


33704 


66296 


34755 


65245 


01050 


98950 


27 


34 


33761 


66239 


34814 


65186 


01053 


98947 


26 


35 


9.33818 


10.66182 


9.34874 


10.65126 


10.01056 


9.98944 


25 


36 


33874 


66126 


34933 


65067 


01059 


98941 


24 


37 


33931 


66069 


34992 


65008 


01062 


98938 


23 


38 


33987 


66013 


35051 


64949 


01064 


98936 


22 


39 


34043 


65957 


35111 


64889 


01067 


98933 


21 


40 


9.34100 


10.65900 


9.35170 


10.64830 


10.01070 


9.98930 


20 


41 


34156 


65844 


35229 


64771 


01073 


98927 


19 


42 


34212 


65788 


35288 


64712 


01076 


98924 


18 


43 


34268 


65732 


35347 


64653 


01079 


98921 


17 


44 


34324 


65676 


35405 


64595 


01081 


98919 


16 


45 


9.34380 


10.65620 


9.35464 


10.64536 


10.01084 


9.98916 


15 


46 


34436 


65564 


35523 


64477 


01087 


98913 


14 


47 


34491 


65509 


35581 


64419 


01090 


98910 


13 


48 


34547 


65453 


35640 


64360 


01093 


98907 


12 


49 


34602 


65398 


35698 


64302 


01096 


98904 


11 


50 


9.34658 


10.65342 


9.35757 


10.64243 


10.01099 


9.98901 


10 


51 


34713 


65287 


35815 


64185 


01102 


98898 


9 


52 


34769 


65231 


35873 


64127 


01104 


98896 


8 


53 


34824 


65176 


35931 


64069 


01107 


98893 


7 


54 


34879 


65121 


35989 


64011 


OHIO 


98890 


6 


55 


9.34934 


10.65066 


9.36047 


10.63953 


10.01113 


9.98887 


5 


56 


34989 


65011 


36105 


63895 


01116 


98884 


4 


57 


35044 


64956 


36163 


63837 


01119 


98881 


3 


58 


35099 


64901 


36221 


63779 


01122 


98878 


2 


59 


35154 


64846 


36279 


63721 


01125 


98875 


1 


60 


35209 


64791 


36336 


63664 


01128 


98872 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



102° 



77° 



192 



Logarithmic Angular Functions. 



13° 



Logarithms. 



166° 



M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent, 


Secant. 


Cosine. 


M. 





9.35209 


10.64791 


9.36336 


10.6: (664 


10.01128 


9.98872 


60 


1 


35263 


64737 


36394 


63606 


01131 


98869 


59 


2 


35318 


64682 


36452 


63548 


011:5:5 


98867 


58 


3 


35373 


64627 


36509 


6:5191 


011:50 


98864 


57 


4 


35427 


64573 


36566 


63434 


011:59 


988(51 


66 


6 


9.85483 


10.64519 


9.: 56621 


10.63:576 


10.01112 


9.9885S 


55 


6 


35536 


64464 


36681 


63319 


01115 


98855 


54 


7 


35590 


64410 


36738 


6:5262 


01148 


98852 


53 


8 


35644 


64356 


36795 


o:5205 


01151 


98819 


52 


9 


35698 


64302 


36852 


63148 


01154 


98846 


51 


10 


9.35752 


10.64248 


9.86909 


10.63091 


10.01157 


9.9881:5 


50 


11 


35806 


64194 


36966 


6:50:5 1 


01100 


98840 


49 


12 


35860 


64140 


37023 


62977 


01163 


98837 


48 


13 


3591 1 


61086 


37080 


62920 


011(5(5 


98834 


47 


14 


35968 


64032 


37137 


6286:5 


01109 


98831 


46 


15 


9.36022 


10.63978 


9.37193 


L0.62807 


10.01172 


9.98828 


45 


16 


36075 


63925 


37250 


(52750 


01175 


98825 


44 


17 


86129 


6887] 


37306 


62(59 1 


01178 


98822 


43 


18 


36182 


63818 


37363 


62(5:57 


01181 


98S19 


42 


19 


36236 


63764 


37419 


(52581 


01184 


9881(5 


41 


20 


9.86289 


10.6371] 


9.37476 


10.62521 


10.011X7 


9.98818 


40 


21 


36342 


63658 


::75:;2 


62468 


01190 


98810 


39 


22 


36395 


63605 


37588 


62 1 1 2 


01193 


98807 


38 


23 


36449 


63551 


37644 


62356 


01196 


98801 


37 


24 


36502 


63498 


37700 


62300 


01199 


98S01 


36 


26 


9.86555 


10.63445 


9.37756 


10.62244 


10.01202 


9.98798 


86 


26 


36608 


63392 


37812 


62188 


01205 


98795 


84 


27 


36660 


633 10 


37868 


62132 


0120S 


98792 


33 


28 


36713 


63287 


37924 


62076 


01211 


987S9 


82 


29 


36766 


63234 


37980 


(52020 


01214 


98786 


81 


30 


9.36819 


L0.63181 


9.38035 


10.61965 


10.01217 


9.9878:5 


30 


31 


3687] 


63 129 


38091 


01909 


01220 


9S780 


29 


32 


36924 


63076 


381 17 


6185:5 


OI22:: 


98777 


28 


33 


86976 


63024 


38202 


61798 


01226 


9877 1 


27 


34 


37028 


62972 


38257 


61743 


01229 


9S771 


26 


35 


9.87081 


10.0291'.) 


9.88313 


L0.61687 


10.01232 


9.9S708 


25 


36 


37133 


62867 


38368 


61632 


01235 


9S705 


21 


37 


37185 


628 If) 


38423 


61577 


01238 


9S702 


23 




:;72:;7 


6276:', 


: 58 179 


61521 


01241 


98759 


22 


89 


87289 


6271] 


38534 


61466 


01211 


98756 


21 


40 


'J.:;?:; 11 


10.62659 


9.38589 


10.61111 


10.01247 


9.98753 


20 


41 


37393 


62607 


38644 


61356 


01250 


9S750 


19 


42 


87445 


62555 


38699 


61301 


01251 


98746 


IS 


4:5 


87497 


62503 


38754 


61210 


01257 


98743 


17 


•U 


87549 


62451 


38808 


61192 


01260 


9S710 


16 


45 


9.87600 


10.62400 


9.: wo:; 


10.611:57 


10.01263 


9.98787 


15 


46 


37652 


62348 


38918 


61082 


01266 


98784 


14 


47 


87703 


62297 


38972 


0102s 


01209 


98731 


13 


48 


87755 


62245 


: 59027 


0097:; 


01272 


98728 


12 


4'J 


37806 


62191 


39082 


60918 


01275 


9S725 


11 


60 


9.87858 


10.62142 


9.89136 


IO.0OS0I 


10.0 127S 


9.9S722 


10 


51 


87909 


62091 


: 59 190 


60S 10 


01281 


98719 


9 


62 


37960 


(12910 




00755 


01285 


9S7I5 


8 




88011 


61989 


39299 


60701 


012S8 


9S712 


7 


64 


88062 


61938 


39353 


00617 


01291 


9S709 


6 




18118 


10.61887 


9.39407 


10.0059:: 


10.01294 


9.9S706 


6 


66 


38164 




39461 


60539 


01297 


98703 


4 


67 




61785 


39515 


80485 


01300 


9S700 


8 






61734 




60431 


01303 


9S097 


2 










60377 


01306 


98694 


1 


60 










01810 


96690 





M. 


Cosine. 


nit. 


< otangent, 


Tangj 


•cant. 




M. 



103° 



Logarithmic Angular Functions. 



193 



14° 






Logarithms. 




165° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.38368 


10.61632 


• 9.39677 


10.60323 


10.01310 


9.98690 


60 


1 


38418 


61582 


39731 


60269 


01313 


98687 


59 


2 


38469 


61531 


89785 


60215 


01316 


98684 


58 


3 


38519 


61481 


39838 


60162 


01319 


98681 


57 


4 


38570 


61430 


39892 


60108 


01322 


98678 


56 


5 


9.88620 


10.61380 


9.39945 


10.60055 


10.01325 


9.98675 


55 


6 


88670 


61330 


39999 


60001 


01329 


98671 


54 


7 


88721 


61279 


40052 


59948 


01332 


98668 


53 


8 


88771 


61229 


40106 


59894 


01335 


98665 


52 


9 


38821 


61179 


40159 


59841 


01338 


98662 


51 


10 


9.38871 


10.61129 


9.40212 


10.59788 


10.01341 


9.98659 


50 


11 


88921 


61079 


40266 


59734 


01344 


98656 


49 


12 


38971 


61029 


40319 


59681 


01348 


98652 


48 


13 


39021 


60979 


40372 


59628 


01351 


98649 


47 


14 


39071 


60929 


40425 


59575 


01354 


98646 


46 


15 


9.39121 


10.60879 


9.40478 


10.59522 


10.01357 


9.98643 


45 


16 


39170 


60830 


40531 


59469 


01360 


98640 


44 


17 


39220 


60780 


40584 


59416 


01364 


98636 


43 


18 


39270 


60730 


40636 


59364 


01367 


98633 


42 


19 


39319 


60681 


40689 


59311 


01370 


98630 


41 


20 


9.39369 


10.6063] 


9.40742 


10.59258 


10.01373 


9.98627 


40 


21 


39418 


60582 


40795 


59205 


01377 


98623 


39 


22 


39467 


60533 


40847 


59153 


01380 


98620 


38 


23 


39517 


60483 


40900 


59100 


01383 


98617 


37 


24 


39566 


60434 


40952 


59048 


01386 


98614 


36 


25 


9.39615 


10.60385 


9.41005 


10.58995 


10.01390 


9.98610 


35 


26 


39664 


60336 


41057 


58943 


01393 


98607 


34 


27 


39713 


60287 


41109 


58891 


01396 


98604 


33 


28 


39762 


60238 


41161 


58839 


01399 


98601 


32 


29 


39811 


60189 


41214 


58786 


01403 


98597 


31 


30 


9.39860 


10.60140 


9.41266 


10.58734 


10.01406 


9.98594 


30 


31 


39909 


60091 


41318 


58682 


01409 


98591 


29 


32 


39958 


60042 


41370 


58630 


01412 


98588 


28 


33 


40006 


59994 


41422 


58578 


01416 


98584 


27 


34 


40055 


59945 


41474 


58526 


01419 


98581 


26 


35 


9.40103 


10.59897 


9.41526 


10.58474 


10.01422 


9.98578 


25 


36 


40152 


59848 


41578 


58422 


01426 


98574 


24 


37 


40200 


59800 


41629 


58371 


01429 


98571 


23 


38 


40249 


59751 


41681 


58319 


01432 


98568 


22 


39 


40297 


59703 


41733 


58267 


01435 


98565 


21 


40 


9.40346 


10.59654 


9.41784 


10.58216 


10.01439 


9.98561 


20 


41 


40394 


59606 


41836 


58164 


01442 


98558 


19 


42 


40442 


59558 


41887 


58113 


01445 


98555 


18 


43 


40490 


59510 


41939 


58061 


01449 


98551 


17 


44 


40538 


59462 


41990 


58010 


01452 


98548 


16 


45 


9.40586 


10.59414 


9.42041 


10.57959 


10.01455 


9.98545 


15 


46 


40634 


59366 


42093 


57907 


01459 


98541 


14 


47 


■10682 


59318 


42144 


57856 


01462 


98538 


13 


48 


40730 


59270 


42195 


57805 


01465 


98535 


12 


49 


40778 


59222 


42246 


57754 


01469 


98531 


11 


50 


9.40825 


10.59175 


9.42297 


10.57703 


10.01472 


9.98528 


10 


51 


40873 


59127 


42348 


57652 


01475 


98525 


9 


52 


4092* 


59079 


42399 


57601 


01479 


98521 


8 


53 


40968 


59032 


42450 


57550 


01482 


98518 


7 


54 


41016 


58984 


42501 


57499 


01485 


98515 


6 


55 


9.41063 


10.58937 


9.42552 


10.57448 


10.01489 


9.98511 


5 


56 


41111 


58889 


42603 


57397 


01492 


98508 


4 


57 


41158 


58842 


42653 


57347 


01495 


98505 


3 


58 


41205 


58795 ! 


42704 


57296 


01499 


98501 


2 


59 


41252 


58748 ! 


42755 


57245 


01502 


98498 


1 


60 


41300 


58700 


42805 


57195 


01506 


98494 





M. 


Cosine. 


Secant, i 


| Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



104° 



75° 



13 



194 



Logarithmic Angular Functions. 



15° 






Logarithms. 






164° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.41300 


10.58700 


9.42805 


10.57195 - 


10.01506 


9.98494 


60 


1 


41347 


58653 


42856 


57144 


01509 


98491 


59 


2 


41394 


58606 


42906 


57094 


01512 


98488 


58 


3 


41441 


58559 


42957 


57043 


01516 


98484 


57 


4 


41488 


58512 


43007 


56993 


01519 


98481 


56 


5 


9.41535 


10.58465 


9.43057 


10.56943 


10.01523 


9.98477 


55 


6 


41582 


58418 


43108 


56892 


01526 


98474 


54 


7 


41628 


58372 


43158 


56842 


01529 


98471 


53 


8 


41675 


58325 


43208 


56792 


01533 


98467 


52 


9 


41722 


58278 


43258 


56742 


01536 


98464 


51 


10 


9.41768 


10.58232 


9.43308 


10.56692 


10.01540 


9.98460 


50 


11 


41815 


58185 


43358 


56642 


01543 


98457 


49 


12 


41861 


58139 


43408 


56592 


01547 


98453 


48 


13 


41908 


58092 


43458 


56542 


01550 


98450 


47 


14 


41954 


58046 


43508 


56492 


01553 


98447 


46 


15 


9.42001 


10.57999 


9.43558 


10.56442 


10.01557 


9.98443 


45 


16 


42047 


57953 


43607 


56393 


01560 


98440 


44 


17 


42093 


57907 


43657 


56343 


01564 


98436 


43 


18 


42140 


57860 


43707 


56293 


01567 


98433 


42 


19 


42186 


57814 


43756 


56244 


01571 


98429 


41 


20 


9.42232 


10.57768 


9.43806 


10.56194 


10.01574 


9.98426 


40 


21 


42278 


57722 


43855 


56145 


01578 


98422 


39 


22 


42324 


57676 


43905 


56095 


01581 


98419 


38 


23 


42370 


57630 


43954 


56046 


01585 


98415 


37 


24 


42416 


57584 


44004 


55996 


01588 


98412 


36 


25 


9.42461 


10.57539 


9.44053 


10.55947 


10.01591 


9.98409 


35 


26 


42507 


57493 


44102 


55898 


01595 


98405 


34 


27 


42553 


57447 


44151 


55849 


01598 


98402 


33 


28 


42599 


57401 


44201 


55799 


01602 


98398 


32 


29 


42644 


57356 


44250 


55750 


01605 


98395 


31 


30 


9.42690 


10.57310 


9.44299 


10.55701 


10.01609 


9.98391 


30 


31 


42735 


57265 


44348 


55652 


01612 


98388 


29 


32 


42781 


57219 


44397 


55603 


01616 


98384 


28 


33 


42826 


57174 


44446 


55554 


01619 


98381 


27 


34 


42872 


57128 


44495 


55505 


01623 


98377 


26 


35 


9.42917 


10.57083 


9.44544 


10.55456 


10.01627 


9.98373 


25 


36 


42962 


57038 


44592 


55408 


01630 


98370 


24 


37 


43008 


56992 


44641 


55359 


01634 


98366 


23 


38 


43053 


56947 


44690 


55310 


01637 


98363 


22 


39 


43098 


56902 


44738 


55262 


01641 


98359 


21 


40 


9.43143 


10.56857 


9.44787 


10.55213 


10.01644 


9.98356 


20 


41 


43188 


56812 


44836 


55164 


01648 


98352 


19 


42 


43233 


56767 


44884 


55116 


01651 


98349 


18 


43 


43278 


56722 


44933 


55067 


01655 


98345 


17 


44 


43323 


56677 


44981 


55019 


01658 


98342 


16 


45 


9.43367 


10.56633 


9.45029 


10.54971 


10.01662 


9.98338 


15 


46 


43412 


56588 


45078 


54922 


01666 


98334 


14 


47 


43457 


56543 


45126 


54874 


01669 


98331 


13 


48 


43502 


56498 


45174 


54826 


01673 


98327 


12 


49 


43546 


56454 


45222 


54778 


01676 


98324 


11 


50 


9.43591 


10.56409 


9.45271 


10.54729 


10.01680 


9.98320 


10 


51 


43635 


56365 


45319 


54681 


01683 


98317 


9 


52 


43680 


56320 


45367 


54633 


01687 


98313 


8 


53 


43724 


56276 


45415 


54585 


01691 


98309 


7 


54 


43769 


56231 


45463 


54537 


01694 


98306 


6 


55 


9.43813 


10.56187 


9.45511 


10.54489 


10.01698 


9.98302 


5 


56 


13857 


561 13 


45559 


54441 


01701 


98299 


4 


57 


13901 


56099 


45606 


54394 


01705 


98295 


3 


58 


13946 


56054 


15654 


54346 


01709 


98291 


2 


59 


43990 


56010 


15702 


54298 


01712 


98288 


1 


60 


44034 


55966 


45750 


54250 


01716 


98284 





M. 


( loaine. 


Secant. 


1 Sotangent, 


Tangent. 


Cosecant. 


Sine. 


M. 



105° 



74° 



Logarithmic Angular Functions. 



195 



16° 






Logarithms. 




163° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 


, 


9.44034 


10.55966 


9.45750 


10.54250 


10.01716 


9.98284 


60 


1 


44078 


55922 


45797 


54203 


01719 


98281 


59 


2 


44122 


55878 


45845 


54155 


01723 


98277 


58 


3 


44166 


55834 


45892 


54108 


01727 


98273 


57 


4 


44210 


55790 


45940 


54060 


01730 


98270 


56 


5 


9.44253 


10.55747 


9.45987 


10.54013 


10.01734 


9.98266 


55 


6 


44297 


55703 


46035 


53965 


01738 


98262 


54 


7 


44341 


55659 


46082 


53918 


01741 


98259 


53 


8 


44385 


55615 


46130 


53870 


01745 


98255 


52 


9 


44428 


55572 


46177 


53823 


01749 


98251 


51 


10 


9.44472 


10.55528 


9.46224 


10.53776 


10.01752 


9.98248 


50 


11 


44516 


55484 


46271 


53729 


01756 


98244 


49 


12 


44559 


55441 


46319 


53681 


01760 


98240 


48 


13 


44602 


55398 


46366 


53634 


01763 


98237 


47 


14 


44646 


55354 


46413 


53587 


01767 


98233 


46 


15 


9.44689 


10.55311 


9.46460 


10.53540 


10.01771 


9.98229 


45 


16 


44733 


55267 


46507 


53493 


01774 


98226 


44 


17 


44776 


55224 


46554 


53446 


01778 


98222 


43 


18 


44819 


55181 


46601 


53399 


01782 


98218 


42 


19 


44862 


55138 


46648 


53352 


01785 


98215 


41 


20 


9.44905 


10.55095 


9.46694 


10.53306 


10.01789 


9.98211 


40 


21 


44948 


55052 


46741 


53259 


01793 


98207 


39 


22 


44992 


55008 


46788 


53212 


01796 


98204 


38 


23 


45035 


54965 


46835 


53165 


01800 


98200 


37 


24 


45077 


54923 


46881 


53119 


01804 


98196 


36 


25 


9.45120 


10.54880 


9.46928 


10.53072 


10.01808 


9.98192 


35 


26 


45163 


54837 


46975 


53025 


01811 


98189 


34 


27 


45206 


54794 


47021 


52979 


01815 


98185 


33 


28 


45249 


54751 


47068 


52932 


01819 


98181 


32 


29 


45292 


54708 


47114 


52886 


01823 


98177 


31 


30 


9.45334 


10.54666 


9.47160 


10.52840 


10.01826 


9.98174 


30 


31 


45377 


54623 


47207 


52793 


01830 


98170 


29 


32 


45419 


54581 


47253 


52747 


01834 


98166 


28 


33 


45462 


54538 


47299 


52701 


01838 


98162 


27 


34 


45504 


54496 


47346 


52654 


01841 


98159 


26 


35 


9.45547 


10.54453 


9.47392 


10.52608 


10.01845 


9.98155 


25 


36 


45589 


54411 


47438 


52562 


01849 


98151 


24 


37 


45632 


54368 


47484 


52516 


01853 


98147 


23 


38 


45674 


54326 


47530 


52470 


01856 


98144 


22 


39 


45716 


54284 


47576 


52424 


01860 


98140 


21 


40 


9.45758 


10.54242 


9.47622 


10.52378 


10.01864 


9.98136 


20 


41 


45801 


54199 


47668 


52332 


01868 


98132 


19 


42 


45843 


54157 


47714 


52286 


01871 


98129 


18 


43 


45885 


54115 


47760 


52240 


01875 


98125 


17 


44 


45927 


54073 


47806 


52194 


01879 


98121 


16 


45 


9.45969 


10.54031 


9.47852 


10.52148 


10.01883 


9.98117 


15 


46 


46011 


53989 


47897 


52103 


01887 


98113 


14 


47 


46053 


53947 


47943 


52057 


01890 


98110 


13 


48 


46095 


53905 


47989 


52011 


01894 


98106 


12 


49 


46136 


53864 


48035 


51965 


01898 


98102 


11 


50 


9.46178 


10.53822 


9.48080 


10.51920 


10.01902 


9.98098 


10 


51 


46220 


53780 


48126 


51874 


01906 


98094 


9 


52 


46262 


53738 


48171 


51829 


01910 


98090 


8 


53 


46303 


53697 


48217 


51783 


01913 


98087 


7 


54 


46345 


53655 


48262 


51738 


01917 


98083 


6 


55 


9.46386 


10.53614 


9.48307 


10.51693 


10.01921 


9.98079 


5 


56 


46428 


53572 


48353 


51647 


01925 


98075 


4 


57 


46469 


53531 


48398 


51602 


01929 


98071 


3 


58 


46511 


53489 


48443 


51557 


01933 


98067 


2 


59 


46552 


53448 


48489 


51511 


01937 


98063 


1 


60 


46594 


53406 


48534 


51466 


01940 


98060 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



73° 



196 



Logarithmic Angular Functions. 



17° 



Logarithms. 



162° 



M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


j Secant. 


Cosine. 


M. 





9.46594 


10.53406 


9.48534 


10.51466 


10.01940 


9.98060 


60 


1 


46635 


53365 


48579 


51421 


01944 


98056 


59 


2 


46676 


53324 


48624 


51376 


01948 


98052 


58 


3 


46717 


53283 


48669 


51331 


01952 


98048 


57 


4 


46758 


53242 


48714 


51286 


01956 


98044 


56 


5 


9.46800 


10.53200 


9.48759 


10.51241 


10.01960 


9.98040 


55 


6 


46841 


53159 


48804 


51196 


01964 


98036 


54 


7 


46882 


53118 


48849 


51151 


01968 


98032 


53 


8 


46923 


53077 


48894 


51106 


01971 


98029 


52 


9 


46964 


53036 


48939 


51061 


01975 


98025 


51 


10 


9.47005 


10.52995 


9.48984 


10.51016 


10.01979 


9.98021 


50 


11 


47045 


52955 


49029 


50971 


01983 


98017 


49 


12 


47086 


52914 


49073 


50927 


01987 


98013 


48 


13 


47127 


52873 


49118 


50882 


01991 


98009 


47 


14 


47168 


52832 


49163 


50837 


01995 


98005 


46 


15 


9.47209 


10.52791 


9.49207 


10.50793 


10.01999 


9.98001 


45 


16 


47249 


52751 


49252 


50748 


02003 


97997 


44 


17 


47290 


52710 


49296 


50704 


02007 


97993 


43 


18 


47330 


52670 


49341 


50659 


02011 


97989 


42 


19 


47371 


52629 


49385 


50615 


02014 


97986 


41 


20 


9.47411 


10.52589 


9.49430 


10.50570 


10.02018 


9.97982 


40 


21 


47452 


52548 


49474 


50526 


02022 


97978 


39 


22 


47492 


52508 


49519 


50481 


02026 


97974 


38 


23 


47533 


52467 


49563 


50437 


02030 


97970 


37 


24 


47573 


52427 


49607 


50393 


02034 


97966 


36 


25 


9.47613 


10.52387 


9.49652 


10.50348 


10.02038 


9.97962 


35 


26 


47654 


52346 


49696 


50304 


02042 


97958 


34 


27 


47694 


52306 


49740 


50260 


02046 


97954 


33 


28 


47734 


52266 


49784 


50216 


02050 


97950 


32 


29 


47774 


52226 


49828 


50172 


02054 


97946 


31 


30 


9.47814 


10.52186 


9.49872 


10.50128 


10.02058 


9.97942 


30 


31 


47854 


52146 


49916 


50084 


02062 


97938 


29 


32 


47894 


52106 


49960 


50040 


02066 


97934 


28 


33 


47934 


52066 


50004 


49996 


02070 


97930 


27 


34 


47974 


52026 


50048 


49952 


02074 


97926 


26 


35 


9.48014 


10.51986 


9.50092 


10.49908 


10.02078 


9.97922 


25 


36 


48054 


51946 


50136 


49864 


02082 


97918 


24 


37 


48094 


51906 


50180 


49820 


02086 


97914 


23 


38 


48133 


51867 


50223 


49777 


02090 


97910 


22 


39 


48173 


51827 


50267 


49733 


02094 


97906 


21 


40 


9.48213 


10.51787 


9.50311 


10.49689 


10.02098 


9.97902 


20 


41 


48252 


51748 


50355 


49645 


02102 


97898 


19 


42 


48292 


51708 


50398 


49602 


02106 


97894 


18 


43 


48332 


51668 


50442 


49558 


02110 


97890 


17 


44 


48371 


51629 


50485 


49515 


02114 


97886 


16 


45 


9.48411 


10.51589 


9.50529 


10.49471 


10.02118 


9.97882 


15 


46 


48450 


51550 


50572 


49428 


02122 


97878 


14 


47 


48490 


51510 


50616 


49384 


02126 


97874 


13 


48 


48529 


51471 


50659 


49341 


02130 


97870 


12 


49 


48568 


51432 


50703 


49297 


02134 


97866 


11 


50 


9. 18607 


10.51393 


•9.50746 


10.49254 


10.02139 


9.97861 


10 


51 


48647 


51353 


50789 


49211 


02143 


97857 


9 


52 


18686 


51314 


50833 


49167 


02147 


97853 


8 


53 


18725 


51275 


50876 


49124 


02151 


97849 


7 


54 


48764 


51236 


50919 


49081 


02155 


97845 


6 


55 


9.48803 


10.51197 


9.50962 


10.49038 


10.02159 


9.97841 


5 


56 


18842 


51158 


51005 


48995 


02163 


97837 


4 


57 


18881 


51111) 


51048 


48952 


02167 


97833 


3 


58 


48920 


51080 


51092 


48908 


02171 


97829 


2 


69 


18959 


51041 


51135 


48865 


02175 


97825 


1 


60 


18998 


51002 


51178 


48822 


02179 


97821 





M. 


Cosine. 


nit. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



107° 



72° 



Logarithmic Angular Functions. 



197 



18° 






Logarithms. 






161° 


M. 


Sine. 


Cosecant. 


' Tangent. 


Cotangent. 


I Secant. 


Cosine. 


M. 





9.48998 


10.51002 


9.51178 


10.48822 


10.02179 


9.97821 


60 


1 


49037 


50963 


51221 


48779 


02183 


97817 


59 


2 


49076 


50924 


51264 


48736 


02188 


97812 


58 


3 


49115 


50885 


51306 


48694 


02192 


97808 


57 


4 


49153 


50847 


51349 


48651 


02196 


97804 


56 


5 


9.49192 


10.50808 


9.51392 


10.48608 


10.02200 


9.97800 


55 


6 


49231 


50769 


51435 


48565 


02204 


97796 


54 


7 


49269 


50731 


51478 


48522 


02208 


97792 


53 


8 


49308 


50692 


51520 


48480 


02212 


97788 


52 


9 


49347 


50653 


51563 


48437 


02216 


97784 


51 


10 


9.49385 


10.50615 


9.51606 


10.48394 


10.02221 


9.97779 


50 


11 


49424 


50576 


51648 


48352 


02225 


97775 


49 


12 


49462 


50538 


51691 


48309 


02229 


97771 


48 


13 


49500 


50500 


51734 


48266 


02233 


97767 


47 


14 


49539 


50461 


51776 


48224 


02237 


97763 


46 


15 


9.49577 


10.50423 


9.51819 


10.48181 


10.02241 


9.97759 


45 


16 


49615 


50385 


51861 


48139 


02246 


97754 


44 


17 


49654 


50346 


51903 


48097 


02250 


97750 


43 


18 


49692 


50308 


51946 


48054 


02254 


97746 


42 


19 


49730 


50270 


51988 


48012 


02258 


97742 


41 


20 


9.49768 


10.50232 


9.52031 


10.47969 


10.02262 


9.97738 


40 


21 


49806 


50194 


52073 


47927 


02266 


97734 


39 


22 


49844 


50156 


52115 


47885 


02271 


97729 


38 


23 


49882 


50118 


52157 


47843 


02275 


97725 


37 


24 


49920 


50080 


52200 


47800 


02279 


97721 


36 


25 


9.49958 


10.50042 


9.52242 


10.47758 


10.02283 


9.97717 


35 


26 


49996 


50004 


52284 


47716 


02287 


97713 


34 


27 


50034 


49966 


52326 


47674 


02292 


97708 


33 


28 


50072 


49928 


52368 


47632 


02296 


97704 


32 


29 


50110 


49890 


52410 


47590 


02300 


97700 


31 


30 


9.50148 


10.49852 


9.52452 


10.47548 ! 


10.02304 


9.97696 


30 


31 


50185 


49815 


52494 


47506 


02309 


97691 


29 


32 


50223 


49777 


52536 


47464 ; 


02313 


97687 


28 


33 


50261 


49739 


52578 


47422 


02317 


97683 


27 


34 


50298 


49702 


52620 


47380 I 


02321 


97679 


26 . 


35 


9.50336 


10.49664 


9.52661 


10.47339 i 


10.02326 


9.97674 


25 


36 


50374 


49626 


52703 


47297 


02330 


97670 


24 


37 


50411 


49589 


52745 


47255 


02334 


97666 


23 


38 


50449 


49551 


52787 


47213 


02338 


97662 


22 


39 


50486 


49514 


52829 


47171 


02343 


97657 


21 


40 


9.50523 


10.49477 


9.52870 


10.47130 


10.02347 


9.97653 


20 


41 


50561 


49439 


52912 


47088 


02351 


97649 


19 


42 


50598 


49402 


52953 


47047 


02355 


97645 


18 


43 


50635 


49365 


52995 


47005 


02360 


97640 


17 


44 


50673 


49327 


53037 


46963 


02364 


97636 


16 


45 


9.50710 


10.49290 


9.53078 


10.46922 


10.02368 


9.97632 


15 


46 


50747 


49253 


53120 


46880 


02372 


97628 


14 


47 


50784 


49216 


53161 


46839 


02377 


97623 


13 


48 


50821 


49179 


53202 


46798 


02381 


97619 


12 


49 


50858 


49142 


53244 


46756 


02385 


97615 


11 


50 


9.50896 


10.49104 


9.53285 


10.46715 


10.02390 


9.97610 


10 


51 


50933 


49067 


53327 


46673 


02394 


97606 


9 


52 


50970 


49030 


53368 


46632 


02398 


97602 


8 


53 


51007 


48993 


53409 


46591 


02403 


97597 


7 


54 


51043 


48957 


53450 


46550 


02407 


97593 


6 


55 


9.51080 


10.48920 


9.53492 


10.46508 


10.02411 


9.97589 


5 


56 


51117 


48883 


53533 


46467 


02416 


97584 


4 


57 


51154 


48846 


53574 


46426 


02420 


97580 


3 


58 


51191 


48809 


53615 


46385 


02424 


97576 


2 


59 


51227 


48773 


53656 


46344 


02429 


97571 


1 


60 


51264 


48736 


53697 


46303 


02433 


97567 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. | 


Cosecant. 


Sine. 


M. 



108° 



71° 



198 



Logarithmic Angular Functions. 



19° 






Logarithms. 




160° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.51264 


10.48736 


9.53697 


10.46303 


10.02433 


9.97567 


60 


1 


51301 


48699 


53738 


46262 


02437 


97563 


59 


2 


51338 


48662 


53779 


46221 


02442 


97558 


58 


3 


51374 


48626 


53820 


46180 


02446 


97554 


57 


4 


51411 


48589 


53861 


46139 


02450 


97550 


56 


5 


9.51447 


10.48553 


9.53902 


10.46098 


10.02455 


9.97545 


55 


6 


51484 


48516 


53943 


46057 


02459 


97541 


54 


7 


51520 


48480 


53984 


46016 


02464 


97536 


53 


8 


51557 


48443 


54025 


45975 


02468 


97532 


52 


9 


51593 


48407 


54065 


45935 


02472 


97528 


51 


10 


9.51629 


10.48371 


9.54106 


10.45894 


10.02477 


9.97523 


50 


11 


51666 


48334 


54147 


45853 


02481 


97519 


49 


12 


51702 


48298 


54187 


45813 


02485 


97515 


48 


13 


51738 


48262 


54228 


45772 


02490 


97510 


47 


14 


51774 


48226 


54269 


45731 


02494 


97506 


46 


15 


9.51811 


10.48189 


9.54309 


10.45691 


10.02499 


9.97501 


45 


16 


51847 


48153 


54350 


45650 


02503 


97497 


44 


17 


51883 


48117 


54390 


45610 


02508 


97492 


43 


18 


51919 


48081 


54431 


45569 


02512 


97488 


42 


19 


51955 


48045 


54471 


45529 


02516 


97484 


41 


20 


9.51991 


10.48009 


9.54512 


10.45488 


10.02521 


9.97479 


40 


21 


52027 


47973 


54552 


45448 


02525 


97475 


39 


22 


52063 


47937 


54593 


45407 


02530 


97470 


38 


23 


52099 


47901 


54633 


45367 


02534 


97466 


37 


24 


52135 


47865 


54673 


45327 


02539 


97461 


36 


25 


9.52171 


10.47829 


9.54714 


10.45286 


10.02543 


9.97457 


35 


26 


52207 


47793 


54754 


45246 


02547 


97453 


34 


27 


52242 


47758 


54794 


45206 


02552 


97448 


33 


28 


52278 


47722 


54835 


45165 


02556 


97444 


32 


29 


52314 


47686 


54875 


45125 


02561 


97439 


31 


30 


9.52350 


10.47650 


9.54915 


10.45085 


10.02565 


9.97435 


30 


31 


52385 


47615 


54955 


45045 


02570 


97430 


29 


32 


52421 


47579 


54995 


45005 


02574 


97426 


28 


33 


52456 


47544 


55035 


44965 


02579 


97421 


27 


34 


52492 


47508 


55075 


44925 


02583 


97417 


26 


35 


9.52527 


10.47473 


9.55115 


10.44885 


10.02588 


9.97412 


25 


36 


52563 


47437 


55155 


44845 


02592 


97408 


24 


37 


52598 


47402 


55195 


44805 


02597 


97403 


23 


38 


52634 


47366 


55235 


44765 


02601 


97399 


22 


39 


52669 


47331 


55275 


44725 


02606 


97394 


21 


40 


9.52705 


10.47295 


9.55315 


10.44685 


10.02610 


9.97390 


20 


41 


52740 


' 47260 


55355 


44645 


02615 


97385 


19 


42 


52775 


47225 


55395 


44605 


02619 


97381 


18 


43 


52811 


47189 


55434 


44566 


02624 


97376 


17 


44 


52846 


47154 


55474 


44526 


02628 


97372 


16 


45 


9.52881 


10.47119 


9.55514 


10.44486 


10.02633 


9.97367 


15 


46 


5291 (i 


47084 


55554 


44446 


02637 


97363 


14 


47 


52951 


47049 


55593 


44407 


02642 


97358 


13 


48 


52986 


47014 


55633 


44367 


02647 


97353 


12 


49 


53021 


46979 


55673 


44327 


02651 


97349 


11 


50 


9.53056 


10.46944 


9.55712 


10.44288 


10.02656 


9.97344 


10 


51 


53092 


46908 


55752 


44248 


02660 


97340 


9 


52 


53126 


46874 


55791 


44209 


02665 


97335 


8 


53 


53161 


46839 


55831 


44169 


02669 


97331 


7 


54 


53196 


16804 


55870 


44130 


02674 


97326 


6 


55 


9.53231 


10.46769 


9.55910 


10.44090 


10.02678 


9.97322 


5 


56 


53266 


4(17:: 1 


55949 


44051 


02683 


97317 


4 


57 


53301 


46699 


55989 


44011 


02688 


97312 


3 


58 


53336 


46664 


56028 


43972 


02692 


97308 


2 


59 


53370 


46630 


56067 


43933 


02697 


97303 


1 


60 


53405 


46595 


56107 


43893 


02701 


97299 





M. 


1 ioedne. 


Secant. 


Cotangent, 


Tangent. 


Cosecant. 


Sine. 


M. 



109° 



LOGARITHMIC ANGULAR FUNCTIONS. 



199 



20° 






Logarithms. 




159° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.53405 


10.46595 


9.56107 


10.43893 


10.02701 


9.97299 


60 


1 


53440 


46560 


56146 


43854 


02706 


97294 


59 


2 


53475 


46525 


56185 


43815 


02711 


97289 


58 


3 


53509 


46491 


56224 


43776 


02715 


97285 


57 


4 


53544 


46456 


56264 


43736 


02720 


97280 


56 


5 


9.53578 


10.46422 


9.56303 


10.43697 


10.02724 


9.97276 


55 


6 


53613 


46387 


56342 


43658 


02729 


97271 


54 


7 


53647 


46353 


56381 


43619 


02734 


97266 


53 


8 


53682 


46318 


56420 


43580 


02738 


97262 


52 


9 


53716 


46284 


56459 


43541 


02743 


97257 


51 


10 


9.53751 


10.46249 


9.56498 


10.43502 


10.02748 


9.97252 


50 


11 


53785 


46215 


56537 


43463 


02752 


97248 


49 


12 


53819 


46181 


56576 


43424 


02757 


97243 


48 


13 


53854 


46146 


56615 


43385 


02762 


97238 


47 


14 


53888 


46112 


56654 


43346 


02766 


97234 


46 


15 


9.53922 


10.46078 


9.56693 


10.43307 


10.02771 


9.97229 


45 


16 


53957 


46043 


56732 


43268 


02776 


97224 


44 


17 


53991 


46009 


56771 


. 43229 


02780 


97220 


43 


18 


54025 


45975 


56810 


43190 


02785 


97215 


42 


19 


54059 


45941 


56849 


43151 


02790 


97210 


41 


20 


9.54093 


10.45907 


9.56887 


10.43113 


10.02794 


9.97206 


40 


21 


54127 


45873 


56926 


43074 


02799 


97201 


39 


22 


54161 


45839 


56965 


43035 


02804 


97196 


38 


23 


54195 


45805 


57004 


42996 


02808 


97192 


37 


24 


54229 


45771 


57042 


42958 


02813 


97187 


36 


25 


9.54263 


10.45737 


9.57081 


10.42919 


10.02818 


9.97182 


35 


26 


54297 


45703 


57120 


42880 


02822 


97178 


34 


27 


54331 


45669 


57158 


42842 


02827 


97173 


33 


28 


54365 


45635 


57197 


42803 


02832 


97168 


32 


29 


54399 


45601 


57235 


42765 


02837 


97163 


31 


30 


9.54433 


10.45567 


9.57274 


10.42726 


10.02841 


9.97159 


30 


31 


54466 


45534 


57312 


42688 


02846 


97154 


29 


32 


54500 


45500 


57351 


42649 


02851 


97149 


28 


33 


54534 


45466 


57389 


42611 


02855 


97145 


27 


34 


54567 


45433 


57428 


42572 


02860 


97140 


26 


35 


9.54601 


10.45399 


9.57466 


10.42534 


10.02865 


9.97135 


25 


36 


54635 


45365 


57504 


42496 


02870 


97130 


24 


37 


54668 


45332 


57543 


42457 


02874 


97126 


23 


38 


54702 


45298 


57581 


42419 


02879 


97121 


22 


39 


54735 


45265 


57619 


42381 


02884 


97116 


21 


40 


9.54769 


10.45231 


9.57658 


10.42342 


10.02889 


9.97111 


20 


41 


54802 


45198 


57696 


42304 


02893 


97107 


19 


42 


54836 


45164 


57734 


42266 


02898 


97102 


18 


43 


54869 


45131 


57772 


-42228 


02903 


97097 


17 


44 


54903 


45097 


57810 


42190 


02908 


97092 


16 


45 


9.54936 


10.45064 


9.57849 


10.42151 


10.02913 


9.97087 


15 


46 


54969 


45031 


57887 


42113 


02917 


97083 


14 


47 


55003 


44997 


57925 


42075 


02922 


97078 


13 


48 


55036 


44964 


57963 


42037 


02927 


97073 


12 


49 


55069 


44931 


58001 


41999 


02932 


97068 


11 


50 


9.55102 


10.44898 


9.58039 


10.41961 


10.02937 


9.97063 


10 


51 


55136 


44864 


58077 


41923 


02941 


97059 


9 


52 


55169 


44831 


58115 


41885 


02946 


97054 


8 


53 


55202 


44798 


58153 


41847 


02951 


97049 


7 


54 


55235 


44765 


58191 


41809 


02956 


97044 


6 


55 


9.55268 


10.44732 


9.58229 


10.41771 


10.02961 


9.97039 


5 


56 


55301 


44699 


58267 


41733 


02965 


97035 


4 


57 


55334 


44666 


58304 


41696 


02970 


97030 


3 


58 


55367 


44633 


58342 


41658 


02975 


97025 


2 


59 


55400 


44600 


58380 


41620 


02980 


97020 


1 


60 


55433 


44567 


58418 


41582 


02985 


97015 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



110° 



69° 



JOO 



Logarithmic Angular Functions. 



21° 






Logarithms. 




158° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


1 Secant. 


Cosine. 


M. 





9.55433 


10.44567 


9.58418 


10.41582 


10.02985 


9.97015 


60 


1 


55466 


44534 


58455 


41545 


02990 


97010 


59 


2 


55499 


44501 


58493 


41507 


02995 


97005 


58 


3 


55532 • 


44468 


58531 


41469 


02999 


97001 


57 


4 


55564 


44436 


58569 


41431 


03004 


96996 


56 


5 


9.55597 


10.44403 


9.58606 


10.41394 


10.03009 


9.96991 


55 


6 


55630 


44370 


58644 


41356 


03014 


96986 


54 


7 


55663 


44337 


58681 


41319 


03019 


96981 


53 


8 


55695 


44305 


58719 


41281 


03024 


96976 


52 


9 


55728 


44272 


58757 


41243 


03029 


96971 


51 


10 


9.55761 


10.44239 


9.58794 


10.41206 


10.03034 


9.96966 


50 


11 


55793 


44207 


58832 


41168 


03038 


96962 


49 


12 


55826 


44174 


58869 


41131 


03043 


96957 


48 


13 


55858 


44142 


58907 


41093 


03048 


96952 


47 


14 


55891 


44109 


58944 


41056 


03053 


96947 


46 


15 


9.55923 


10.44077 


9.58981 


10.41019 


10.03058 


9.96942 


45 


16 


55956 


44044 


59019 


40981 


03063 


96937 


44 


17 


55988 


44012 


59056 


40944 


03068 


96932 


43 


18 


56021 


43979 


59094 


40906 


03073 


96927 


42 


19 


56053 


43947 


59131 


40869 


03078 


96922 


41 


20 


9.56085 


10.43915 


9.59168 


10.40832 


10.03083 


9.96917 


40 


21 


56118 


43882 


59205 


40795 


03088 


96912 


39 


22 


56150 


43850 


59243 


40757 


03093 


96907 


38 


23 


56182 


43818 


59280 


40720 


03097 


96903 


37 


24 


56215 


43785 


59317 


40683 


03102 


96898 


36 


25 


9.56247 


10.43753 


9.59354 


10.40646 


10.03107 


9.96893 


35 


26 


56279 


43721 


59391 


40609 


03112 


96888 


34 


27 


56311 


43689 


59429 


40571 


03117 


96883 


33 


28 


56343 


43657 


59466 


40534 


03122 


96878 


32 


29 


56375 


43625 


59503 


40497 


03127 


96873 


31 


30 


9.56408 


10.43592 


9.59540 


10.40460 


10.03132 


9.96868 


30 


31 


56440 


43560 


59577 


40423 


03137 


96863 


29 


32 


56472 


43528 


59614 


40386 


03142 


96858 


28 


33 


56504 


43496 


59651 


40349 


03147 


96853 


27 


34 


56536 


43464 


59688 


40312 


03152 


96848 


26 


35 


9.56568 


10.43432 


9.59725 


10.40275 


10.03157 


9.96843 


25 


36 


56599 


43401 


59762 


40238 


03162 


96838 


24 


37 


56631 


43369 


59799 


40201 


03167 


96833 


23 


38 


56663 


43337 


59835 


40165 


03172 


96828 


22 


39 


56695 


43305 


59872 


40128 


03177 


96823 


21 


40 


9.56727 


10.43273 


9.59909 


10.40091 


10.03182 


9.96818 


20 


41 


56759 


43241 


59946 


40054 


03187 


96813 


19 


42 


56790 


43210 


59983 


40017 


03192 


96808 


18 


43 


56822 


43178 


60019 


39981 


03197 


96803 


17 


44 


56854 


43146 


60056 


39944 


03202 


96798 


16 


45 


9.56886 


10.43114 


9.60093 


10.39907 


10.03207 


9.96793 


15 


46 


56917 


43083 


60130 


39870 


03212 


96788 


14 


47 


56949 


43051 


60166 


39834 


03217 


96783 


13 


48 


56980 


43020 


60203 


39797 


03222 


96778 


12 


49 


57012 


42988 


60240 


39760 


03228 


96772 


11 


50 


9.57044 


10.42956 


9.60276 


10.39724 


10.03233 


9.96767 


10 


51 


57075 


42925 


60313 


39687 


03238 


96762 


9 


52 


57107 


42893 


60349 


39651 


03243 


96757 


8 


53 


57138 


42862 


60386 


39614 


03248 


96752 


7 


54 


57169 


42831 


60422 


39578 


03253 


96747 


6 


55 


9.57201 


10.42799 


9.60459 


10.39541 


10.03258 


9.96742 


5 


56 


f>72:;2 


42768 


60495 


39505 


03263 


96737 


4 


57 


57264 


42736 


60532 


39468 


03268 


96732 


3 


58 


57295 


42705 


60568 


39432 


03273 


96727 


2 


59 


57326 


42674 


60605 


39395 


03278 


96722 


1 


60 


57358 


42642 


60641 


39359 


03283 


96717 





M. 


Cosine. 


Secft&t. 


| Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



111° 



68° 



Logarithmic Angular Functions. 



201 



22° 






Logarithms. 






157° 


M. 


Sine. 


Cosecant. 


Tangent. 


1 Cotangent. 


Secant. 


Cosine. 


M. 





9.57358 


10.42642 


9.60641 


10.39359 


10.03283 


9.96717 


60 


1 


57389 


42611 


60677 


39323 


03289 


96711 


59 


2 


57420 


42580 


60714 


39286 


03294 


96706 


58 


3 


57451 


42549 


60750 


39250 


03299 


96701 


57 


4 


57482 


42518 


60786 


39214 


03304 


96696 


56 


5 


9.57514 


10.42486 


9.60823 


10.39177 


10.03309 


9.96691 


55 


6 


57545 


42455 


60859 


39141 


03314 


96686 


54 


7 


57576 


42424 


60895 


39105 


03319 


96681 


53 


8 


57607 


42393 


60931 


39069 


03324 


96676 


52 


9 


57638 


42362 


60967 


39033 


03330 


96670 


51 


10 


9.57669 


10.42331 


9.61004 


10.38996 


10.03335 


9.96665 


50 


11 


57700 


42300 


61040 


38960 


03340 


96660 


49 


12 


57731 


42269 


61076 


38924 


03345 


96655 


48 


13 


57762 


42238 


61112 


38888 


03350 


96650 


47 


14 


57793 


42207 


61148 


38852 


03355 


96645 


46 


15 


9.57824 


10.42176 


9.61184 


10.38816 


10.03360 


9.96640 


45 


16 


57855 


42145 


61220 


38780 


03366 


96634 


44 


17 


57885 


42115 


61256 


38744 


03371 


96629 


43 


18 


57916 


42084 


61292 


38708 


03376 


96624 


42 


19 


57947 


42053 


61328 


38672 


03381 


96619 


41 


20 


9.57978 


10.42022 


9.61364 


10.38636 


10.03386 


9.96614 


40 


21 


58008 


41992 


61400 


38600 


03392 


96608 


39 


22 


58039 


41961 


61436 


38564 


03397 


96603 


38 


23 


58070 


41930 


61472 


38528 


03402 


96598 


37 


24 


58101 


41899 


61508 


38492 


03407 


96593 


36 


25 


9.58131 


10.41869 


9.61544 


10.38456 


10.03412 


9.96588 


35 


26 


58162 


41838 


61579 


38421 


03418 


96582 


34 


27 


58192 


41808 


61615 


38385 


03423 


96577 


33 


28 


58223 


41777 


61651 


38349 


03428 


96572 


32 


29 


58253 


41747 


61687 


38313 


03433 


96567 


31 


30 


9.58284 


10.41716 


9.61722 


10.38278 


10.03438 


9.96562 


30 


31 


58314 


41686 


61758 


38242 


03444 


96556 


29 


32 


58345 


41655 


61794 


38206 


03449 


96551 


28 


33 


58375 


41625 


61830 


38170 


03454 


96546 


27 


34 


58406 


41594 


61865 


38135 


03459 


96541 


26 


35 


9.58436 


10.41564 


. 9.61901 


10.38099 


10.03465 


9.96535 


25 


36 


58467 


41533 


61936 


38064 


03470 


96530 


24 


37 


58497 


41503 


61972 


38028 


03475 


96525 


23 


38 


58527 


41473 


62008 


37992 


03480 


96520 


22 


39 


58557 


41443 


62043 


37957 


03486 


96514 


21 


40 


9.58588 


10.41412 


9.62079 


10.37921 


10.03491 


9.96509 


20 


41 


58618 


41382 


62114 


37886 


03496 


96504 


19 


42 


58648 


41352 


62150 


37850 


03502 


96498 


18 


43 


58678 


41322 


62185 


37815 


03507 


96493 


17 


44 


58709 


41291 


62221 


37779 


03512 


96488 


16 


45 


9.58739 


10.41261 


9.62256 


10.37744 


10.03517 


9.96483 


15 


46 


58769 


41231 


62292 


37708 


03523 


96477 


14 


47 


58799 


41201 


62327 


37673 


03528 


96472 


13 


48 


58829 


41171 


62362 


37638 


03533 


96467 


12 


49 


58859 


41141 


62398 


37602 


03539 


96461 


11 


50 


9.58889 


10.41111 


9.62433 


10.37567 


10.03544 


9.96456 


10 


51 


58919 


41081 


62468 


37532 


03549 


96451 


9 


52 


58949 


41051 


62504 


37496 


03555 


96445 


8 


53 


58979 


41021 


62539 


37461 


03560 


96440 


7 


54 


59009 


40991 


62574 


37426 


03565 


96435 


6 


55 


9.59039 


10.40961 


9.62609 


10.37391 


10.03571 


9.96429 


5 


56 


59069 


40931 


62645 


37355 


03576 


96424 


4 


57 


59098 


40902 


62680 


37320 


03581 


96419 


3 


58 


59128 


40872 


62715 


37285 


03587 


96413 


2 


59 


59158 


40842 


62750 


37250 


03592 


96408 


1 


60 


59188 


40812 


62785 


37215 


03597 


96403 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



112° 



67° 



202 



Logarithmic Angular Functions. 



23° 






Logarithms. 




156° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.59188 


10.40812 


9.62785 


10.37215 


10.03597 


9.96403 


60 


1 


59218 


40782 


62820 


37180 


03603 


96397 


59 


2 


59247 


40753 


62855 


37145 


03608 


96392 


58 


3 


59277 


40723 


62890 


37110 


03613 


96387 


57 


4 


59307 


40693 


62926 


37074 


03619 


96381 


56 


5 


9.59336 


10.40664 


9.62961 


10.37039 


10.03624 


9.96376 


55 


6 


59366 


40634 


62996 


37004 


03630 


96370 


54 


7 


59396 


40604 


63031 


36969 


03635 


96365 


53 


8 


59425 


40575 


63066 


36934 


03640 


96360 


52 


9 


59455 


40545 


63101 


36899 


03646 


96354 


51 


10 


9.59484 


10.40516 


9.63135 


10.36865 


10.03651 


9.96349 


50 


11 


59514 


40486 


63170 


36830 


03657 


96343 


49 


12 


59543 


40457 


63205 


36795 


03662 


96338 


48 


13 


59573 


40427 


63240 


36760 


03667 


96333 


47 


14 


59602 


40398 


63275 


36725 


03673 


96327 


46 


15 


9.59632 


10.40368 


9.63310 


10.36690 


10.03678 


9.96322 


45 


16 


59661 


40339 


63345 


36655 


03684 


96316 


44 


17 


59690 


40310 


63379 


36621 


03689 


96311 


43 


18 


59720 


40280 


63414 


36586 


03695 


96305 


42 


19 


59749 


40251 


63449 


36551 


03700 


96300 


41 


20 


9.59778 


10.40222 


9.63484 


10.36516 


10.03706 


9.96294 


40 


21 


59808 


40192 


63519 


36481 


03711 


96289 


39 


22 


59837 


40163 


63553 


36447 


03716 


96284 


38 


23 


59866 


40134 


63588 


36412 


03722 


96278 


37 


24 


59895 


40105 


63623 


36377 


03727 


96273 


36 


25 


9.59924 


10.40076 


9.63657 


10.36343 


10.03733 


9.96267 


35 


26 


59954 


40046 


63692 


36308 


03738 


96262 


34 


27 


59983 


40017 


63726 


36274 


03744 


96256 


33 


28 


60012 


39988 


63761 


36239 


03749 


96251 


32 


29 


60041 


39959 


63796 


36204 


03755 


96245 


31 


30 


9.60070 


10.39930 


9.63830 


10.36170 


10.03760 


9.96240 


30 


31 


60099 


39901 


63865 


36135 


03766 


96234 


29 


32 


60128 


39872 


63899 


36101 


03771 


96229 


28 


33 


60157 


39843 


63934 


36066 


03777 


96223 


27 


34 


60186 


39814 


63968 


36032 


03782 


96218 


26 


35 


9.60215 


10.39785 


9.64003 


10.35997 


10.03788 


9.96212 


25 


36 


60244 


39756 


64037 


35963 


03793 


96207 


24 


37 


60273 


39727 


64072 


35928 


03799 


96201 


23 


38 


60302 


39698 


64106 


35894 


03804 


96196 


22 


39 


60331 


39669 


64140 


35860 


03810 


96190 


21 


40 


9.60359 


10.39641 


9.64175 


10.35825 


10.03815 


9.96185 


20 


41 


60388 


39612 


64209 


35791 


03821 


96179 


19 


42 


60417 


39583 


64243 


35757 


03826 


96174 


18 


43 


60446 


39554 


64278 


35722 


03832 


96168 


17 


44 


60474 


39526 


64312 


35688 


03838 


96162 


16 


45 


9.60503 


10.39497 


9.64346 


10.35654 


10.03843 


9.96157 


15 


46 


60532 


39468 


64381 


35619 


03849 


96151 


14 


47 


60561 


39439 


64415 


35585 


03854 


96146 


13 


48 


60589 


39411 


64449 


35551 


03860 


96140 


12 


49 


60618 


393S2 


64483 


35517 


03865 


96135 


11 


50 


9.60646 


10.39354 


9.64517 


10.35483 


10.03871 


9.96129 


10 


51 


60675 


39325 


64552 


35448 


03877 


96123 


9 


52 


60704 


39296 


64586 


35414 


03882 


96118 


8 


53 


60732 


39268 


64620 


35380 


03888 


96112 


7 


54 


60761 


39239 


64654 


35346 


03893 


96107 


6 


55 


9.80789 


10.39211 


9.64688 


10.35312 


10.03899 


9.96101 


5 


56 


60818 


39182 


64722 


35278 


03905 


96095 


4 


57 


60846 


39154 


64756 


35244 


03910 


96090 


3 


58 


60875 


39125 


64790 


35210 


03916 


96084 


2 


59 


60903 


39097 


64824 


35176 


03921 


96079 


1 


60 


60931 


39069 


64858 


35142 


03927 


96073 





M. 


Cosine. 


Si cant. 


Cotangent 


Tangent. 


Cosecant. 


Sine. 


M. 



113° 



LOGAEITHMIC ANGULAR FUNCTIONS. 



203 



24° 






Logarithms. 






155° 


M. 


Sine. 


Cosecant. 


| Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.60931 


10.39069 


9.64858 


10.35142 


10.03927 


9.96073 


60 


1 


60960 


39040 


64892 


35108 


03933 


96067 


59 


o 


60988 


39012 


64926 


35074 


03938 


96062 


58 


3 


61016 


38984 


64960 


35040 


03944 


96056 


57 


4 


61045 


38955 


64994 


35006 


03950 


96050 


56 


5 


9.61073 


10.38927 


9.65028 


10.34972 


10.03955 


9.96045 


55 


6 


61101 


38899 


65062 


34938 


03961 


96039 


54 


7 


61129 


38871 


65096 


34904 


03966 


96034 


53 


8 


61158 


38842 


65130 


34870 


03972 


96028 


52 


9 


61186 


38814 


65164 


34836 


03978 


96022 


51 


10 


9.61214 


10.38786 


9.65197 


10.34803 


10.03983 


9.96017 


50 


11 


61242 


38758 


65231 


34769 


03989 


96011 


49 


12 


61270 


38730 


65265 


34735 


03995 


96005 


48 


13 


61298 


38702 


65299 


34701 


04000 


96000 


47 


14 


61326 


38674 


65333 


34667 


04006 


95994 


46 


15 


9.61354 


10.38646 


9.65366 


10.34634 


10.04012 


9.95988 


45 


16 


61382 


38618 


65400 


34600 


04018 


95982 


44 


17 


61411 


38589 


65434 


34566 


04023 


95977 


43 


18 


61438 


38562 


65467 


34533 


04029 


95971 


42 


19 


61466 


38534 


65501 


34499 


04035 


95965 


41 


20 


9.61494 


10.38506 


9.65535 


10.34465 


10.04040 


9.95960 


40 


21 


61522 


38478 


65568 


31432 


04046 


95954 


39 


22 


61550 


38450 


65602 


34398 


04052 


95948 


38 


23 


61578 


38422 


65636 


34364 


04058 


95942 


37 


24 


61606 


38394 


65669 


34331 


04063 


95937 


36 


25 


9.61634 


10.38366 


9.65703 


10.34297 


10.04069 


9.95931 


35 


26 


61662 


38338 


65736 


34264 


04075 


95925 


34 


27 


61689 


38311 


65770 


34230 


04080 


95920 


33 


28 


61717 


38283 


65803 


34197 


04086 


95914 


32 


29 


61745 


38255 


65837 


34163 


04092 


95908 


31 


30 


9.61773 


10.38227 


9.65870 


10.34130 


10.04098 


9.95902 


30 


31 


61800 


38200 


65904 


34096 


04103 


95897 


29 


32 


61828 


38172 


65937 


34063 


04109 


95891 


28 


33 


61856 


38144 


65971 


34029 


04115 


95885 


27 


34 


61883 


38117 


66004 


33996 


04121 


95879 


26 


35 


9.61911 


10.38089 


9.66038 


10.33962 


10.04127 


9.95873 


25 


36 


61939 


38061 


66071 


33929 


04132 


95868 


24 


37 


61966 


38034 


66104 


33896 


04138 


95862 


23 


38 


61994 


38006 


66138 


33862 


04144 


95856 


22 


39 


62021 


37979 


66171 


33829 


04150 


95850 


21 


40 


9.62049 


10.37951 


9.66204 


10.33796 


10.04156 


9.95844 


20 


41 


62076 


37924 


66238 


33762 


04161 


95839 


19 


42 


62104 


37896 


66271 


33729 


04167 


95833 


18 


43 


62131 


37869 


66304 


33696 


04173 


95827 


17 


44 


62159 


37841 


66337 


33663 


04179 


95821 


16 


45 


9.62186 


10.37814 


9.66371 


10.33629 


10.04185 


9.95815 


15 


46 


62214 


37786 


66404 


33596 


04190 


95810 


14 


47 


62241 


37759 


66437 


33563 


04196 


95804 


13 


48 


62268 


37732 


66470 


33530 


04202 


95798 


1.2 


49 


62296 


37704 


66503 


33497 


04208 


95792 


11 


50 


9.62323 


10.37677 


9.66537 


10.33463 


10.04214 


9.95786 


10 


51 


62350 


37650 


66570 


33430 


04220 


95780 


9 


52 


62377 


37623 


66603 


33397 


04225 


95775 


8 


53 


62405 


37595 


66636 


33364 


04231 


95769 


7 


54 


62432 


37568 


66669 


33331 


04237 


95763 


6 


55 


9.62459 


10.37541 


9.66702 


10.33298 


10.04243 


9.95757 


5 


56 


62486 


37514 


66735 


33265 


04249 


95751 


4 


57 


62513 


37487 


66768 


33232 


04255 


95745 


3 


58 


62541 


37459 


66801 


33199 


04261 


95739 


2 


59 


62568 


37432 


66834 


33166 


04267 


95733 


1 


60 


62595 


37405 


66867 


33133 


04272 


95728 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Qosecant. 


Sine. 


M. 



114° 



65° 



204 



Logarithmic Angular Functions. 



25° 






Logarithms. 




154° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.62595 


10.37405 


9.66867 


10.33133 


10.04272 


9.95728 


60 


1 


62622 


37378 


66900 


33100 


04278 


95722 


59 


2 


62649 


37351 


66933 


33067 


04284 


95716 


58 


3 


62676 


37324 


66966 


33034 


04290 


95710 


57 


4 


62703 


37297 


66999 


33001 


04296 


95704 


56 


5 


9.62730 


10.37270 


9.67032 


10.32968 


10.04302 


9.95698 


55 


6 


62757 


37243 


67065 


32935 


04308 


95692 


54 


7 


62784 


37216 


67098 


32902 


04314 


95686 


53 


8 


62811 


37189 


67131 


32869 


04320 


95680 


52 


9 


62838 


37162 


67163 


32837 


04326 


95674 


51 


10 


9.62865 


10.37135 


9.67196 


10.32804 


10.04332 


9.95668 


50 


11 


62892 


37108 


67229 


32771 


04337 


95663 


49 


12 


62918 


37082 


67262 


32738 


04343 


95657 


48 


13 


62945 


37055 


67295 


32705 


04349 


95651 


47 


14 


62972 


37028 


67327 


32673 


04355 


95645 


46 


15 


9.62999 


10.37001 


9.67360 


10.32640 


10.04361 


9.95639 


45 


16 


63026 


36974 


67393 


32607 


04367 


95633 


44 


17 


63052 


36948 


67426 


32574 


04373 


95627 


43 


18 


63079 


36921 


67458 


32542 


04379 


95621 


42 


19 


63106 


36894 


67491 


32509 


04385 


95615 


41 


20 


9.63133 


10.36867 


9.67524 


10.32476 


10.04391 


9.95609 


40 


21 


63159 


36841 


67556 


32444 


04397 


95603 


39 


22 


63186 


36814 


67589 


32411 


04403 


95597 


38 


23 


63213 


36787 


67622 


32378 


04409 


95591 


37 


24 


63239 


36761 


67654 


32346 


04415 


95585 


36 


25 


9.63266 


10.36734 


9.67687 


10.32313 


10.04421 


9.95579 


35 


26 


63292 


36708 


67719 


32281 


04427 


95573 


34 


27 


63319 


36681 


67752 


32248 


04433 


95567 


33 


28 


63345 


36655 


67785 


32215 


04439 


95561 


32 


29 


63372 


36628 


67817 


32183 


04445 


95555 


31 


30 


9.63398 


10.36602 


9.67850 


10.32150 


10.04451 


9.95549 


30 


31 


63425 


36575 


67882 


32118 


04457 


95543 


29 


32 


63451 


36549 


67915 


32085 


04463 


95537 


28 


33 


63478 


36522 


67947 


32053 


04469 


95531 


27 


34 


63504 


36496 


67980 


32020 


04475 


95525 


26 


35 


9.63531 


10.36469 


9.68012 


10.31988 


10.04481 


9.95519 


25 


36 


63557 


36443 


68044 


31956 


04487 


95513 


24 


37 


63583 


36417 


68077 


31923 


04493 


95507 


23 


38 


63610 


36390 


68109 


31891 


04500 


95500 


22 


39 


63636 


36364 


68142 


31858 


04506 


95494 


21 


40 


9.63662 


10.36338 


9.68174 


10.31826 


10.04512 


9.95488 


20 


41 


63689 


36311 


68206 


31794 


04518 


95482 


19 


42 


63715 


36285 


68239 


31761 


04524 


95476 


18 


43 


63741 


36259 


68271 


31729 


04530 


95470 


17 


44 


63767 


36233 


68303 


31697 


04536 


95464 


16 


45 


9.63794 


10.36206 


9.68336 


10.31664 


10.04542 


9.95458 


15 


46 


63820 


36180 


68368 


31632 


04548 


95452 


14 


47 


63846 


36154 


68400 


31600 


04554 


95446 


13 


48 


63872 


36128 


68432 


31568 


04560 


95440 


12 


49 


63898 


36102 


68465 


31535 


04566 


95434 


11 


50 


9.63924 


10.36076 


9.68497 


10.31503 


10.04573 


9.95427 


10 


51 


63950 


36050 


68529 


31471 


04579 


95421 


9 


52 


63976 


36024 


68561 


31439 


04585 


95415 


8 


53 


64002 


35998 


(is.-)!);} 


31407 


04591 


95409 


7 


54 


64028 


35972 


68626 


31374 


04597 


95403 


6 


55 


9.64054 


10.35946 


9.68658 


10.31342 


10.04603 


9.95397 


5 


56 


64080 


36920 


68690 


31310 


04609 


95391 


4 


57 


64106 


35894 


68722 


31278 


04616 


95384 


3 


58 


64132 


:frsos 


68754 


31246 


04622 


95378 


2 


59 


64158 


85842 


68786 


31214 


04628 


95372 


1 


60 


64184 


35816 


68818 


31182 


04634 


95366 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



115° 



Logarithmic Angular Functions. 



205 



26° 






Logarithms. 




1 


153° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.64184 


10.35816 


9.68818 


10.31182 


10.04634 


9.95366 


60 


1 


64210 


35790 


68850 


31150 


04640 


95360 


59 


2 


64236 


35764 


68882 


31118 


04646 


95354 


58 


3 


64262 


35738 


68914 


31086 


04652 


95348 


57 


4 


64288 


35712 


68946 


31054 


04659 


95341 


56 


5 


9.64313 


10.35687 


9.68978 


10.31022 


10.04665 


9.95335 


55 


6 


64339 


35661 


69010 


30990 


04671 


95329 


54 


7 


64365 


35635 


69042 


30958 


04677 


95323 


53 


8 


64391 


35609 


69074 


30926 


04683 


95317 


52 


9 


64417 


35583 


69106 


30894 


04690 


95310 


51 


10 


9.64442 


10.35558 


9.69138 


10.30862 


10.04696 


9.95304 


50 


11 


64468 


35532 


69170 


30830 


04702 


95298 


49 


12 


64494 


35506 


69202 


30798 


04708 


95292 


48 


13 


64519 


35481 


69234 


30766 


04714 


95286 


47 


14 


64545 


35455 


69266 


30734 


04721 


95279 


46 


15 


9.64571 


10.35429 


9.69298 


10.30702 


10.04727 


9.95273 


45 


16 


64596 


35404 


69329 


30671 


04733 


95267 


44 


17 


64622 


35378 


69361 


30639 


04739 


95261 


43 


18 


64647 


35353 


69393 


30607 


04746 


95254 


42 


19 


64673 


35327 


69425 


30575 


04752 


95248 


41 


20 


9.64698 


10.35302 


9.69457 


10.30543 


10.04758 


9.95242 


40 


21 


64724 


35276 


69488 


30512 


04764 


95236 


39 


22 


64749 


35251 


69520 


30480 


04771 


95229 


38 


23 


64775 


35225 


69552 


30448 


04777 


95223 


37 


24 


64800 


35200 


69584 


30416 


04783 


95217 


36 


25 


9.64826 


10.35174 


9.69615 


10.30385 


10.04789 


9.95211 


35 


26 


64851 


35149 


69647 


30353 


04796 


95204 


34 


27 


64877 


35123 


69679 


30321 


04802 


95198 


33 


28 


64902 


35098 


69710 


30290 


04808 


95192 


32 


29 


64927 


35073 


69742 


30258 


04815 


95185 


31 


30 


9.64953 


10.35047 


9.69774 


10.30226 


10.04821 


9.95179 


30 


31 


64978 


35022 


69805 


30195 


04827 


95173 


29 


32 


65003 


34997 


69837 


30163 


04833 


95167 


28 


33 


65029 


34971 


69868 


30132 


04840 


95160 


27 


34 


65054 


34946 


69900 


30100 


04846 


95154 


26 


35 


9.65079 


10.34921 


9.69932 


10.30068 


10.04852 


9.95148 


25 


36 


65104 


34896 


69963 


30037 


04859 


95141 


24 


37 


65130 


34870 


69995 


30005 


04865 


95135 


23 


38 


65155 


34845 


70026 


29974 


04871 


95129 


22 


39 


65180 


34820 


70058 


29942 


04878 


95122 


21 


40 


9.65205 


10.34795 


9.70089 


10.29911 


10.04884 


9.95116 


20 


41 


65230 


34770 


70121 


29879 


04890 


95110 


19 


42 


65255 


34745 


70152 


29848 


04897 


95103 


18 


43 


65281 


34719 


70184 


29816 


04903 


95097 


17 


44 


65306 


34694 


70215 


29785 


04910 


95090 


16 


45 


9.65331 


10.34669 


9.70247 


10.29753 


10.04916 


9.95084 


15 


46 


65356 


34644 


70278 


29722 


04922 


95078 


14 


47 


65381 


34619 


70309 


29691 


04929 


95071 


13 


48 


65406 


34594 


70341 


29659 


04935 


95065 


12 


49 


65431 


34569 


70372 


29628 


04941 


95059 


11 


50 


9.65456 


10.34544 


9.70404 


10.29596 


10.04948 


9.95052 


10 


51 


65481 


34519 


70435 


29565 


04954 


95046 


9 


52 


65506 


34494 


70466 


29534 


04961 


95039 


8 


53 


65531 


34469 


70498 


29502 


04967 


95033 


7 


54 


65556 


34444 


70529 


29471 


04973 


95027 


6 


55 


9.65580 


10.34420 


9.70560 


10.29440 


10.04980 


9.95020 


5 


56 


65605 


34395 


70592 


29408 


04986 


95014 


4 


57 


65630 


34370 


70623 


29377 


04993 


95007 


3 


58 


65665 


34345 


70654 


29346 


04999 


95001 


2 


59 


65680 


34320 


70685 


29315 


05005 


94995 


1 


60 


65705 


34295 


70717 


29283 


05012 


94988 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



116° 



63° 



206 



Logarithmic Angular Functions. 



27° 



Logarithms. 



152° 



M. 


Sine. 


Cosecant. 


Tangent. 
9.70717 


Cotangent. 
, 10.29283 


Secant. 


Cosine. 


M. 





9.65705 


10.34295 


10.05012 


9.949S8 


60 


1 


65729 


34271 


70748 


29252 


05018 


94982 


59 


2 


65754 


34246 


70779 


29221 


05025 


94975 


58 


3 


65779 


34221 


70810 


29190 


05031 


94969 


57 


4 


65S04 


34196 


70841 


29159 


05038 


94962 


56 


5 


9.65828 


10.34172 


9.70873 


10.29127 


10.05044 


9.94956 


55 


6 


65853 


34147 


70904 


29096 


05051 


94949 


54 


7 


65878 


34122 


70935 


29065 


05057 


94943 


53 


8 


65902 


34098 


70966 


29034 


05064 


94936 


52 


9 


65927 


34073 ! 


70997 


29003 


05070 


94930 


51 


10 


9.65952 


10.34048 


9.71028 


10.28972 


10.05077 


9.94923 


50 


11 


65976 


34024 


71059 


28941 ' 


05083 


94917 


49 


12 


66001 


33999 


71090 


28910 | 


05089 


94911 


48 


13 


66025 


33975 


71121 


28879 ' 


05096 


94904 


47 


14 


66050 


33950 ! 


71153 


28847 


05102 


94898 


46 


15 


9.66075 


10.33925 


9.71184 


10.28816 


10.05109 


9.94891 


45 


16 


66099 


33901 j 


71215 


28785 j 


05115 


94885 


44 


17 


66124 


33876 


71246 


28754 


05122 


94878 


43 


18 


66148 


33852 


71277 


28723 ! 


05129 


94871 


42 


19 


66173 


3:3827 


71308 


28692 | 


05135 


94865 


41 


20 


9.66197 


10.33808 


9.71339 


10.28661 


10.05142 


9.94858 


40 


21 


66221 


33779 


71370 


28630 I 


05148 


94852 


39 


22 


66246 


33754 


71401 


28599 | 


05155 


94845 


38 


23 


66270 


33730 


71431 


28569 


05161 


94839 


37 


24 


66295 


33705 


71462 


28538 


05168 


94832 


36 


25 


9.66319 


10.33681 


9.71493 


10.28507 


10.05174 


9.94826 


35 


26 


66343 


33657 


71524 


28476 


05181 


94819 


34 


27 


66368 


33632 


71555 


28445 


05187 


94813 


33 


28 


66392 


33608 


71586 


28414 


05194 


94806 


32 


29 


66416 


33584 


71617 


28383 


05201 


94799 


31 


30 


9.66441 


10.33559 


9.71648 


10.28352 


10.05207 


9.94793 


30 


31 


66465 


33535 


71679 


28321 


05214 


94786 


29 


32 


66489 


33511 


71709 


28291 


05220 


94780 


28 


33 


66513 


33487 


71740 


28260 


05227 


94773 


27 


34 


66537 


33463 


71771 


28229 


05233 


94767 


26 


35 


9.66562 


10.33438 


9.71802 


10.28198 


10.05240 


9.94760 


25 


36 


66586 


33414 


71833 


28167 


05247 


94753 


24 


37 


66610 


33390 


71863 


28137 


05253 


94747 


23 


38 


66634 


33366 


71894 


28106 


05260 


94740 


22 


39 


66658 


33342 


71925 


28075 


05266 


94734 


21 


40 


9.66682 


10.33318 


9.71955 


10.28045 


10.05273 


9.94727 


20 


41 


66706 


33294 


71986 


28014 


05280 


94720 


19 


42 


66731 


33269 


72017 


27983 


05286 


94714 


18 


43 


66755 


33245 


72048 


27952 


05293 


94707 


17 


44 


66779 


33221 


72078 


27922 


05300 


94700 


16 


45 


9.66803 


10.33197 


9.72109 


10.27891 


10.05306 


9.94694 


15 


46 


66827 


33173 


72140 


27860 


05313 


94687 


14 


47 


66851 


33149 


72170 


27830 


05320 


94680 


13 


48 


66875 


33125 


72201 


27799 


05326 


94674 


12 


49 


66899 


33101 


72231 


27769 


05333 


94667 


11 


50 


9.66922 


10.: 


9.72262 


10.27738 


10.05340 


9.94660 


10 


51 


66946 


33054 


72293 


27707 


05346 


94654 


9 


52 


66970 




72323 


27677 


05353 


94647 


8 


53 


66994 


33006 


72354 


27646 


05360 


94640 


7 


54 


67018 




72384 


27616 


05366 


94634 


6 


55 


9.67042 


10.32958 


9.72415 


10.27585 


10.05373 


9.94627 


5 


56 


67066 


32934 


72445 


27555 


05380 


94620 


4 


57 


67090 


32910 


72476 


2752 1 


05386 


94614 


3 


58 


67113 


32887 


72506 


27494 


05393 


94607 


2 


59 


67137 




72537 


27463 


05400 


94600 


1 


60 


67161 




72567 


271:;:; 


05407 


94593 





M. 


Cosin<-. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



117° 



62° 



Logarithmic Angular Functions. 



207 



28° 






Logarithms. 




151° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.67161 


10.32839 


9.72567 


10.27433 


10.05407 


9.94593 


60 


1 


67185 


32815 


72598 


27402 


05413 


94587 


59 


2 


67208 


32792 


72628 


27372 


05420 


94580 


58 


3 


67232 


32768 


72659 


27341 


05427 


94573 


57 


4 


67256 


32744 


72689 


27311 


05433 


94567 


56 


5 


9.67280 


10.32720 


9.72720 


10.27280 


10.05440 


9.94560 


55 


6 


67303 


32697 


72750 


27250 


05447 


94553 


54 


7 


67327 


32673 


72780 


27220 


05454 


94546 


53 


8 


67350 


32650 


72811 


27189 


05460 


94540 


52 


9 


67374 


32626 


72841 


27159 


05467 


94533 


51 


10 


9.67398 


10.32602 


9.72872 


10.27128 


10.05474 


9.94526 


50 


11 


67421 


32579 


72902 


27098 


05481 


94519 


49 


12 


67445 


32555 


72932 


27068 


05487 


94513 


48 


13 


67468 


32532 


72963 


27037 


05494 


94506 


47 


14 


67492 


32508 


72993 


27007 


05501 


94499 


46 


15 


9.67515 


10.32485 


9.73023 


10.26977 


10.05508 


9.94492 


45 


16 


67539 


32461 


73054 


26946 


05515 


94485 


44 


17 


67562 


32-438 


73084 


26916 


05521 


94479 


43 


18 


67586 


32414 


73114 


26886 


05528 


94472 


42 


19 


67609 


32391 


73144 


26856 


05535 


94465 


41 


20 


9.67633 


10.32367 


9.73175 


10.26825 


10.05542 


9.94458 


40 


21 


67656 


32344 


73205 


26795 


05549 


94451 


39 


22 


67680 


32320 


73235 


26765 


05555 


94445 


38 


23 


67703 


32297 


73265 


26735 


05562 


94438 


37 


24 


67726 


32274 


73295 


26705 


05569 


94431 


36 


25 


9.67750 


10.32250 


9.73326 


10.26674 


10.05576 


9.94424 


35 


26 


67773 


32227 


73356 


26644 


05583 


94417 


34 


27 


67796 


32204 


73386 


26614 


05590 


94410 


33 


28 


67820 


32180 


73416 


26584 


05596 


94404 


32 


29 


67843 


32157 


73446 


26554 


05603 


94397 


31 


30 


9.67866 


10.32134 


9.73476 


10.26524 


10.05610 


9.94390 


30 


31 


67890 


32110 


73507 


26493 


05617 


94383 


29 


32 


67913 


32087 


73537 


26463 


05624 


94376 


28 


33 


67936 


32064 


73567 


26433 


05631 


94369 


27 


34 


67959 


32041 


73597 


26403 


05638 


94362 


26 


35 


9.67982 


10.32018 


9.73627 


10.26373 


10.0o645 


9.9i355 


25 


36 


68006 


31994 


73657 


26343 


05651 


94349 


24 


37 


68029 


31971 


73687 


26313 


0o65S 


9-i342 


23- 


38 


68052 


31948 


73717 


26283 


05665 


94335 


22 


39 


68075 


31925 


73747 


26253 


0o672 


9*328 


21 


40 


9.68098 


10.31902 


9.73777 


10.26223 


10.0o679 


9.94321 


20 


41 


68121 


31879 


73807 


26193 


O0686 


94314 


19 


42 


6S144 


31856 


73837 


26163 


0o693 


94307 


18 


43 


68167 


31833 


73S67 


26133 


05700 


94300 


17 


44 


68190 


31810 


73897 


26103 


05707 


94293 


16 


45 


9.68213 


10.31787 


9.73927 


10.26073 


10.0o714 


9.94286 


15 


46 


68237 


31763 


73957 


26043 


05721 


94279 


14 


47 


68260 


31740 


73987 


26013 


05727 


94273 


13 


48 


68283 


31717 


74017 


25983 


05734 


94266 


12 


49 


68305 


31695 


74047 


25953 


05741 


94259 


11 


50 


9.68328 


10.31672 


9.74077 


10.25923 


10.05748 


9.94252 


10 


51 


68351 


31649 


74107 


25893 


05755 


94245 


9 


52 


68374 


31626 


74137 


25863 


05762 


94238 


8 


53 


68397 


31603 


74166 


25834 


05769 


94231 


7 


54 


68420 


31580 


74196 


25804 


05776 


94224 


6 


55 


9.68443 


10.31557 


9.74226 


10.25774 


10.05783 


9.94217 


5 


56 


68466 


31534 


74256 


25744 


05790 


94210 


4 


57 


68489 


31511 


74286 


25714 


05797 


94203 


3 


58 


68512 


314S8 


74316 


25684 


05804 


94196 


2 


59 


68534 


31466 


74345 


25655 


05811 


94189 


1 


60 


68557 


31443 


74375 


25625 


05818 


94182 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



118° 



61° 



208 



Logarithmic Angular Functions. 



29° 






Logarithms. 




150° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.68557 


10.31443 


9.74375 


10.25625 


10.05818 


9.94182 


60 


1 


68580 


31420 


74405 


25595 


05825 


94175 


59 


2 


68603 


31397 


74435 


25565 


05832 


94168 


58 


3 


68625 


31375 


74465 


25535 


05839 


94161 


57 


4 


68648 


31352 


74494 


25506 


05846 


94154 


56 


5 


9.68671 


10.31329 


9.74524 


10.25476 


10.05853 


9.94147 


55 


6 


68694 


31306 


74554 


25446 


05860 


94140 


54 


7 


68716 


31284 


74583 


25417 


05867 


94133 


53 


8 


68739 


31261 


74613 


25387 


05874 


94126 


52 


9 


68762 


31238 


74643 


25357 


05881 


94119 


51 


10 


9.68784 


10.31216 


9.74673 


10.25327 


10.05888 


9.94112 


50 


11 


68807 


31193 


74702 


25298 


05895 


94105 


49 


12 


68829 


31171 


74732 


25268 


05902 


94098 


48 


13 


68852 


31148 


74762 


25238 


05910 


94090 


47 


14 


68875 


31125 


74791 


25209 


05917 


94083 


46 


15 


9.68897 


10.31103 


9.74821 


10.25179 


10.05924 


9.94076 


45 


16 


68920 


31080 


74851 


25149 


05931 


94069 


44 


17 


68942 


31058 


74880 


25120 


05938 


94062 


43 


18 


68965 


31035 


74910 


25090 


05945 


94055 


42 


19 


68987 


31013 


74939 


25061 


05952 


94048 


41 


20 


9.69010 


10.30990 


9.74969 


10.25031 


10.05959 


9.94041 


40 


21 


69032 


30968 


74998 


25002 


05966 


94034 


39 


22 


69055 


30945 


75028 


24972 


05973 


94027 


38 


23 


69077 


30923 


75058 


24942 


05980 


94020 


37 


24 


69100 


30900 


75087 


24913 


05988 


94012 


36 


25 


9.69122 


10.30878 


9.75117 


10.24883 


10.05995 


9.94005 


35 


26 


69144 


30856 


75146 


24854 


06002 


93998 


34 


27 


69167 


30833 


75176 


24824 


06009 


93991 


33 


28 


69189 


30811 


75205 


24795 


06016 


93984 


32 


29 


69212 


30788 


75235 


24765 


06023 


93977 


31 


30 


9.69234 


10.30766 


9.75264 


10.24736 


10.06030 


9.93970 


30 


31 


69256 


30744 


75294 


24706 


06037 


93963 


29 


32 


69279 


30721 


75323 


24677 


06045 


93955 


28 


33 


69301 


30699 


75353 


24647 


06052 


93948 


27 


34 


69323 


30677 


75382 


24618 


06059 


93941 


26 


35 


9.69345 


10.30655 


9.75411 


10.24589 


10.06066 


9.93934 


25 


36 


69368 


30632 


75441 


24559 


06073 


93927 


24 


37 


69390 


30610 


75470 


24530 


06080 


93920 


23 


38 


69412 


30588 


75500 


24500 


06088 


93912 


22 


39 


69434 


30566 


75529 


24471 


06095 


93905 


21 


40 


9.69456 


10.30544 


9.75558 


10.24442 


10.06102 


9.93898 


20 


41 


69479 


30521 


75588 


24412 


06109 


93891 


19 


42 


69501 


30499 


75617 


24383 


06116 


93884 


18 


43 


69523 


30477 


75647 


24353 


06124 


93876 


17 


44 


69545 


30455 


75676 


24324 


06131 


93869 


16 


45 


9.69567 


10.30433 


9.75705 


10.24295 


10.06138 


9.93862 


15 


46 


69589 


30411 


75735 


24265 


06145 


93855 


14 


47 


69611 


30389 


75764 


24236 


06153 


93847 


13 


48 


69633 


30367 


75793 


24207 


06160 


93840 


12 


49 


69655 


30345 


75822 


24178 


06167 


93833 


11 


50 


9.69677 


10.30323 


9.75852 


10.24148 


10.06174 


9.93826 


10 


51 


69699 


30301 


75881 


24119 


06181 


93819 


9 


52 


69721 


30279 


75910 


24090 


06189 


93811 


8 


53 


69743 


30257 


75939 


24061 


06196 


93804 


7 


54 


69765 


30235 


75969 


24031 


06203 


93797 


6 


55 


9.697S7 


10.30213 


9.75998 


10.24002 


10.06211 


9.93789 


5 


56 


69809 


30191 


76027 


23973 


06218 


93782 


4 


57 


69831 


30169 


76056 


23944 


06225 


93775 


3 


58 


69853 


30147 


76086 


23914 


06232 


93768 


2 


59 


69875 


30125 


76115 


23885 


06240 


93760 


1 


60 


69897 


30103 


76144 


23856 


06247 


93753 





M. 


Cosine. 


Secant. 


Cotangent, 


Tangent. 


Cosecant. 


Sine. 


M. 



119° 



60° 



Logarithmic Angular Functions. 



209 



30° 






Logarithms. 




149° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent, 


Secant. 


Cosine. 


M. 





9.69897 


10.30103 


9.76144 


10.23856 


10.06247 


9.93753 


60 


1 


69919 


30081 


76173 


23827 


06254 


93746 


59 


2 


69941 


30059 


76202 


23798 


06262 


93738 


58 


3 


69963 


30037 


76231 


23769 


06269 


93731 


57 


4 


69984 


30016 


76261 


23739 


06276 


93724 


56 


5 


9.70006 


10.29994 


9.76290 


10.23710 


10.06283 


9.93717 


55 


6 


70028 


29972 


76319 


23681 


06291 


93709 


54 


7 


70050 


29950 


76348 


23652 


06298 


93702 


53 


8 


70072 


29928 


76377 


23623 


06305- 


93695 


52 


9 


70093 


29907 


76406 


23594 


06313 


93687 


51 


10 


9.70115 


10.29885 


9.76435 


10.23565 


10.06320 


9.93680 


50 


11 


70137 


29863 


76464 


23536 


06327 


93673 


49 


12 


70159 


29841 


76493 


23507 


06335 


93665 


48 


13 


70180 


29820 


76522 


23478 


06342 


93658 


47 


14 


70202 


29798 


76551 


23449 


06350 


93650 


46 


15 


9.70224 


10.29776 


9.76580 


10.23420 


10.06357 


9.93643 


45 


16 


70245 


29755 


76609 


23391 


06364 


93636 


44 


17 


70267 


29733 


76639 


23361 


06372 


93628 


43 


18 


70288 


29712 


76668 


23332 


06379 


93621 


42 


19 


70310 


29690 


76697 


23303 


06386 


93614 


41 


20 


9.70332 


10.29668 


9.76725 


10.23275 


10.06394 


9.93606 


40 


21 


70353 


29647 


76754 


23246 


06401 


93599 


39 


22 


70375 


29625 


76783 


23217 


06409 


93591 


38 


23 


70396 


29604 


76812 


23188 


06416 


93584 


37 


24 


70418 


29582 


76841 


23159 


06423 


93577 


36 


25 


9.70439 


10.29561 


9.76870 


10.23130 


10.06431 


9.93569 


35 


26 


70461 


29539 


76899 


23101 


06438 


93562 


34 


27 


70482 


29518 


76928 


23072 


06446 


93554 


33 


28 


70504 


29496 


76957 


23043 


06453 


93547 


32 


29 


70525 


29475 


76986 


23014 


06461 


93539 


31 


30 


9.70547 


10.29453 


9.77015 


10.22985 


10.06468 


9.93532 


30 


31 


70568 


29432 


77044 


22956 


06475 


93525 


29 


32 


70590 


29410 


77073 


22927 


06483 


93517 


28 


33 


70611 


29389 


77101 


22899 


06490 


93510 


27 


34 


70633 


29367 


77130 


22870 


06498 


93502 


26 


35 


9.70654 


10.29346 


9.77159 


10.22841 


10.06505 


9.93495 


25 


36 


70675 


29325 


77188 


22812 


06513 


93487 


24 


37 


70697 


29303 


77217 


22783 


06520 


93480 


23 


38 


70718 


29282 


77246 


22754 


06528 


93472 


22 


39 


70739 


29261* 


77274 


22726 


06535 


93465 


21 


40 


9.70761 


10.29239 


9.77303 


10.22697 


10.06543 


9.93457 


20 


41 


70782 


29218 


77332 


22668 


06550 


93450 


19 


42 


70803 


29197 


77361 


22639 


06558 


93442 


18 


43 


70824 


29176 


77390 


22610 


06565 


93435 


17 


44 


70846 


29154 


77418 


22582 


06573 


93427 


16 


45 


9.70867 


10.29133 


9.77447 


10.22553 


10.06580 


9.93420 


15 


46 


70888 


29112 


77476 


22524 


06588 


93412 


14 


47 


70909 


29091 


77505 


22495 


06595 


93405 


13 


48 


70931 


29069 


77533 


22467 


06603 


93397 


12 


49 


70952 


29048 


77562 


22438 


06610 


93390 


11 


50 


9.70973 


10.29027 


9.77591 


10.22409 


10.06618 


9.93382 


10 


51 


70994 


29006 


77619 


22381 


06625 


93375 


9 


52 


71015 


28985 


77648 


22352 


06633 


93367 


8 


53 


71036 


28964 


77677 


22323 


06640 


93360 


7 


54 


71058 


28942 


77706 


22294 


06648 


93352 


6 


55 


9.71079 


10.28921 


9.77734 


10.22266 


10.06656 


9.93344 


5 


56 


71100 


28900 


77763 


22237 


06663 


93337 


4 


57 


71121 


28879 


77791 


22209 


06671 


93329 


3 


58 


71142 


28858 


77820 


22180 


06678 


93322 


2 


59 


71163 


28837 


77849 


22151 


06686 


93314 


1 


60 


71184 


28816 


77877 


22123 


06693 


93307 





M. 


Cosine. 


Secant. 


Cotangent, 


Tangent. 


Cosecant. 


Sine. 


M. 



120° 



59° 



14 



210 



Logarithmic Angular Functions. 



31° 






Logarithms. 




148° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.71184 


10.28816 


9.77877 


10.22123 


10.06693 


9.93307 


60 


1 


71205 


28795 


77906 


22094 


06701 


93299 


59 


2 


71226 


28774 


77935 


22065 


06709 


93291 


58 


3 


71247 


28753 


77963 


22037 


06716 


93284 


57 


4 


71268 


28732 


77992 


22008 


06724 


93276 


56 


5 


9.71289 


10.28711 


9.78020 


10.21980 


10.06731 


9.93269 


55 


6 


71310 


28690 


78049 


21951 


06739 


93261 


54 


7 


71331 


28669 


78077 


21923 


06747 


93253 


53 


8 


71352 


28648 


78106 


21894 


06754 


93246 


52 


9 


71373 


28627 


78135 


21865 


06762 


93238 


51 


10 


9.71393 


10.28607 


9.78163 


10.21837 


10.06770 


9.93230 


50 


11 


71414 


28586 


78192 


21808 


06777 


93223 


49 


12 


71435 


28565 


78220 


21780 


06785 


93215 


48 


13 


71456 


28544 


78249 


21751 


06793 


93207 


47 


14 


71477 


28523 


78277 


21723 


06800 


93200 


46 


15 


9.71498 


10.28502 


9.78306 


10.21694 


10.06808 


9.93192 


45 


16 


71519 


28481 


78334 


21666 


06816 


93184 


44 


17 


71539 


28461 


78363 


21637 


06823 


93177 


43 


18 


71560 


28440 


78391 


21609 


06831 


93169 


42 


19 


71581 


28419 


78419 


21581 


06839 


93161 


41 


20 


9.71602 


10.28398 


9.78448 


10.21552 


10.06846 


9.93154 


40 


21 


71622 


28378 


78476 


21524 


06854 


93146 


39 


22 


71643 


28357 


78505 


21495 


06862 


93138 


38 


23 


71664 


28336 


78533 


21467 


06869 


93131 


37 


24 


71685 


28315 


78562 


21438 


06877 


93123 


36 


25 


9.71705 


10.28295 


9:78590 


10.21410 


10.06885 


9.93115 


35 


26 


71726 


28274 


78618 


21382 


06892 


93108 


34 


27 


71747 


28253 


78647 


21353 


06900 


93100 


33 


28 


71767 


28233 


78675 


21325 


06908 


93092 


32 


29 


71788 


28212 


78704 


21296 


06916 


93084 


31 


30 


9.71809 


10.28191 


9.78732 


10.21268 


10.06923 


9.93077 


30 


31 


71829 


28171 


78760 


21240 


06931 


93069 


29 


32 


718*0 


28150 


78789 


21211 


06939 


93061 


28 


33 


71870 


28130 


78817 


21183 


06947 


93053 


27 


34 


71891 


28109 


78845 


21155 


06954 


93046 


26 


35 


9.71911 


10.28089 


9.78874 


10.21126 


10.06962 


9.93038 


25 


36 


71932 


28068 


78902 


21098 


06970 


93030 


24 


37 


71952 


28048 


78930 


21070 


06978 


93022 


23 


38 


71973 


28027 


78959 


21041 


06986 


93014 


22 


39 


71994 


28006 


78987 


21013 


06993 


93007 


21 


40 


9.72014 


10.27986 


9.79015 


10.20985 


10.07001 


9.92999 


20 


41 


72034 


27966 


79043 


20957 


07009 


92991 


19 


42 


72055 


27945 


79072 


20928 


07017 


92983 


18 


43 


72075 


27925 


79100 


20900 


07024 


92976 


17 


44 


72096 


27904 


79128 


20872 


07032 


92968 


16 


45 


9.72116 


10.27884 


9.79156 


10.20844 


10.07040 


9.92960 


15 


46 


72137 


27863 


79185 


20815 


07048 


92952 


14 


47 


72157 


27843 


79213 


20787 


07056 


92944 


13 


48 


72177 


27823 


79241 


20759 


07064 


92936 


12 


49 


72198 


27802 


79269 


20731 


07071 


92929 


11 


50 


9.72218 


10.27782 


9.79297 


10.20703 


10.07079 


9.92921 


10 


51 


72238 


27762 


79326 


20674 


07087 


92913 


9 


52 


72259 


27741 


79354 


20646 


07095 


92905 


8 


53 


72279 


27721 


79382 


20618 


07103 


92897 


7 


54 


72299 


27701 


79410 


20590 


07111 


92889 


6 


55 


9.72320 


10.27680 


9.79438 


10.20562 


10.07119 


9.92881 


5 


56 


72340 


27660 


79466 


20534 


07126 


92874 


4 


57 


72360 


27640 


79495 


20505 


07134 


92866 


3 


58 


72381 


27619 


79523 


20477 


07142 


92858 


2 


59 


72101 


27599 


79551 


20449 


07150 


92850 


1 


60 


7242] 


27579 


79579 


20421 


07158 


92842 





M. 


1 "sine. 


Secant. 


Cotangent, 


Tangent. 


Cosecant. 


Sine. 


M. 



121 c 



Logarithmic Angular Functions. 



211 



32° 






Logarithms. 




147° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 


* 


9.72421 


10.27579 


9.79579 


10.20421 


10.07158 


9.92842 


60 


1 


72441 


27559 


79607 


20393 


07166 


92834 


59 


2 


72461 


27539 


79635 , 


20365 


07174 


92826 


58 


3 


72482 


27518 


79663 


20337 


07182 


92818 


57 


4 


72502 


27498 


79691 


20309 


07190 


92810 


56 


5 


9.72522 


10.27478 


9.79719 


10.20281 


10.07197 


9.92803 


55 


6 


72542 


27458 


79747 


20253 


07205 


92795 


54 


7 


72562 


27438 


79776 


20224 


07213 


92787 


53 


8 


72582 


27418 


79804 


20196 


07221 


92779 


52 


9 


72602 


27398 


79832 


20168 


07229 


92771 


51 


10 


9.72622 


10.27378 


9.79860 


10.20140 


10.07237 


9.92763 


50 


11 


72643 


27357 


79888 


20112 


07245 


92755 


49 


12 


72663 


27337 


79916 


20084 


07253 


92747 


48 


13 


72683 


27317 


79944 


20056 


07261 


92739 


47 


14 


72703 


27297 


79972 


20028 


07269 


92731 


46 


15 


9.72723 


10.27277 


9.80000 


10.20000 


10.07277 


9.92723 


45 


16 


72743 


27257 


80028 


19972 


07285 


92715 


44 


17 


72763 


27237 


80056 


19944 


07293 


92707 


43 


18 


72783 


27217 


80084 


19916 


07301 


92699 


42 


19 


72803 


27197 


80112 


19888 


07309 


92691 


41 


20 


9.72823 


10.27177 


9.80140 


10.19860 


10.07317 


9.92683 


40 


21 


72843 


27157 


80168 


19832 


07325 


92675 


39 


22 


72863 


27137 


80195 


19805 


07333 


92667 


38 


23 


72883 


27117 


80223 


19777 


07341 


92659 


37 


24 


72902 


27098 


80251 


19749 


07349 


92651 


36 


25 


9.72922 


10.27078 


9.80279 


10.19721 


10.07357 


9.92643 


35 


26 


72942 


27058 


80307 


19693 


07365 


92635 


34 


27 


72962 


27038 


80335 


19665 


07373 


92627 


33 


28 


72982 


27018 


80363 


19637 


07381 


92619 


32 


29 


73002 


26998 


80391 


19609 


07389 


92611 


31 


30 


9.73022 


10.26978 


9.80419 


10.19581 


10.07397 


9.92603 


30 


31 


73041 


26959 


80447 


19553 


07405 


92595 


29 


32 


73061 


26939 


80474 


19526 


07413 


92587 


28 


33 


73081 


26919 


80502 


19498 


07421 


92579 


27 


34 


73101 


26899 


80530 


19470 


07429 


92571 


26 


35 


9.73121 


10.26879 


9.80558 


10.19442 


10.07437 


9.92563 


25 


36 


73140 


26860 


80586 


19414 


07445 


92555 


24 


37 


73160 


26840 


80614 


19386 


07454 


92546 


23 


38 


73180 


26820 


80642 


19358 


07462 


92538 


22 


39 


73200 


26800 


80669 


19331 


07470 


92530 


21 


40 


9.73219 


10.26781 


9.80697 


10.19303 


10.07478 


9.92522 


20 


41 


73239 


26761 


80725 


19275 


07486 


92514 


19 


42 


73259 


26741 


80753 


19247 


07494 


92506 


18 


43 


73278 


26722 


80781 


19219 


07502 


92498 


17 


44 


73298 


26702 


80808 


19192 


07510 


92490 


16 


45 


9.73318 


10.26682 


9.80836 


10.19164 


10.07518 


9.92482 


15 


46 


73337 


26663 


80864 


19136 


07527 


92473 


14 


47 


73357 


26643 


80892 


19108 


07535 


92465 


13 


48 


73377 


26623 


80919 


19081 


07543 


92457 


12 


49 


73396 


26604 


80947 


19053 


07551 


92449 


11 


50 


9.73416 


10.26584 


9.80975 


10.19025 


10.07559 


9.92441 


10 


51 


73435 


26565 


81003 


.18997 


07567 


92433 


9 


52 


73455 


26545 


81030 


18970 


07575 


92425 


8 


53 


73474 


26526 


81058 


18942 


07584 


92416 


7 


54 


73494 


26506 


81086 


18914 


07592 


92408 


6 


55 


9.73513 


10.26487 


9.81113 


10.18887 


10.07600 


9.92400 


5 


56 


73533 


26467 


81141 


18859 


07608 


92392 


4 


57 


73552 


26448 


81169 


18831 


07616 


92384 


3 


58 


73572 


26428 


81196 


18804 


07624 


92376 


2 


59 


73591 


26409 


81224 


18776 


07633 


92367 


1 


60 


73611 


26389 


81252 


18748 


07641 


92359 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



57° 



212 



Logarithmic Angular Functions. 



33° 






Logarithms. 




146° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.73611 


10.26389 


9.81252 


10.18748 


10.07641 


9.92359 


60 


1 


73630 


26370 


81279 


18721 


07649 


92351 


59 


2 


73650 


26350 


81307 


18693 


07657 


92343 


58 


3 


73669 


26331 


81335 


18665 


07665 


92335 


57 


4 


73689 


26311 


81362 


18638 


07674 


92326 


56 


5 


9.73708 


10.26292 


9.81390 


10.18610 


10.07682 


9.92318 


55 


6 


73727 


26273 


81418 


18582 


07690 


92310 


54 


7 


73747 


26253 


81445 


18555 


07698 


92302 


53 


8 


73766 


26234 


81473 


18527 


07707 


92293 


52 


9 


73785 


26215 


81500 


18500 


07715 


92285 


51 


10 


9.73805 


10.26195 


9.81528 


10.18472 


10.07723 


9.92277 


50 


11 


73824 


26176 


81556 


18444 


07731 


92269 


49 


12 


73843 


26157 


81583 


18417 


07740 


922o0 


48 


13 


73863 


26137 


81611 


18389 


07748 


92252 


47 


14 


73882 


26118 


81638 


18362 


07756 


92244 


46 


15 


9.73901 


10.26099 


9.81666 


10.18334 


10.07765 


9.92235 


45 


16 


73921 


26079 


81693 


18307 


07773 


92227 


44 


17 


73940 


26060 


81721 


18279 


07781 


92219 


43 


18 


73959 


26041 


81748 


18252 


07789 


92211 


42 


19 


73978 


26022 


81776 


18224 


07798 


92202 


41 


20 


9.73997 


10.26003 


9.81803 


10.18197 


10.07806 


9.92194 


40 


21 


74017 


25983 


81831 


18169 


07814 


92186 


39 


22 


74036 


25964 


81858 


18142 


07823 


92177 


38 


23 


74055 


25945 


81886 


18114 


07831 


92169 


37 


24 


74074 


25926 


81913 


18087 


07839 


92161 


36 


25 


9.74093 


10.25907 


9.81941 


10.18059 


10.07848 


9.92152 


35 


26 


74113 


25887 


81968 


18032 


07856 


92144 


34 


27 


74132 


25868 


81996 


18004 


07864 


92136 


33 


28 


74151 


25849 


82023 


17977 


07873 


92127 


32 


29 


74170 


25830 


82051 


17949 


07881 


92119 


31 


30 


9.74189 


10.25811 


9.82078 


10.17922 


10.07889 


9.92111 


30 


31 


74208 


25792 


82106 


17894 


07898 


92102 


29 


32 


74227 


25773 


82133 


17867 


07906 


92094 


28 


33 


74246 


25754 


82161 


17839 


07914 


92086 


27 


34 


74265 


25735 


82188 


17812 


07923 


92077 


26 


35 


9.74284 


10.25716 


9.82215 


10.17785 


10.07931 


9.92069 


25 


36 


74303 


25697 


82243 


17757 


07940 


92060 


24 


37 


74322 


25678 


82270 


17730 


07948 


92052 


23 


38 


74341 


25659 


82298 


17702 


07956 


92044 


22 


39 


74360 


25640 


82325 


17675 


07965 


92035 


21 


40 


9.74379 


10.25621 


9.82352 


10.17648 


10.07973 


9.92027 


20 


41 


74398 


25602 


82380 


17620 


07982 


92018 


19 


42 


74417 


25583 


82407 


17593 


07990 


92010 


18 


43 


74436 


25564 


82435 


17565 


07998 


92002 


17 


44 


74455 


25545 


82462 


17538 


08007 


91993 


16 


45 


9.74474 


10.25526 


9.82489 


10.17511 


10.08015 


9.91985 


15 


46 


74493 


25507 


82517 


17483 


08024 


91976 


14 


47 


74512 


25488 


82544 


17456 


08032 


91968 


13 


48 


74531 


25469 


82571 


17429 


08041 


91959 


12 


49 


74549 


25451 


82599 


17401 


08049 


91951 


11 


50 


9.74568 


10.25432 


9.82626 


10.17374 


10.08058 


9.91942 


10 


51 


74587 


25413 


82653 


17347 


08066 


91934 


9 


52 


74606 


25394 


82681 


17319 


08075 


91925 


8 


53 


74625 


25375 


82708 


17292 


08083 


91917 


7 


54 


74644 


25356 


82735 


17265 


08092 


91908 


6 


55 


9.74662 


10.25338 


9.82762 


10.17238 


10.08100 


9.91900 


5 


56 


74681 


25319 


82790 


17210 


08109 


91891 


4 


57 


71700 


25300 


82817 


17183 


08117 


91883 


3 


58 


71719 


25281 


82844 


17156 


08126 


91874 


2 


59 


74737 


25263 


82871 


17129 


08134 


91866 


1 


60 


74756 


25244 


82899 


17101 


08143 


91857 





M. 


Cosiiif. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



123° 



56° 



Logarithmic Angular Functions. 



213 



34° 






Logarithms. 




145° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 


• 


9.74756 


10.25244 


9.82899 


10.17101 


10.08143 


9.91857 


60 


1 


74775 


25225 


82926 


17074 


08151 


91849 


59 


2 


74794 


25206 


82953 


17047 


08160 


91840 


58 


3 


74812 


25188 


82980 


17020 


08168 


91832 


57 


4 


74831 


25169 


83008 


16992 


08177 


91823 


56 


5 


9.74850 


10.25150 


9.83035 


10.16965 


10.08185 


9.91815 


55 


6 


74868 


25132 


83062 


16938 


08194 


91806 


54 


7 


74887 


25113 


83089 


16911 


08202 


91798 


53 


8 


74906 


25094 


83117 


16883 


08211 


91789 


52 


9 


74924 


25076 


83144 


16856 


08219 


91781 


51 


10 


9.74943 


10.25057 


9.83171 


10.16829 


10.08228 


9.91772 


50 


11 


74961 


25039 


83198 


16802 


08237 


91763 


49 


12 


74980 


25020 


83225 


16775 


08245 


91755 


48 


13 


74999 


25001 


83252 


16748 


08254 


91746 


47 


14 


75017 


24983 


83280 


16720 


08262 


91738 


46 


15 


9.75036 


10.24964 


9.83307 


10.16693 


10.08271 


9.91729 


45 


16 


75054 


24946 


83334 


16666 


08280 


91720 


44 


17 


75073 


24927 


83361 


16639 


08288 


91712 


43 


18 


75091 


24909 


83388 


16612 


08297 


91703 


42 


19 


75110 


24890 


83415 


16585 


08305 


91695 


41 


20 


9.75128 


10.24872 


9.83442 


10.16558 


10.08314 


9.91686 


40 


21 


75147 


24853 


83470 


16530 


08323 


91677 


39 


22 


75165 


24835 


83497 


16503 


08331 


91669 


38 


23 


75184 


24816 


83524 


16476 


08340 


91660 


37 


24 


75202 


24798 


83551 


16449 


08349 


91651 


36 


25 


9.75221 


10.24779 


9.83578 


10.16422 


10.08357 


9.91643 


35 


26 


75239 


24761 


83605 


16395 


08366 


91634 


34 


27 


75258 


24742 


83632 


16368 


08375 


91625 


33 


28 


75276 


24724 


83659 


16341 


08383 


91617 


32 


29 


75294 


24706 


83686 


16314 


08392 


91608 


31 


30 


9.75313 


10.24687 


9.83713 


10.16287 


10.08401 


9.91599 


30 


31 


75331 


24669 


83740 


16260 


08409 


91591 


29 


32 


75350 


24650 


83768 


16232 


08418 


91582 


28 


33 


75368 


24632 


83795 


16205 


08427 


91573 


27 


34 


75386 


24614 


83822 


16178 


08435 


91565 


26 


35 


9.75405 


10.24595 


9.83849 


10.16151 


10.08444 


9.91556 


25 


36 


75423 


24577 


83876 


16124 


08453 


91547 


24 


37 


75441 


24559 


83903 


16097 


08462 


91538 


23 


38 


75459 


24541 


83930 


16070 


08470 


91530 


22 


39 


75478 


24522 


83957 


16043 


08479 


91521 


21 


40 


9.75496 


10.24504 


9.83984 


10.16016 


10.08488 


9.91512 


20 


41 


75514 


24486 


84011 


15989 


08496 


91504 


19 


42 


75533 


24467 


84038 


15962 


08505 


91495 


18 


43 


75551 


24449 


84065 


15935 


08514 


91486 


17 


44 


75569 


24431 


84092 


15908 


08523 


91477 


16 


45 


9.75587 


10.24413 


9.84119 


10.15881 


10.08531 


9.91469 


15 


46 


75605 


24395 


84146 


15854 


08540 


91460 


14 


47 


75624 


24376 


84173 


15827 


08549 


91451 


13 


48 


75642 


24358 


84200 


15800 


08558 


91442 


12 


49 


75660 


24340 


84227 


15773 


08567 


91433 


11 


50 


9.75678 


10.24322 


9.84254 


10.15746 


10.08575 


9.91425 


10 


51 


75696 


24304 


84280 


15720 


08584 


91416 


9 


52 


75714 


24286 


84307 


15693 


08593 


91407 


8 


53 


75733 


24267 


84334 


15666 


08602 


91398 


7 


54 


75751 


24249 


84361 


15639 


08611 


91389 


6 


55 


9.75769 


10.24231 


9.84388 


10.15612 


10.08619 


9.91381 


5 


56 


75787 


24213 


84415 


15585 


08628 


91372 


4 


57 


75805 


24195 


84442 


15558 


08637 


91363 


3 


58 


75823 


24177 


84469 


15531 


08646 


91354 


2 


59 


75841 


24159 


84496 


15504 


08655 


91345 


1 


60 


75859 


24141 


84523 


15477 


08664 


91336 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



124° 



55° 



214 



Logarithmic Angular Functions. 



35° 






Logarithms. 






144° 


M. 


Sine. 


Cosecant. 


I Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.75859 


10.24141 


9.84523 


10.15477 


10.08664 


9.91336 


60" 


1 


75877 


24123 


84550 


15450 


08672 


91328 


59 


2 


75895 


24105 


84576 


15424 


08681 


91319 


58 


3 


75913 


24087 


84603 


15397 


08690 


91310 


57 


4 


75931 


24069 


84630 


15370 


08699 


91301 


56 


5 


9.75949 


10.24051 


9.84657 


10.15343 


10.08708 


9.91292 


55 


6 


75967 


24033 


84684 


15316 


08717 


91283 


54 


7 


75985 


24015 


84711 


15289 


08726 


91274 


53 


8 


76003 


23997 


84738 


15262 


08734 


91266 


52 


9 


76021 


23979 


84764 


15236 


08743 


91257 


51 


10 


9.76039 


10.23961 


9.84791 


10.15209 


10.08752 


9.91248 


50 


11 


76057 


23943 


84818 


15182 


08761 


91239 


49 


12 


76075 


23925 


84845 


15155 


08770 


91230 


48 


13 


76093 


23907 


84872 


15128 


08779 


91221 


47 


14 


76111 


23889 


84899 


15101 


08788 


91212 


46 


15 


9.76129 


10.23871 


9.84925 


10.15075 


10.08797 


9.91203 


45 


16 


76146 


23854 


84952 


15048 


08806 


91194 


44 


17 


76164 


23836 


84979 


15021 


08815 


91185 


43 


18 


76182 


23818 


85006 


14994 


08824 


91176 


42 


19 


76200 


23800 


85033 


14967 


08833 


91167 


41 


20 


9.76218 


10.23782 


9.85059 


10.14941 


10.08842 


9.91158 


40 


21 


76236 


23764 


85086 


14914 


08851 


91149 


39 


22 


76253 


23747 


85113 


14887 


08859 


91141 


38 


23 


76271 


23729 


85140 


14860 


08868 


91132 


37 


24 


76289 


23711 


85166 


14834 


08877 


91123 


36 


25 


9.76307 


10.23693 


9.85193 


10.14807 


10.08886 


9.91114 


35 


26 


76324 


23676 


85220 


14780 


08895 


91105 


34 


27 


76342 


23658 


85247 


14753 


08904 


91096 


33 


28 


76360 


23640 


85273 


14727 


08913 


91087 


32 


29 


76378 


23622 


85300 


14700 


08922 


91078 


31 


30 


9.76395 


10.23605 


9.85327 


10.14673 


10.08931 


9.91069 


30 


31 


76413 


23587 


85354 


14646 


08940 


91060 


29 


32 


76431 


23569 


85380 


14620 


08949 


91051 


28 


33 


76448 


23552 


85407 


14593 


08958 


91042 


27 


34 


76466 


23534 


85434 


14566 


08967 


91033 


26 


35 


9.76484 


10.23516 


9.85460 


10.14540 


10.08977 


9.91023 


25 


36 


76501 


23499 


85487 


14513 


08986 


91014 


24 


37 


76519 


23481 


85514 


14486 


08995 


91005 


23 


38 


76537 


23463 


85540 


14460 


09004 


90996 


22 


39 


76554 


23446 


85567 


14433 


09013 


. 90987 


21 


40 


9.76572 


10.23428 


9.85594 


10.14406 


10.09022 


9.90978 


20 


41 


76590 


23410 


86620 


14380 


09031 


90969 


19 


42 


76607 


23393 


85647 


14353 


09040 


90960 


18 


43 


76625 


23375 


85674 


14326 


09049 


90951 


17 


44 


76642 


23358 


85700 


14300 


09058 


90942 


16 


45 


9.76660 


10.23340 


9.85727 


10.14273 


10.09067 


9.90933 


15 


46 


76677 


23323 


85754 


14246 


09076 


90924 


14 


47 


76695 


23305 


85780 


14220 


09085 


90915 


13 


48 


76712 


23288 


85807 


14193 


09094 


90906 


12 


49 


76730 


23270 


85834 


14166 


09104 


90896 


11 


50 


9.76747 


10.23253 


9.85860 


10.14140 


10.09113 


9.90887 


10 


51 


76765 


23235 


85887 


14113 


09122 


90878 


9 


52 


76782 


23218 


85913 


14087 


09131 


90869 


8 


53 


76800 


23200 


85940 


14060 


09140 


90860 


7 


54 


76817 


23183 


85967 


14033 


09149 


90851 


6 


55 


9.76835 


10.23165 


9.85993 


10.14007 


10.09158 


9.90842 


5 


56 


76852 


23148 


86020 


13980 


09168 


90832 


4 


57 


76870 


23130 


86046 


13954 


09177 


90823 


3 


58 


76887 


23113 


86073 


13927 


09186 


90814 


2 1 


59 


76904 


23096 


86100 


13900 


09195 


90805 


l 1 


60 


76922 


23078 


86126 


13874 


09204 


90796 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. ] 


M. 



125° 



54° 



Logarithmic Angular Functions. 



215 



36° 






Logarithms. 






143° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.76922 


10.23078 


9.86126 


10.13874 


10.09204 


9.90796 


60 


1 


76939 


23061 


86153 


13847 


09213 


90787 


59 


2 


76957 


23043 


86179 


13821 


09223 


90777 


58 


3 


76974 


23026 


86206 


13794 


09232 


90768 


57 


4 


76991 


23009 


86232 


13768 


09241 


90759 


56 


5 


9.77009 


10.22991 


9.86259 


10.13741 


10.09250 


9.90750 


55 


6 


77026 


22974 


86285 


13715 


09259 


90741 


54 


7 


77043 


22957 


86312 


13688 


09269 


90731 


53 


8 


77061 


22939 


86338 


13662 


09278 


90722 


52 


9 


77078 


22922 


86365 


13635 


09287 


90713 


51 


10 


9.77095 


10.22905 


9.86392 


10.13608 


10.09296 


9.90704 


50 


11 


77112 


22888 


86418 


13582 


09306 


90694 


49 


12 


77130 


22870 


86445 


13555 


09315 


90685 


48 


13 


77147 


22853 


86471 


13529 


09324 


90676 


47 


14 


77164 


22836 


86498 


13502 


09333 


90667 


46 


15 


9.77181 


10.22819 


9.86524 


10.13476 


10.09343 


9.90657 


45 


16 


77199 


22801 


86551 


13449 


09352 


90648 


44 


17 


77216 


22784 


86577 


13423 


09361 


90639 


43 


18 


77233 


22767 


86603 


13397 


09370 


90630 


42 


19 


77250 


22750 


86630 


13370 


09380 


90620 


41 


20 


9.77268 


10.22732 


9.86656 


10.13344 


10.09389 


9.90611 


40 


21 


77285 


22715 


86683 


13317 


09398 


90602 


39 


22 


77302 


22698 


86709 


13291 


09408 


90592 


38 


23 


77319 


22681 


86736 


13264 


09417 


90583 


37 


24 


77336 


22664 


86762 


13238 


09426 


90574 


36 


25 


9.77353 


10.22647 


9.86789 


10.13211 


10.09435 


9.90565 


35 


26 


77370 


22630 


86815 


13185 


09445 


90555 


34 


27 


77387 


22613 


86842 


13158 


09454 


90546 


33 


28 


77405 


. 22595 


86868 


13132 


09463 


90537 


32 


29 


77422 


22578 


86894 


13106 


09473 


90527 


31 


30 


9.77439 


10.22561 


9.86921 


10.13079 


10.09482 


9.90518 


30 


31 


77456 


22544 


86947 


13053 


09491 


90509 


29 


32 


77473 


22527 


86974 


13026 


09501 


90499 


28 


33 


77490 


22510 


87000 


13000 


09510 


90490 


27 


34 


77507 


22493 


87027 


12973 


09520 


90480 


26 


35 


9.77524 


10.22476 


9.87053 


10.12947 


10.09529 


9.90471 


25 


36 


77541 


22459 


87079 


12921 


09538 


90462 


24 


37 


77558 


22442 


87106 


12894 


09548 


90452 


23 


38 


77575 


22425 


87132 


12868 


09557 


90443 


22 


39 


77592 


22408 


87158 


12842 


09566 


90434 


21 


40 


9.77609 


10.22391 


9.87185 


10.12815 


10.09576 


9.90424 


20 


41 


77626 


22374 


87211 


12789 


09585 


90415 


19 


42 


77643 


22357 


87238 


12762 


09595 


90405 


18 


43 


77660 


22340 


87264 


12736 


09604 


90396 


17 


44 


77677 


22323 


87290 


12710 


09614 


90386 


16 


45 


9.77694 


10.22306 


9.87317 


10.12683 


10.09623 


9.90377 


15 


46 


77711 


22289 


87343 


12657 


09632 


90368 


14 


47 


77728 


22272 


87369 


12631 


09642 


90358 


13 


48 


77744 


22256 


87396 


12604 


09651 


90349 


12 


49 


77761 


22239 


87422 


12578 


09661 


90339 


11 


50 


9.77778 


10.22222 


9.87448 


10.12552 


10.09670 


9.90330 


10 


51 


77795 


22205 


87475 


12525 


09680 


90320 


9 


52 


77812 


22188 


87501 


12499 


09689 


90311 


8 


53 


77829 


22171 


87527 


12473 


09699 


90301 


7 


54 


77846 


22154 


87554 


12446 


09708 


90292 


6 


55 


9.77862 


10.22138 


9.87580 


10.12420 


10.09718 


9.90282 


5 


56 


77879 


22121 


87606 


12394 


09727 


90273 


4 


57 


77896 


22104 


87633 


12367 


09737 


90263 


3 


58 


77913 


22087 


87659 


12341 


09746 


90254 


2 


59 


77930 


22070 


87685 


12315 


09756 


90244 


1 


60 


77946 


22054 


87711 


12289 


09765 


90235 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



126° 



53° 



216 



Logarithmic Angular Functions. 



37° 



Logarithms. 



142° 



M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.77946 


10.22054 


9.87711 


10.12289 


10.09765 


9.90235 


60 


1 


77963 


22037 


87738 


12262 


09775 


90225 


59 


2 


77980 


22020 


87764 


12236 


09784 


90216 


58 


3 


77997 


22003 


87790 


12210 


09794 


90206 


57 


4 


78013 


21987 


87817 


12183 


09803 


90197 


56 


5 


9.78030 


10.21970 


9.87843 


10.12157 


10.09813 


9.90187 


55 


6 


78047 


21953 


87869 


12131 


09822 


90178 


54 


7 


78063 


21937 


87895 


12105 


09832 


90168 


53 


8 


78080 


21920 


87922 


12078 


09841 


90159 


52 


9 


78097 


21903 


87948 


12052 


09851 


90149 


51 


10 


9.78113 


10.21887 


9.87974 


10.12026 


10.09861 


9.90139 


50 


11 


78130 


21870 


88000 


12000 


09870 


90130 


49 


12 


78147 


21853 


88027 


11973 


09880 


90120 


48 


13 


78163 


21837 


88053 


11947 


09889 


90111 


47 


14 


78180 


21820 


88079 


11921 


09899 


90101 


46 


15 


9.78197 


10.21803 


9.88105 


10.11895 


10.09909 


9.90091 


45 


16 


78213 


21787 


88131 


11869 


09918 


90082 


44 


17 


78230 


21770 


88158 


11842 


09928 


90072 


43 


18 


78246 


21754 


88184 


11816 


09937 


90063 


42 


19 


78263 


21737 


88210 


11790 


09947 


90053 


41 


20 


9.78280 


10.21720 


9.88236 


10.11764 


10.09957 


9.90043 


40 


21 


78296 


21704 


88262 


11738 


09966 


90034 


39 


22 


78313 


21687 


88289 


11711 


09976 


90024 


38 


23 


78329 


21671 


88315 


11685 


09986 


90014 


37 


24 


78346 


21654 


88341 


11659 


09995 


90005 


36 


25 


9.78362 


10.21638 


9.88367 


10.11633 


10.10005 


9.89995 


35 


26 


78379 


21621 


88393 


11607 


10015 


89985 


34 


27 


78395 


21605 


88420 


11580 


10024 


89976 


33 


28 


78412 


21588 


88446 


11554 


10034 


89966 


32 


29 


78428 


21572 


88472 


11528 


10044 


89956 


31 


30 


9.78445 


10.21555 


9.88498 


10.11502 


10.10053 


9.89947 


30 


31 


78461 


21539 


88524 


11476 


10063 


89937 


29 


32 


78478 


21522 


88550 


11450 


10073 


89927 


28 


33 


78494 


21506 


88577 


11423 


10082 


89918 


27 


34 


78510 


21490 


88603 


11397 


10092 


89908 


26 


35 


9.78527 


10.21473 


9.88629 


10.11371 


10.10102 


9.89898 


25 


36 


78543 


21457 


88655 


11345 


10112 


89888 


24 


37 


78560 


21440 


88681 


11319 


10121 


89879 


23 


38 


78576 


21424 


88707 


11293 


10131 


89869 


22 


39 


78592 


21408 


88733 


11267 


10141 


89859 


21 


40 


9.78609 


10.21391 


9.88759 


10.11241 


10.10151 


9.89849 


20 


41 


78625 


21375 


88780 


11214 


10160 


89840 


19 


42 


78642 


21358 


88812 


11188 


10170 


89830 


18 


43 


78658 


21342 


88838 


11162 


10180 


89820 


17 


44 


78674 


21326 


88864 


11136 


10190 


89810 


16 


45 


9.78691 


10.21309 


9.88890 


10.11110 


10.10199 


9.89801 


15 


46 


78707 


21293 


88916 


11084 


10209 


89791 


14 


47 


78723 


21277 


88942 


11058 


10219 


89781 


13 


48 


78739 


21261 


88968 


11032 


10229 


89771 


12 


49 


78756 


21244 


88994 


11006 


10239 


89761 


11 


50 


9.78772 


10.21228 


9.89020 


10.10980 


10.10248 


9.89752 


10 


51 


78788 


21212 


89046 


10954 


10258 


89742 


9 


52 


78805 


21195 


89073 


10927 


10268 


89732 


8 


53 


7882] 


21179 


89099 


10901 


10278 


89722 


7 


54 


78837 


21163 


89125 


10875 


10288 


89712 


6 


55 


9.78858 


10.21147 


9.89151 


10.10849 


10.10298 


9.89702 


5 


56 


78869 


21131 


89177 


10823 


10307 


89693 


4 


57 


78886 


21114 


89203 


10797 


10317 


89683 


3 


58 


78902 


21098 


89229 


10771 


10327 


89673 


2 


59 


78918 


21082 


89255 


10745 


10337 


89663 


1 


60 


78934 


21066 


89281 


10719 


10347 


89653 





M. 


Cosine. 


Secant. 


< lotangent, 


Tangent. 


Cosecant. 


Sine. 


M. 



127° 



Logarithmic Angular Functions. 



217 



38° 






Logarithms. 




141° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


| Secant. 


Cosine. 


M. 





9.78934 


10.21066 


9.89281 


10.10719 


10.10347 


9.89653 


60 


1 


78950 


21050 


89307 


10693 


10357 


89643 


59 


2 


78967 


21033 


89333 


10667 


10367 


89633 


58 


3 


78983 


21017 


89359 


10641 


10376 


89624 


57 


4 


78999 


21001 


89385 


10615 


10386 


89614 


56 


5 


9.79015 


10.20985 


9.89411 


10.10589 


10.10396 


9.89604 


55 


6 


79031 


20969 


89437 


10563 


10406 


89594 


54 


7 


79047 


20953 


89463 


10537 


10416 


89584 


53 


8 


79063 


20937 


89489 


10511 


10426 


89574 


52 


9 


79079 


20921 


89515 


10485 


10436 


89564 


51 


10 


9.79095 


10.20905 


9.89541 


10.10459 


10.10446 


9.89554 


50 


11 


79111 


20889 


89567 


10433 


10456 


89544 


49 


12 


79128 


20872 


89593 


10407 


10466 


89534 


48 


13 


79144 


20856 


89619 


10381 


10476 


89524 


47 


14 


79160 


20840 


89645 


10355 


10486 


89514 


46 


15 


9.79176 


10.20824 


9.89671 


10.10329 


10.10496 


9.89504 


45 


16 


79192 


20808 


89697 


10303 


10505 


89495 


44 


17 


79208 


20792 


89723 


10277 


10515 


89485 


43 


18 


79224 


20776 


89749 


10251 


10525 


89475 


42 


19 


79240 


20760 


89775 


10225 


10535 


89465 


41 


20 


9.79256 


10.20744 


9.89801 


10.10199 


10.10545 


9.89455 


40 


21 


79272 


20728 


89827 


10173 


10555 


89445 


39 


22 


79288 


20712 


89853 


10147 


10565 


89435 


38 


23 


79304 


20696 


89879 


10121 


10575 


89425 


37 


24 


79319 


20681 


89905 


10095 


10585 


89415 


36 


25 


9.79335 


10.20665 


9.89931 


10.10069 


10.10595 


9.89405 


35 


26 


79351 


20649 


89957 


10043 


10605 


89395 


34 


27 


79367 


20633 


89983 


10017 


10615 


89385 


33 


28 


79383 


20617 


90009 


09991 


10625 


89375. 


32 


29 


79399 


20601 


90035 


09965 


10636 


89364 


31 


30 


9.79415 


10.20585 


9.90061 


10.09939 


10.10646 


9.89354 


30 


31 


79431 


20569 


90086 


09914 


10656 


89344 


29 


32 


79447 


20553 


90112 


09888 


10666 


89334 


28 


33 


79463 


20537 


90138 


09862 


10676 


89324 


27 


34 


79478 


20522 


90164 


09836 


10686 


89314 


26 


35 


9.79494 


10.20506 


9.90190 


10.09810 


10.10696 


9.89304 


25 


36 


79510 


20490 


90216 


09784 


10706 


89294 


24 


37 


79526 


20474 


90242 


09758 


10716 


89284 


23 


38 


79542 


20458 


90268 


09732 


10726 


89274 


22 


39 


79558 


20442 


90294 


09706 


10736 


89264 


21 


40 


9.79573 


10.20427 


9.90320 


10.09680 


10.10746 


9.89254 


20 


41 


79589 


20411 


90346 


09654 


10756 


89244 


19 


42 


79605 


20395 


90371 


09629 


10767 


89233 


18 


43 


79621 


20379 


90397 


09603 


10777 


89223 


17 


44 


79636 


20364 


90423 


09577 


10787 


89213 


16 


45 


9.79652 


10.20348 


9.90449 


10.09551 


10.10797 


9.89203 


15 


46 


79668 


20332 


90475 


09525 


10807 


89193 


14 


47 


79684 


20316 


90501 


09499 


10817 


89183 


13 


48 


79699 


20301 


90527 


09473 


10827 


89173 


12 


49 


79715 


20285 


90553 


09447 


10838 


89162 


11 


50 


9.79731 


10.20269 


9.90578 


10.09422 


10.10848 


9.89152 


10 


51 


79746 


20254 


90604 


09396 


10858 


89142 


9 


52 


79762 


20238 


90630 


09370 


10868 


89132 


8 


53 


79778 


20222 


90656 


09344 


10878 


89122 


7 


54 


79793 


20207 


90682 


09318 


10888 


89112 


6 


55 


9.79809 


10.20191 


9.90708 


10.09292 


10.10899 


9.89101 


5 


56 


79825 


20175 


90734 


09266 


10909 


89091 


4 


57 


79840 


20160 


90759 


09241 


10919 


89081 


3 


58 


79856 


20144 


90785 


09215 


10929 


89071 


2 


59 


79872 


20128 


90811 


09189 


10940 


89060 


1 


60 


79887 


20113 


90837 


09163 


10950 


89050 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



128° 



51° 



218 



Logarithmic Angular Functions. 



39° 






Logarithms. 




140° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.79887 


10.20113 


9.90837 


10.09163 


10.10950 


9.89050 


60 


1 


79903 


20097 


90863 


09137 


10960 


89040 


59 


2 


79918 


20082 


90889 


09111 


10970 


89030 


58 


3 


79934 


20066 


90914 


09086 


10980 


89020 


57 


4 


79950 


20050 


90940 


09060 


10991 


89009 


56 


5 


9.79965 


10.20035 


9.90966 


10.09034 


10.11001 


9.88999 


55 


6 


79981 


20019 


90992 


09008 


11011 


88989 


54 


7 


79996 


20004 


91018 


08982 


11022 


88978 


53 


8 


80012 


19988 


91043 


08957 


11032 


88968 


52 


9 


80027 


19973 


91069 


08931 


11042 


88958 


51 


10 


9.80043 


10.19957 


9.91095 


10.08905 


10.11052 


9.88948 


50 


11 


80058 


19942 


91121 


08879 


11063 


88937 


49 


12 


80074 


19926 


91147 


08853 


11073 


88927 


48 


13 


80089 


19911 


91172 


08828 


11083 


88917 


47 


14 


80105 


19895 


91198 


08802 


11094 


88906 


46 


15 


9.80120 


10.19880 


9.91224 


10.08776 


10.11104 


9.88896 


45 


16 


80136 


19864 


91250 


08750 


11114 


88886 


44 


17 


80151 


19849 


91276 


08724 


11125 


88875 


43 


18 


80166 


19834 


91301 


08699 


11135 


88865 


42 


19 


80182 


19818 


91327 


08673 


11145 


88855 


41 


20 


9.80197 


10.19803 


9.91353 


10.08647 


10.11156 


9.88844 


40 


21 


80213 


19787 


91379 


08621 


11166 


88834 


39 


22 


80228 


19772 


91404 


08596 


11176 


88824 


38 


23 


80244 


19756 


91430 


08570 


11187 


88813 


37 


24 


80259 


19741 


91456 


08544 


11197 


88803 


36 


25 


9.80274 


10.19726 


9.91482 


10.08518 


10.11207 


9.88793 


35 


26 


80290 


19710 


91507 


08493 


11218 


88782 


34 


27 


80305 


19695 


91533 


08467 


11228 


88772 


33 


28 


80320 


19680 


91559 


08441 


11239 


88761 


32 


29 


80336 


19664 


91585 


08415 


11249 


88751 


31 


30 


9.80351 


10.19649 


9.91610 


10.08390 


10.11259 


9.88741 


30 


31 


80366 


19634 


91636 


08364 


11270 


88730 


29 


32 


80382 


19618 


91662 


08338 


11280 


88720 


28 


33 


80397 


19603 


91688 


08312 


11291 


88709 


27 


34 


80412 


19588 


91713 


08287 


11301 


88699 


26 


35 


9.80428 


10.19572 


9.91739 


10.08261 


10.11312 


9.88688 


25 


36 


80443 


19557 


91765 


08235 


11322 


88678 


24 


37 


80458 


19542 


91791 


08209 


11332 


88668 


23 


38 


80473 


19527 


91816 


08184 


11343 


88657 


22 


39 


80489 


19511 


91842 


08158 


11353 


88647 


21 


40 


9.80504 


10.19496 


9.91868 


10.08132 


10.11364 


9.88636 


20 


41 


80519 


19481 


91893 


08107 


11374 


88626 


19 


42 


80534 


19466 


91919 


08081 


11385 


88615 


18 


43 


80550 


19450 


91945 


08055 


11395 


88605 


17 


44 


80565 


19435 


91971 


08029 


11406 


88594 


16 


45 


9.80580 


10.19420 


9.91996 


10.08004 


10.11416 


9.88584 


15 


46 


80595 


19405 


92022 


07978 


11427 


88573 


14 


47 


80610 


19390 


92048 


07952 


11437 


88563 


13 


48 


80625 


19375 


92073 


07927 


11448 


88552 


12 


49 


80641 


19359 


92099 


07901 


11458 


88542 


11 


50 


9.80656 


10.19344 


9.92125 


10.07875 


10.11469 


9.88531 


10 


51 


N0671 


19329 


92150 


07850 


11479 


88521 


9 


52 


80686 


19314 


92176 


07824 


11490 


88510 


8 


53 


80701 


19299 


92202 


07798 


11501 


88499 


7 


51 


80716 


19284 


92227 


07773 


11511 


88489 


6 


55 


9.80731 


10.19269 


9.92253 


10.077 J 7 


10.11522 


9.88478 


5 


56 


so; 16 


19254 


92279 


07721 


11532 


88468 


4 


57 


80762 


L9238 


92304 


07696 


11543 


88457 


3 


58 


80777 


19223 


92330 


07670 


11553 


88447 


2 


59 


80792 


19208 


92356 


07644 


11564 


88436 


1 


60 


80807 


19193 


92381 


07619 


11575 


88425 





M. 


1 osine. 


Secant. 


Cotangent, 


Tangent. 


Cosecant. 


Sine. 


M. 



129° 



50° 



Logarithmic Angular Functions. 



219 



40° 






Logarithms. 




139° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.80807 


10.19193 


9.92381 


10.07619 


10.11575 


9.88425 


60 


1 


80822 


19178 


92407 


07593 


11585 


88415 


59 


2 


80837 


19163 


92433 


07567 


11596 


88404 


58 


3 


80852 


19148 


92458 


07542 


11606 


88394 


57 


4 


80867 


19133 


92484 


07516 


11617 


88383 


56 


5 


9.80882 


10.19118 


9.92510 


10.07490 


10.11628 


9.88372 


55 


6 


80897 


19103 


92535 


07465 


11638 


88362 


54 


7 


80912 


19088 


92561 


07439 


11649 


88351 


53 


8 


80927 


19073 


92587 


07413 


11660 


88340 


52 


9 


80942 


19058 


92612 


07388 


11670 


88330 


51 


10 


9.80957 


10.19043 


9.92638 


10.07362 


10.11681 


9.88319 


50 


11 


80972 


19028 


92663 


07337 


11692 


88308 


49 


12 


80987 


19013 


92689 


07311 


11702 


88298 


48 


13 


81002 


18998 


92715 


07285 


11713 


88287 


47 


14 


81017 


18983 


92740 


07260 


11724 


88276 


46 


15 


9.81032 


10.18968 


9.92766 


10.07234 


10.11734 


9.88266 


45 


16 


81047 


18953 


92792 


07208 


11745 


88255 


44 


17 


81061 


18939 


92817 


07183 


11756 


88244 


43 


18 


81076 


18924 


92843 


07157 


11766 


88234 


42 


19 


81091 


18909 


92868 


07132 


11777 


88223 


41 


20 


9.81106 


10.18894 


9.92894 


10.07106 


10.11788 


9.88212 


40 


21 


81121 


18879 


92920 


07080 


11799 


88201 


39 


22 


81136 


18864 


92945 


07055 


11809 


88191 


38 


23 


81151 


18849 


92971 


07029 


11820 


88180 


37 


24 


81166 


18834 


92996 


07004 


11831 


88169 


36 


25 


9.81180 


10.18820 


9.93022 


10.06978 


10.11842 


9.88158 


35 


26 


81195 


18805 


93048 


06952 


11852 


88148 


34 


27 


81210 


18790 


93073 


06927 


11863 


88137 


33 


28 


81225 


18775 


93099 


06901 


11874 


88126 


32 


29 


81240 


18760 


93124 


06876 


11885 


88115 


31 


30 


9.81254 


10.18746 


9.93150 


10.06850 


10.11895 


9.88105 


30 


31 


81269 


18731 


93175 


06825 


11906 


88094 


29 


32 


81284 


18716 


93201 


06799 


11917 


88083 


28 


33 


81299 


18701 


93227 


06773 


11928 


88072 


27 


34 


81314 


18686 


93252 


06748 


11939 


88061 


26 


35 


9.81328 


10.18672 


9.93278 


10.06722 


10.11949 


9.88051 


25 


36 


81343 


18657 


93303 


06697 


11960 


88040 


24 


37 


81358 


18642 


93329 


06671 


11971 


88029 


23 


38 


81372 


18628 


93354 


06646 


11982 


88018 


22 


39 


81387 


18613 


93380 


06620 


11993 


88007 


21 


40 


9.81402 


10.18598 


9.93406 


10.06594 


10.12004 


9.87996 


20 


41 


81417 


18583 


93431 


06569 


12015 


87985 


19 


42 


81431 


18569 


93457 


06543 


12025 


87975 


18 


43 


81446 


18554 


93482 


06518 


12036 


87964 


17 


44 


81461 


18539 


93508 


06492 


12047 


87953 


16 


45 


9.81475 


10.18525 


9.93533 


10.06467 


10.12058 


9.87942 


15 


46 


81490 


18510 


93559 


06441 


12069 


87931 


14 


47 


81505 


18495 


93584 


06416 


12080 


87920 


13 


48 


81519 


18481 


93610 


06390 


12091 


87909 


12 


49 


81534 


18466 


93636 


06364 


12102 


87898 


11 


50 


9.81549 


10.18451 


9.93661 


10.06339 


10.12113 


9.87887 


10 


51 


81563 


18437 


93687 


06313 


12123 


87877 


9 


52 


81578 


18422 


93712 


06288 


12134 


87866 


8 


53 


81592 


18408 


93738 


06262 


12145 


87855 


7 


54 


81607 


18393 


93763 


06237 


12156 


87844 


6 


55 


9.81622 


10.18378 


9.93789 


10.06211 


10.12167 


9.87833 


5 


56 


81636 


18364 


93814 


06186 


12178 


87822 


4 


57 


81651 


18349 


93840 


06160 


12189 


87811 


3 


58 


81665 


18335 


93865 


06135 


12200 


87800 


2 


59 


81680 


18320 


93891 


06109 


12211 


87789 


1 


60 


81694 


18306 


93916 


06084 


12222 


87778 





M, 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


1 Cosecant. 


Sine. 


M. 



130° 



49° 



220 



Logarithmic Angular Functions. 



41° 






Logarithms. 




138° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.81694 


10.18306 


9.93916 


10.06084 


10.12222 


9.87778 


60 


1 


81709 


18291 


93942 


06058 


12233 


87767 


59 


2 


81723 


18277 


93967 


06033 


12244 


87756 


58 


3 


81738 


18262 


93993 


06007 


12255 


87745 


57 


4 


81752 


18248 


94018 


05982 


12266 


87734 


56 


5 


9.81767 


10.18233 


9.94044 


10.05956 


10.12277 


9.87723 


55 


6 


81781 


18219 


94069 


05931 


12288 


87712 


54 


7 


81796 


18204 


94095 


05905 


12299 


87701 


53 


8 


81810 


18190 


94120 


05880 


12310 


87690 


52 


9 


81825 


18175 


94146 


05854 


12321 


87679 


51 


10 


9.81839 


10.18161 


9.94171 


10.05829 


10.12332 


9.87668 


50 


11 


81854 


18146 


94197 


05803 


12343 


87657 


49 


12 


81868 


18132 


94222 


05778 


12354 


87646 


48 


13 


81882 


18118 


94248 


05752 


12365 


87635 


47 


14 


81897 


18103 


94273 


05727 


12376 


87624 


46 


15 


9.81911 


10.18089 


9.94299 


10.05701 


10.12387 


9.87613 


45 


16 


81926 


18074 


94324 


05676 


12399 


87601 


44 


17 


81940 


18060 


94350 


05650 


12410 


87590 


43 


18 


81955 


18045 


94375 


05625 


12421 


87579 


42 


19 


81969 


18031 


94401 


05599 


12432 


87568 


41 


20 


9.81983 


10.18017 


9.94426 


10.05574 


10.12443 


9.87557 


40 


21 


81998 


18002 


94452 


05548 


12454 


87546 


39 


22 


82012 


17988 


94477 


05523 


12465 


87535 


38 


23 


82026 


17974 


94503 


05497 


12476 


87524 


37 


24 


82041 


17959 


94528 


05472 


12487 


87513 


36 


25 


9.82055 


10.17945 


9.94554 


10.05446 


10.12499 


9.87501 


35 


26 


82069 


17931 


94579 


05421 


12510 


87490 


34 


27 


82084 


17916 


94604 


05396 


12521 


87479 


33 


28 


82098 


17902 


94630 


05370 


12532 


87468 


32 


29 


82112 


17888 


94655 


05345 


12543 


87457 


31 


30 


9.82126 


10.17874 


9.94681 


10.05319 


10.12554 


9.87446 


30 


3i 


82141 


17859 


94706 


05294 


12566 


87434 


29 


32 


82155 


17845 


94732 


05268 


12577 


87423 


28 


33 


82169 


17831 


94757 


05243 


12588 


87412 


27 


34 


82184 


17816 


94783 


05217 


12599 


87401 


26 


35 


9.82198 


10.17802 


9.94808 


10.05192 


10.12610 


9.87390 


25 


36 


82212 


17788 


94834 


05166 


12622 


87378 


24 


37 


82226 


17774 


94859 


05141 


12633 


87367 


23 


38 


82240 


17760 


94884 


05116 


12644 


87356 


22 


39 


82255 


17745 


94910 


05090 


12655 


87345 


21 


40 


9.82269 


10.17731 


9.94935 


10.05065 


10.12666 


9.87334 


20 


41 


82283 


17717 


94961 


05039 


12678 


87322 


19 


42 


82297 


17703 


94986 


05014 


12689 


87311 


18 


43 


82311 


17689 


95012 


04988 


12700 


87300 


17 


44 


82326 


17674 


95037 


04963 


12712 


87288 


16 


45 


9.82310 


10.17660 


9.95062 


10.04938 


10.1272:', 


9.87277 


15 


46 


82354 


17646 


95088 


04912 


12734 


87266 


14 


47 


82368 


17632 


95113 


04887 


12745 


87255 


13 


48 


82382 


17618 


95139 


04861 


12757 


87243 


12 


49 


82:', 96 


17604 


95164 


04836 


12768 


87232 


11 


50 


9.82410 


10.17590 


9.95190 


10.04810 


10.12779 


9.87221 


10 


51 


82424 


17576 


95215 


04785 


12791 


87209 


9 


52 


82439 


17561 


95240 


04760 


12802 


87198 


8 


68 


82453 


17.". 17 


95266 


04734 


12813 


87187 


7 


54 


82467 


17533 


95291 


04709 


12825 


87175 


6 


55 


9.82481 


10.17519 


9.95317 


10.04683 


10.12836 


9.87164 


5 


56 


82495 


17505 


95342 


04658 


12847 


87153 


4 


57 


82509 


17491 


95368 


04632 


12859 


87141 


3 


58 


82523 


17477 


95393 


04607 


12870 


87130 


2 


59 


82537 


17463 


95418 


04582 


L2881 


87119 


1 


60 


82553 


17449 


95444 


04556 


12893 


87107 





M. 


< lotAne. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



131° 



Logarithmic Angular Functions. 



221 



42° 






Logarithms. 




1 


37° 


M. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.82551 


10.17449 


9.95444 


10.04556 


10.12893 


9.87107 


60 


1 


82565 


17435 


95469 


04531 


12904 


87096 


59 


2 


82579 


17421 


95495 


04505 


12915 


87085 


58 


3 


82593 


17407 


95520 


04480 


12927 


87073 


57 


4 


82607 


17393 


95545 


04455 


12938 


87062 


56 


5 


9.82621 


10.17379 


9.95571 


10.04429 


10.12950 


9 87050 


55 


6 


82635 


17365 


95596 


04404 


12961 


87039 


54 


7 


82649 


17351 


95622 


04378 


12972 


87028 


53 


8 


82663 


17337 


95647 


04353 


12984 


87016 


52 


9 


82677 


17323 


95672 


04328 


12995 


87005 


51 


10 


9.82691 


10.17309 


9.95698 


10.04302 


10.13007 


9.86993 


50 


11 


82705 


17295 


95723 


04277 


13018 


86982 


49 


12 


82719 


17281 


95748 


04252 


13030 


86970 


48 


13 


82733 


17267 


95774 


04226 


13041 


86959 


47 


14 


82747 


17253 


95799 


04201 


13053 


86947 


46 


15 


9.82761 


10.17239 


9.95825 


10.04175 


10.13064 


9.86936 


45 


16 


82775 


17225 


95850 


04150 


13076 


86924 


44 


17 


82788 


17212 


95875 


04125 


13087 


86913 


43 


18 


82802 


17198 


95901 


04099 


13098 


86902 


42 


19 


82816 


17184 


95926 


04074 


13110 


86890 


41 


20 


9.82830 


10.17170 


9.95952 


10.04048 


10.13121 


9.86879 


40 


21 


82844 


17156 


95977 


04023 


13133 


86867 


39 


22 


82858 


17142 


96002 


03998 


13145 


86855 


38 


23 


82872 


17128 


96028 


03972 


13156 


86844 


37 


24 


82885 


17115 


96053 


03947 


13168 


86832 


36 


25 


9.82899 


10.17101 


9.96078 


10.03922 


10.13179 


9.86821 


35 


26 


82913 


17087 


96104 


03896 


13191 


86809 


34 


27 


82927 


17073 


96129 


03871 


13202 


86798 


33 


28 


82941 


17059 


96155 


03845 


13214 


86786 


32 


29 


82955 


17045 


96180 


03820 


13225 


86775 


31 


30 


9.82968 


10.17032 


9.96205 


10.03795 


10.13237 


9.86763 


30 


31 


82982 


17018 


96231 


03769 


13248 


86752 


29 


32 


82996 


17004 


96256 


03744 


13260 


86740 


28 


33 


83010 


16990 


96281 


03719 


13272 


86728 


27 


34 


83023 


16977 


96307 


03693 


13283 


86717 


26 


35 


9.83037 


10.16963 


9.96332 


10.03668 


10.13295 


9.86705 


25 


36 


83051 


16949 


96357 


03643 


13306 


86694 


24 


37 


83065 


16935 


96383 


03617 


13318 


86682 


23 


38 


83078 


16922 


96408 


03592 


13330 


86670 


22 


39 


83092 


16908 


96433 


03567 


13341 


86659 


21 


40 


9.83106 


10.16894 


9.96459 


10.03541 


10.13353 


9.86647 


20 


41 


83120 


16880 


96484 


03516 


13365 


86635 


19 


42 


83133 


16867 


96510 


03490 


13376 


86624 


18 


43 


83147 


16853 


96535 


03465 


13388 


86612 


17 


44 


83161 


16839 


96560 


03440 


13400 


86600 


16 


45 


9.83174 


10.16826 


9.96586 


10.03414 


10.13411 


9.86589 


15 


46 


83188 


16812 


96611 


03389 


13423 


86577 


14 


47 


83202 


16798 


96636 


03364 


13435 


86565 


13 


48 


83215 


16785 


96662 


03338 


13446 


86554 


12 


49 


83229 


16771 


96687 


03313 


13458 


86542 


11 


50 


9.83242 


10.16758 


9.96712 


10.03288 


10.13470 


9.86530 


10 


51 


83256 


16744 


96738 


03262 


13482 


86518 


9 


52 


83270 


16730 


96763 


03237 


13493 


86507 


8 


53 


83283 


16717 


96788 


03212 


13505 


86495 


7 


54 


83297 


16703 


96814 


03186 


13517 


86483 


6 


55 


9.83310 


10.16690 


9.96839 


10.03161 


10.13528 


9.86472 


5 


56 


83324 


16676 


96864 


03136 


13540 


86460 


4 


57 


83338 


16662 


96890 


03110 


13552 


86448 


3 


58 


83351 


16649 


96915 


03085 


13564 


86436 


2 


59 


83365 


16635 


96940 


03060 


13575 


86425 


1 


60 


83378 


16622 


96966 


03034 


13587 


86413 





M. 


Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



132° 



47° 



222 



Logarithmic Angular Functions. 



43° 






Logarithms. 






136° 


M. 


Sine. 
9.83378 


Cosecant. 
10.16622 


Tangent 


Cotangent. 


Secant. 


Cosine. 


M. 





9.96966 


10.03034 


10.18.8S7 


9.86413 


60 


1 


83392 


16608 


96991 


03009 


18599 


S6401 


59 


2 


83405 


16595 


9701(5 


0298 1 


13611 


86889 


58 


3 


83419 


16581 


97042 


0295S 


18623 


8(5877 


57 


4 


83432 


16568 


97067 


0298.3 


13634 


86866 


56 


5 


9.83446 


10.16554 


9.97092 


10.0290S 


10.13646 


9.86854 


55 


6 


83459 


16541 


971 IS 


02SS2 


13658 


86842 


54 


7 


83478 


16527 


97148 


02S57 


18(570 


868.80 


53 


8 


83486 


16514 


97168 


02S82 


13682 


868.18 


52 


9 


83500 


16500 


97198 


02S07 


18694 


8680(5 


51 


10 


9.83513 


10.16487 


9.97219 


10.027S1 


10.18705 


9.86295 


50 


11 


83527 


16473 


97244 


0275t5 


13717 


8(5288 


49 


12 


83540 


16460 


97269 


02781 


18729 


86271 


48 


13 


83554 


16446 


97295 


02705 


13741 


86259 


47 


14 


83567 


16433 


97820 


02680 


18758 


S6247 


46 


15 


9.83581 


10.16419 


9.97345 


10.02(555 


10.13765 


9.86285 


45 


16 


83594 


16406 


97371 


02(529 


18777 


8(5228 


44 


17 


8360S 


16892 


9789(5 


0260 1 


13789 


86211 


43 


18 


83621 


16879 


97421 


02579 


18800 


86200 


42 


19 


83634 


16366 


97447 


025.\; 


13812 


86188 


41 


20 


9.83648 


10.16352 


9.97472 


10.02528 


10.13824 


9.86176 


40 


21 


83661 


168.8.9 


97497 


02508. 


18886 


86164 


39 


22 


83074 


16326 


97523 


02477 


13848 


86152 


38 


23 


83688 


16312 


975 IS 


02152 


13860 


86140 


37 


24 


83701 


16299 


97578 


02427 


18S72 


86128 


36 


25 


9.83715 


10.16285 


9.97598 


10.02102 


10.13884 


9.86116 


35 


26 


83728 


16272 


97624 


028.7(5 


13896 


86104 


34 


27 


83741 


16259 


97t519 


02351 


13908 


8(5092 


33 


28 


83755 


16245 


97674 


02826 


13920 


86080 


32 


29 


83768 


16232 


97700 


02800 


13932 


860(58 


31 


30 


9.83781 


10.16219 


9.97725 


10.02275 


10.18944 


9.8605(5 


30 


31 


83795 


16205 


97750 


02250 


13956 


86044 


29 


32 


83808 


16192 


97776 


02221 


13968 


86082 


28 


33 


83821 


16179 


97S01 


02199 


13980 


8(5020 


27 


34 


83834 


16166 


97826 


02174 


18992 


8(5008 


26 


35 


9.83848 


10.16152 


9.97S51 


10.02149 


10.14004 


9.S5996 


25 


36 


83861 


16189 


97S77 


02128. 


14016 


85984 


24 


37 


83874 


16126 


97902 


02098 


14028 


88972 


23 


38 


83887 


16113 


97927 


02073 


14040 


859(50 


22 


39 


83901 


16099 


v:w 


02047 


1 1052 


85948 


21 


40 


9.83914 


10.16986 


9.97978 


10.02022 


10.140(51 


9.85936 


20 


41 


83927 


16073 


9S008 


01997 


14076 


85924 


19 


42 


83940 


16060 


9S029 


01971 


14088 


85912 


18 


43 


83954 


L6046 


98054 


01946 


14100 


85900 


17 


44 


83967 


16033 


\)so:^ 


01921 


14112 


85S88 


16 


45 


9.83980 


10.1(5020 


9.98101 


10.01896 


10.11121 


9.85S76 


15 


46 


83993 


16007 


98130 


01 870 


14136 


8586 1 


14 


47 


8 1006 


15994 


\)^\^ 


01815 


1 1119 


85S51 


13 


48 


8 1020 


159S0 


98 ISO 


01820 


14161 


85889 


12 


49 


84033 


15987 


98206 


01791 


14178 


S5S27 


11 


50 


9.84046 


10.15954 


9.98281 


10.01769 


10.14185 


9.85815 


10 


51 


8 1059 


15941 


98256 


01744 


14197 


85803 


9 


52 


84072 


1592S 


98281 


01719 


14209 


85791 


8 


53 


84085 


L5915 


98307 


01693 


1 1221 


85779 


7 


54 


84098 


15902 


98332 


01668 


1 128,4 


85766 


6 


55 


9.84112 


10.15888 


9.98357 


10.01643 


10.1121(5 


9.85754 


5 


56 


84125 


L5875 


98383 


01(517 


1 12.-.S 


S8712 


4 


57 


84138 


15862 


98408 


01592 


14270 


8578.0 


3 


58 


84151 


15849 


98433 


01867 


L4282 


857 IS 


2 


59 


84164 


15836 


98458 


01542 


L4294 


85706 


1 


60 


84177 


15823 


98484 


0181(5 


1 1307 


85693 





M. 


( 'osine. 


Seoant. 


Cotan^'iit. 


Tangent. 


Co secant. 


Sine. 


M. 



133° 



Logarithmic Angular Functions. 



223 



44° 






Logarithms. 






35° 


M. 


Sine. 


Cosecant. 


1 Tangent. 


Cotangent. 


Secant. 


Cosine. 


M. 





9.84177 


10.15823 


9.98484 


10.01510 


10.14307 


9.85693 


60 


1 


81190 


15810 


98509 


01491 


14319 


85681 


59 


2 


84203 


15797 


98534 


01466 


14331 


85669 


58 


3 


84216 


15781 


98500 


01440 


14343 


85657 


57 


4 


81229 


15771 


98585 


01415 


14355 


85645 


56 


5 


9.84212 


10.157:»s 


9.98610 


10.01390 


10.14368 


9.85632 


55 


C) 


81255 


15745 


98635 


01365 


14380 


85620 


54 


7 


84269 


15731 


98661 


01339 


14392 


85608 


53 


8 


84282 


15718 


98686 


01314 


14404 


85596 


52 


9 


81295 


15705 


98711 


01289 


14417 


85583 


51 


10 


9.84308 


10.15692 


9.98737 


10.01263 


10.14429 


9.85571 


50 


11 


84321 


15679 


98762 


01238 


14441 


85559 


49 


12 


84334 


15666 


98787 


01213 


14453 


85547 


48- 


13 


84347 


15653 


98812 


01188 


14466 


85534 


47 


14 


84360 


15010 


98838 


01162 


14478 


S5522 


46 


15 


9.84373 


10.15027 


9.98863 


10.01137 


10.14490 


9.85510 


45 


16 


84385 


15015 


98888 


01112 


14503 


85497 


44 


17 


84398 


15002 


98913 


01087 


14515 


8.5485 


43 


18 


84411 


15589 


98939 


01061 


14527 


85473 


42 


19 


84424 


15576 


98964 


01036 


14540 


85460 


41 


20 


9.84437 


10.1550:5 


9.98989 


10.01011 


10.14552 


9.85448 


40 


21 


84450 


15550 


99015 


00985 


14564 


85436 


39 


22 


84463 


15537 


99040 


00960 


14577 


85423 


38 


23 


84476 


15524 


99005 


00935 


14589 


85411 


37 


24 


84489 


15511 


99090 


00910 


14601 


85399 


36 


25 


9.84502 


10.15498 


9.99116 


10.00884 


10.14614 


9.85386 


35 


26 


84515 


15485 


99141 


00859 


14626 


85374 


34 


27 


84528 


15472 


99166 


00834 


14639 


85361 


33 


28 


84540 


15460 


99191 


00809 


14651 


85349 


32 


29 


84553 


15447 


99217 


00783 


14663 


85337 


31 


30 


9.84566 


10.15434 


9.99242 


10.00758 


10.14676 


9.85324 


30 


31 


84579 


15421 


99267 


00733 


14688 


85312 


29 


32 


84592 


15408 


99293 


00707 


14701 


85299 


28 


33 


84605 


15395 


99318 


00682 


14713 


85287 


27 


34 


84618 


15:582 


99343 


00657 


14726 


85274 


26 


35 


9.84630 


10.15370 


9.99368 


10.00032 


10.14738 


9.85262 


25 


36 


84643 


15357 


99394 


00606 


14750 


85250 


24 


37 


84656 


15344 


99419 


00581 


14763 


85237 


23 


38 


84669 


15331 


99444 


00556 


14775 


85225 


22 


39 


84682 


15318 


99469 


00531 


14788 


85212 


21 


40 


9.84694 


10.15306 


9.99495 


10.00505 


10.14800 


9.85200 


20 


41 


84707 


15293 


99520 


00480 


14813 


85187 


19 


42 


84720 


15280 


99545 


00455 


14825 


85175 


18 


43 


84733 


15267 


99570 


00430 


14838 


85162 


17 


44 


84745 


15255 


99596 


00404 j 


14850 


85150 


16 


45 


9.84758 


10.15242 


9.99621 


10.00379 


10.14863 


9.85137 


15 


46 


84771 


15229 


99646 


00354 


14875 


85125 


14 


47 


84784 


15216 


99672 


00328 


14888 


85112 


13 


48 


84796 


15204 


99097 


00303 


14900 


85100 


12 


49 


84809 


15191 


99722 


00278 


14913 


85087 


11 


50 


9.84822 


10.15178 


9.99747 


10.00253 


10.14926 


9.85074 


10 


51 


84835 


15165 


99773 


00227 


14938 


85062 


9 


52 


84847 


15153 


99798 


00202 


14951 


85049 


8 


53 


84860 


15140 


99823 


00177 


14963 


85037 


7 


54 


84873 


15127 


99848 


00152 


14976 


85024 


6 


55 


9.84885 


10.15115 


9.99874 


10.00126 


10.14988 


9.85012 


5 


56 


84898 


15102 


99899 


00101 


15001 


84999 


4 


57 


84911 


15089 


99924 


00076 


15014 


84986 


3 


58 


S 192:5 


15077 


99949 


00051 


15026 


84974 


2 


59 


84936 


15064 


99975 


00025 


15039 


84961 


1 


60 


84949 


15051 


10.00000 


00000 


15051 


84949 





M. 


Cosine. 


Secant. | 


| Cotangent. 


Tangent. 


Cosecant. 


Sine. 


M. 



134° 



45° 



224 



Plane Triangles. 



Formulas for Right- Angled Triangles. 




**4 



1. 

2. 



a = 


-f/6 2 + c 2 . 


a = 


c 
sin C' 


a = 


b 
cos C 



a= 2- 



: "\sin2C" 



5. 


b = a cos C. 


6. 


b = c cot C. 


7. 


b = a sin 5. 


8. 


b = c tan 2?. 



b = 



2Q 



\tan C 



10. 

11. 

12. 

13. 



a 2 sin 2(7 
«- 4 '" 

q = i^& 2 tan C. 

Q = l^c 2 COt C. 



sin C 



Q = V^V (a + c)(a — c). 
c 



15. cos C = 



16. tan C=-r. 
o 



17. sin 2C= 



18. tan C = 



4Q 
a 2 ' 

2Q 



Say the angle to be C — 60°. In the first column of the table of sines, 
60° corresponds with 0.86602 in the next column, which is the length of 
sin 60°, when the radius of the circle is one, or the unit, and the expression 
sin 60° X 36 means 0.86602 X 36 = 31.17672, and likewise with all the other 
trigonometrical expressions. 

In a triangle the functions of an angle have a certain relation to the 
opposite side ; it is this relationship which enables us to solve the triangle 
by the application of simple arithmetic. 

In triangles the sides are denoted by the letters a, b, and c; their respec- 
tive opposite angles are denoted by A, B, and C, and the area by Q. 

Example. The side c in a right-angled triangle being 365 feet, and the 
angle C— 39° 20', how long is the side a = ? 



Formula 2. a = 



365 



:;i;r, 
sin C sin 89° 20' 0.63:5*3 



= 575.86 feet, the answer. 



Or, by logarithms, 



log. a = log. 365 — log. sin 39° 20' 
= 2.56229 — 9.80197 
= 2.76032, num. = 575.86. 



Plane Triangles. 



225 



2. 



Formulas for Oblique-Angled Triangles. 





a : b = sin A : sin B, and b : c = sin B : sin C. 
a: c = sin A : sin C, and Q : ez& = sin C : 2. 



a = 


csin J. 
sin C 


a = 


csin A 


sin (J. + B) 


a = 


2Q 
b sin C * 


b = 


c sin i? 

sin C 


b = 


2Q 
c sin J. " 


sin C = 


csin B 
b ' 


sin C = 


csin A 



sin ^4. = 



2Q 
be' 



. a sin (7 
9. sin ^4 = . 



10. 



11. 



a = "j/fr 2 + c 2 — 26c cos J.. 



_ 2Q sin J, 

a ~~ \sin J5sin(^4 + £)' 



12. 



S=%(a + b + c). 



(s-b)(s-c) 



be 



13. sin y z A = -\/- 

14. ^x,.^!^ 

15. Zz % A = ^^. 

16. cos ~%B -- 



■V 



'g(g — 6) 
ac 



17. 



18. 



Q = 



5c sin J[ 



a& sin C 



c 2 sin J. sin B 

~~ 2~sin (^4 + B)' 



20. Q= j/(S— a)(S — b){S—c)S. 



21. 6 = 



/2Q_si 
\ sir 



sin (A + C) 



sin ^4 sin C 



22. c 



= Vsi 



2Q sin 



sin A sin (^4 + O) 



15 



226 



Spherical Triangles. 



Right-Angled Spherical Triangle. 








b 


1. 


sin b = sin a sin B. 


12. 


2. 


tan c — tan a cos B. 




3. 


cot C = cos a tan J5. 


13. 


4. 


tan c = sin b tan C. 




5. 


cos a = cos 6 cos c. 




6. 


cos 5 = cos b sin C. 


14. 


7. 


tan b 

tan a = -=. 

cos 


15. 


8. 


tan fr 

sin c = 5 . 

tan B 


16. 


9. 


sin 5 
sm a = — — =-. 

sm5 




LO. 


_, cos 2? 

sin C = r- 

cos b 


17. 


LI. 


cos a 

cos c = j-. 

cos b 


18. 



sin B = 


sin 6 
sin a' 


cos C = 


tan & 
tana* 


tan (7 = 


tan c 

sin b ' 


tan 5 = 


tan b 
sin c 


cos c = 


cos C 
sin 2?' 


cos 6 =■ 


cos 5 
sin C* 


cos a = 


cot (7 



tan £' 



The sum of the three angles in a spherical triangle is greater than two 
right angles and less than six right angles. 

By spherical trigonometry we ascertain distances and courses on the 
surface of the earth, positions and motions of the heavenly bodies, etc., 
etc. Examples will be furnished in geography and astronomy. 

Example. In a right-angled spherical triangle the side or hypothenuse 
a = 30° 20', the angle B = (>8° 50'. How long is the side b = ? 

Formula 1. sin b = sin a sin B = sin 36° 20' X sin 68° 60'. 
a log. sin 36° 20' = 9:77267 

B log. sin 68° 50' = 9:96966 



The answer, 



log. sin 33° 32' = 9:74233 or, b = 33° 32'. 



Spherical Triangles. 



227 



Oblique-Angled Spherical Triangle. 




19. sin a : sin b = sin A : sin B. 



20. sin b : sin c = sin B : sin C. 



sin b -- 



sin b sin 


A 


sin B 




sin c sin 


B 



sin C 



21. 



tan y 2 (a +.b) = tan ^ C08 ^7p| . 
/2V y /2 cos%(J. + J5) 



22. 



hn X («-») -^j Cxil+3 - 



23. 



tanK(B + C) = cot^ ^|g + g . 



24. 



-.^/t, ^s *. f • * sin %(b — c) 

tan fc(* - C) = cot^ s . n ^ + c j . 



25. 



26. 



COt%A 



tan ^c 



= tan%(£— C) 
= tan %(a — b} 



sin 3^(6 + c) 
sin ^(b — c) ' 

sin %{A + £) 
sin%(^4 — JB)' 



Example. Oblique-angled spherical triangle, c = 72° 30', B = 17° 30', 
C = 79° 50'. How long is the side 6 = ? 



Formula 20. sin 5 = 



sin c sin B sin 72° 30' X sin 17° 30' 



sin C 



sin 79° 50' 



c 
B 



C 
The answer, 



+ log. sin 72° 30' = 9:97942 
+ log. sin 17° 30' = 9:47812 

+ = 1:45754 

+ log. sin 79° 50' = 9:99312 



log. sin 16° 56' = 9:46442 or, b = 16° 56'. 



228 



Spherical Triangles. 



Oblique=Angled Spherical Triangle. 




[B refers to the whole angle between a and c, and b to the whole line 
opposite B.] 

27. tan %{m + n) tan y(m — n) = tan %(a + c) tan %(a — c) tan m = 
tan c cos J.. 



28. 



29. 



30. 



31. 



tan C = 



cos n = 



sin ?/i tan A 




sin (6 — m)' 




cos c cos (6 - 


-m) 


cos m 




cos a cos m 





COt 771 = 



COS C 

b = m ± n. 

cos c tan A 



tan a 
a — b — c 



^ + £ + e 



32. 



sin %^1 = 



/sin (g — c) sin (s — 6) 



sin 6 sin c 



sin %a = 



'cos 5 cos (S — ^4) 
\ sin 2? sin C 



To find the area of a spherical triangle : 

Let Q be the area of the triangle in square degrees. If R = radius of 
the sphere, the length of one degree will 

IP 



= ||, or one square degree = -3, 



1. 
2. 



co t^Q = 
sin y 2 Q = 



cot y^c cot ya + cos B 
sin B 

sin %c sin %q sin B 

cos yp 



1. 


sin (a ± jS) 


2. 


COS (a ± |8) 


3. 


sin 2a 


4. 


sin 3a 


5. 


COS 2a 


6. 


COS 3a 


7. 


sin a + sin 


8. 


sin a — sin 



Trigonometrical Formulas. 229 

Trigonometrical Formulas. 

= sin a cos £ ± cos a sin £. 
= cos a cos jS + sin a sin £. 

= 2 sin a COS a. 

= 3 sin a — 4 sin a 3 = sin a(4 cos a 2 — 1). 

= cos a 2 — sin a 2 = 2 cos a 2 — 1 = 1 — 2 sin a 2 . 

= 4 COS a 3 — 3 COS a = COS a(l — 4 sin a 2 ). 



i — ^-^ sm 



2 • 

a _j_ « a — ft 
9. COS a + COS jS = 2 COS =-*- COS = • 

a _L ft ft a 

10. cos a — cos = 2 sin — — - sin ! —-^ — . 

11. sin a 2 =3^(1 — cos 2a). 

12. COS a 2 = %{1 + COS 2a). 

13. sin a 3 = M(3 sin a — sin 3a). 

14. COS a 3 = 3^(3 COS a -f COS 3a). 

, tan a ± tan « 

15. tan (a ± |8) = . 7 — ^. 

It tan a tan £ 

,. , , , _. COt a COt j8 T 1 

16. cot (a + ft) = r , . „ . 

V r/ ± COt a + COt j3 



17. tan 2a 

18. cot 2a 



2 tan a 

~~ 1 — tan a 2 * 

COt a 2 — 1 
2 COt a 



/l — cos 2a _ sin 2a 2tan%a 

~ \ 1 + cos 2a ~~ 1 + 2 cos a ~~ 1 — tan %< 



20. COt a 

21. tan a ± tan = 

22. COt a ± COt /3 = 



-V4 



+ cos 2a sin 2a cot 3^a 2 — 1 



COS 2a 1 — COS 2a 2 COt Y^a. 

sin (a± /3) 

COS a COS £' 

sin (jS ± a) 
sin a sin |8* 



^ sin a + sin /3 = tan 3^(a + p) 
sin a — sin p tan%(a-— /3)' 



230 Differential and Integral Calculus. 

Differential and Integral Calculus. 

When one quantity depends upon another, so that the variations of one | 
produce certain variations in the other, the one quantity is said to be a* 
function of the other. There are many such functional relations occurring I 
in mechanics and engineering ; as, for example, those between time and 
distance in falling bodies, the expansion of steam in a cylinder, etc. 

Thus, in the case of a falling body, we know that the motion begins 
slowly and grows quicker and quicker, so that after a body has been fall- 
ing for several seconds it passes over a much greater distance in each later 
second than it did in the first second. By observing the spaces passed 
over by a falling body in several consecutive seconds we will find, as did 
Galileo, that the distances increase proportionally to the squares of the 
times, or, in modern notation, 

* = V^v. 

The closer together the successive observations are taken, the more 
nearly the truth will the deduction be. Thus, if we platted the values of 
s for a number of values of t, — taking the time intervals as one second, — 
and joined the points by straight lines, we should have a broken line, indi- 
cating roughly the curve. By taking the time intervals closer, the broken 
character of the line becomes less apparent. When the time intervals are 
taken so very close to each other that the broken character of the line can 
no longer be distinguished, it appears as a smooth curve, the equation of 
which gives the law connecting the two interdependent variables. 

The object of the calculus is to discuss the immediately consecutive 
values of variables, in order that their relations may be reduced to expres- 
sions suitable for use in computation. 

The method used is to take any single relation between the variable 
quantities under consideration, make a small increase in one of them, 
and compute what the corresponding increase will become in the other. 
Then, by deducing the ratio of the two increases in value, we get an alge- 
braic expression corresponding to the geometrical construction giving the 
broken line instead of the curve, as described above. By a simple trans- 
position in the equation the actual value of the increment may be made 
equal to zero, and, at the same time, permit the ratio of the variations at 
that instant to be determined. 

Thus, let y = ax 2 ; then let x be increased by a quantity, Ax or h, and y 
will have a corresponding increase, Ay, and we have 



Subtracting 

Dividing by 
we get 

Now, when Ax is equal to zero, Ay is also equal to zero; and thus, 
when the increment, A* = h, is zero, we have 

£ = 20*. 

That is, the ratio between the increments of x and of y at their zero values 
is equal to 2ax. It is usually stated in this demonstration that the value, 
lax, is reached when the increment is made infinitely small, so that its 
square, hr, may be considered still smaller, and hence negligible ; but this 
manoeuvring is altogether unnecessary, as there is no reason to objeci to 
the determination of the value of % as the true and exact ratio of the two 
increments. 

This is seen by an example in falling bodies. In the equation y = ax 2 , 
let a — 16, and substitute s for y and t for x— s representing space, and t, 
time,— and we have s — 16£-\ and 2ax becomes 32£, the well-known formula 
for the velocity at the end of t seconds. 

The integral calculus is the reverse of the differential calculus, being 
the study of the methods of finding the quantities and expressions which 
correspond to given differentials. The differential of a quantity is indi- 



y + Ay = 

y = 

Ay = 

A* = 


= a(x + h) 2 

= ax 2 + 2axh + ah 2 

= ax 2 

= 2axh + ah 2 

= h 


A2/_ 
Ax 


= 2ax + ah 



Differential and Integral Calculus. 231 

cated "by prefixing the letter d, as dx (read differential of x), and the inte- 
gral of an expression is indicated by the symbol /, an old-fashioned form 
of the letter s (signifying summation). 

The usual working method of applying the calculus to a technical 
problem is first to state, in the form of an equation, the relation existing 
between two immediately consecutive states of the functions under con- 
sideration, these being found from observation or from their known rela- 
tions by differentiation. This statement being made, both sides of the 
equation are integrated simultaneously, the result being a general alge- 
braic statement of the relation between the varying quantities within the 
limits for which the integration is made. 

As an example, we may give the determination of the law of barometric 
pressure, or the formula for computing differences of altitude by observing 
differences in atmospheric pressure by the barometer. 

Taking a column of air with a base equal to a unit of area (say 1 square 
metre) and an unknown height, x, and calling the pressure at the bottom 
of the column = p, and letting the weight of a unit volume of air (say 1 
cubic metre) at the bottom = q, we have the following : 

Let the height of the column, x, be increased by a very small quantity, 
dx, so that it becomes x + dx. We then have the pressure on the base in- 
creased by some small quantity, dp. But this is also equal to qdx, or the 
weight per cubic metre times the portion of a cubic metre which has been 
added ; hence, we have 

dp = qdx. 

According to Mariotte's law, the weights of given volumes of air are 
proportional to the pressures; or, for another pair of pressures and vol- 
umes, p' and q f , we have 

q' p' * p' 

which, in the above equation, gives 

a' 
dp = -^-rpdx; 
p' y 

or, calling the constant quantity -™ = c, we have 

dp = cpdx. 
Dividing both sides by p, we get 

<^ = edx; 
P 

and integrating ( — =Jcdx t or log. p = ex -f C, 



rd% 



when x = 0, p = p f , and log. p' = C. 
Subtracting, we have log. p — log. p' = ex; 

but log. p — log. p' = log. ~, and hence ex = log. ~, 

1 i P 
or x = — log. -=--. 

c p' 

and we have thus derived a formula giving the value of x for any two 
pressures,— that is, for the vertical height between two points at which 
the air pressures are p and p' '. 

The whole art of using the calculus in engineering or applied science 
lies in the framing of the equations and the use of the observed relations 
of the quantities, the processes of differentiation and integration being 
almost as much matters of routine as the use of logarithms. 

For those desiring to refresh their recollection, reference may be made 
to Professor Perry's "Calculus for Engineers," Professor R. H. Smith's 
"Calculus for Engineers and Physicists," and Autenheimer's " Elemen- 
tarbuch der differential und integral Rechnung," the latter being espe- 
cially rich in numerical applications to mechanics, physics, and practical 
science. 



232 



Differential Calculus Formulas. 



Formulas. 

1. y = x 

2. y = ax 2 

3. y — x* 

4. Zabx 3 

5. 4a& 2 #n 

6. a -f x 3 

7. (a + 6)z2 

8. 6a& 4 z 3 — c 

9. x + 3^2 — v 

10. 6z3 + 4aa;2 
— 3aa; 



11. ^2 



12. ZVZ 



13. x(x* — bx) 



14. i* 



15. 



16. 



17. (a + y^s = 



Differentials. 

dy = dx. 

dy = 2axdx. 

dy = nar^- 1 ^. 
= 9abx 2 dx. 
= 4nab 2 x n - l dx. 
= Sx 2 dx. 
= 2a;(a + 6) da;. 
= 18ab 4 a;2cfcc. 
= da; + 6zdz — dv. 

= (18x2 + 8aa; — 3) die. 
= -ydx -f 2xvdv. 

{dx dv , dz\ 
= xvz[ h— ). 

\ X ' v z J 

-- (3x 2 __ 2Wx)dx. 
2xvdx — x 2 dv 



adx 

nax n - l dx 
a> * 

3(a + V~x) 2 dx 
2j/a7 



Formulas. Differentials. 

21. a* = a x ladx. 



22. 


d'l'x 


dx 

X 


23. 


xVx 


- (1 + J-a;)da;. 


24. 


l-x 
x n 


_ (l — lx)dx 

x n + 1 


25. 


X 

Tx 


(l-x — l)dx 
(l-x) 2 • 



26. 



27. 



ay 



__ ayxda; — ax 2 dy 

V&-\-y 2 j/{30 + ^2)3 

a — 26a; _2&2^e 

(a + bx) 2 ~ (a + fre)*' 



28. j/; 



x = x = 



dx 

2\/x' 



29. (ax + a*)" = ^(q^ + &) n -\ 

(a -f 2a;) dx. 



30. |/ a 2 + 6a;2 = ■ 



bxdx 



18. (a + y «)*» = m(a + y^m-i 



1 x -i^ 
— a;« da;. 



yV + bx 2 

31. d-2(aa^) = 6oa;da;2. ' 

32. d3(aa^) = 6ada* 

33. d 4 (oa;3) = 6aa;o-ida; = o. 

34. sin v = -j- cos vdv. 



19. 



20. 



da; 



n(a — x) n 



2\/2ax — x 2 



(a — x) n ~i' 

'lad x 
\/2ax — x^' 



35. cos v 

36. tan v 

37. cot v 

38. sec v 

39. cosec v 



40. Tan for any 
curve, t 



= — sin vdv. 
dv 



= + 



= + 



cos 2 V 

dv 

sin 2 v ' 

cos vdv 

cos 2 v 

cos vdv 
sin 2 v 



7 /-. . d * 2 



I I 



Integral Calculus Formulas. 



233 



Differentials. Integrals. 

1. fdx = x + c Jxdx — — + C. 

2. /4ax 3 dx = 4a/x 3 dx = ax* -f C. 

3. jxndx = -^ii + G 

71+1 

4. ./V 'xdxjx^dx =/x^dx. 

5. f-^= =Jx^ A dx = 2j/s + a 

6. f — f- =Jxr^dx =Jx-^dx. 
8./-^=/^ 



/x- 2 dx = h CI 



-2S+ a 



ax 4 



9 -/H + ^) da 



10. 



12. 



adx 



bdx 

3ax 2 dx 
b -f- ax 3 



&l/ x + <?• 
= al'x + a 

= &^(a + x) + a 

= Z-(6 + ax 3 ) + a 



/ad, 

/ 

13. /axdx + 3x 2 dx nT 2 

— b 2 dx = -^- + x 3 — 5 2 x 

2 + a 

14. /(a* + & 2 ) = /(a 2 + 62)^. 

15. f(ax— %&)2dx = a?(j — a % + ~\ 

+ a 

16. /3(ax — x 2 ) 2 

(a — 2x)dx = (ax — x 2 ) 3 + C. 



"■J 5 



'n(x n ~ 1 dx) 

\/ a -}- x n 

2adx 
a 2 — x 2 



■■ 2"|/ a 2 + x" + C. 



a — x 



19. TV a 2 + x 2 dx = -gl/aa + ^2 + 
a 2 . 



?-(x + yV+a 



20. /|/a + bx*dx = — (a + 5x) 3 + C. 



Differentials. Integrals. 

dx 



J i/a 2 



j/a 2 + ; 



22. /3rax 2 dx 



23. Jmxdx 



:£•(# + 1/ a 2 +z 2 ). 



= raft 3 — ma 3 . 



:-f(& 2 -« 2 ). 



24. 



25. 



/dX _ 7T 

a 2 + a 2 — "2a"* 



|/a 2 — x 2 



2' 



6 a c b c 

26. Jdx = -Jdx =/ = /+/. 
a 6 a a 6 

27. /sin xdx = — cos x + C. 

28. / cos xdx = sin x + C. 

29. / tan xdx = — V cos x + C. 

30. / cot xdx = — I- sin x + C. 



31. 



32. 



= Z- tan — -f C. 



/ *fx 
sin x 

J cos x \ 4 2 / 



+ a 



33. / sin x cos xdx = % sin 2 x + C. 



i cos 6x 

35. ( dx 

x 



36. 



37. 







dt 



- Z 2 
dx 



00. 



= circle arc, of 
which t = tan. 



_ = circle arc, of 
y 2x — x 2 which x = sin 

versus. 



38. ff/6adx3 = 

XT6axdx 2 = /3ax 2 dx = ax 3 + a 

39. f/2(a + &)dx 2 =(a + 5)x 2 + 

Oox + Ci- 

40. jOT 2i; 2 dx 2 + 8vxdxdv + 2x 2 di; 2 == x%2. 



234 Mechanics and Statics. 

MECHANICS. 

In considering the action of force upon matter, it is important to have 
a clear understanding of the terms as used in the following pages. 

Without going deeply into the theoretical considerations of analytical 
mechanics, we will discuss briefly those relations commonly used by the 
engineer in daily practice, leaving the profounder questions for the elabo- 
rate theoretical treatises, such as those of Rankine, Hertz, Lagrange, Du 
Bois, and others. 

There are three elementary quantities used in mechanics, from which 
numerous compound quantities are derived : 

1. Force, usually expressed in units of weight, as pounds, tons, kilo- 
grammes, etc. 

2. Distance, expressed in linear units, feet, yards, metres, etc. 

3. Time, expressed in hours, minutes, or seconds. 

From these we derive a number of compound expressions, some of 
which are given here, others will be used as occasion requires. 

Thus, we have 

Work, which is the product of force by distance, and expressed by a 
combination of units of weight and distance, as foot-pounds, kilogramme- 
tres, etc. 

Power, which is the product of force by distance, divided by time, or 
the performance of a given amount of work in a given time, expressed as 
foot-pounds per minute, kilogrammetres per second, etc. 

Velocity is distance divided by time, as feet per minute, metres per 
second, miles per hour. 

Acceleration is the time-rate of change of the velocity of a body, 
expressed as a velocity divided by time, feet per second per second, miles 
per hour per second, etc. 

Forces may be conveniently represented by straight lines, the position 
of the line showing the direction of action of the force, and the length of 
the line indicating the magnitude of the force on some convenient scale. 
The convenience of the graphical method of solving problems in statics 
and mechanics renders it most useful, and in the following pages it will be 
'extensively employed. 

So far as precision is concerned, it is quite as practicable to construct 
force diagrams with a high degree of precision as it is to make the draw- 
ings of the structures to which they are to be applied, while the accuracy 
of the work is materially increased by the possibility of examining the 
relations of all the forces at once. 

STATICS. 

Statical problems are those which deal with the equilibrium of forces 
acting upon bodies at rest. 

It is customary to consider the bodies upon which the forces act as 
being rigid, although it is well understood that all substances are more or 
less elastic : it being found more practicable to determine the relations of 
the forces first, and then to modify these, when necessary, for the influence 
of the elasticity of the material under consideration. 

In order that a body or a structure shall remain at rest, it is necessary 
that all the forces acting upon it should balance each other. If this were 
not the case, the body would move in a direction dominated by the pre- 
ponderating force. This fact is used to aid in the determination of statical 
problems. The influence of the combined action of all the known forces 
acting on a body enables the magnitude and direction of the remaining 
force which holds them in equilibrium to be determined. 

The most convenient, rapid, and accurate method of combining and 
resolving the action of forces is the graphical method. 

A single, force may be indicated by a straight line, the length of which, 
on any convenient scale, shows the magnitude of the force. 

Thus, a force of 10 pounds may be represented as a straight line 10 
inches long, in which case the scale is 1 inch to the pound. The direc- 
tion of the action of the force is shown by the direction of the line, and, 



Statics. 



235 



if unopposed, the body upon which the force acts will move in the direc- 
tion of the line of action of the force. 

If the body does not move, equilibrium must be maintained by reason of 
the action of a force of equal magnitude to the first force, acting in the 
opposite direction. 

Thus, a weight of 10 pounds, suspended from a cord, hangs stationary. 
There must, therefore, be produced in the cord a reacting force of 10 
pounds, acting upward, otherwise the weight would fall. The upward 
reaction in the cord cannot be greater than 10 pounds, or the weight would 
move upward ; hence, we know that the reaction in the cord is exactly 
equal to the force of the weight, but acts in the opposite direction. 

When more than one force is to be considered, the question becomes 
more complicated, but the principle is the same. 





Thus, if we have two forces, 10, 20, acting upon the point, 0, we find 
the magnitude and direction of the opposing force, which just balances 
and holds them in equilibrium, as follows : 

At any convenient place on the paper draw a line, 1, parallel to 10, and 
of a length corresponding, on any convenient scale, to the force, 10. 
Thus, if 10 is 5 pounds, the line, 1, may be 5 inches, or 5 centimetres, or 5 
feet long. From the extremity of 1 draw 2, parallel to 20, and of a length 
equal to the force, 20, on the same scale as used for 1. Then join the ex- 
tremities of 1 and 2 by a line, 3. This last line will then be equal in length 
to the desired force, which holds 10 and 20 in equilibrium, and it will also 
be parallel to it in direction. By drawing 03 parallel to 3 we have the 
balancing force fully determined. 

For more than two forces we may proceed in a similar manner. 




Thus, if we have five forces acting upon the point, 0, we draw the poly- 
gon, having the sides 1, 2, 3, 4, and 5, respectively, parallel to the forces 
and, proportional to their magnitude, upon the same scale, and then close 
the polygon by the dotted line, 6, which gives the magnitude and direc- 
tion of the resultant, 06, which will hold the other forces in equilibrium. 



236 



Statics. 



If the polygon closes of itself, the system of forces is already in equi- 
librium ; if it does not close of itself, the length and direction of the side 
necessary to close it will give the required result. 

The foregoing discussion has assumed that the forces under considera- 
tion all act at the same point. When, however, the forces act at various 
points in a body which may be assumed as rigid, the resultant may be 
found as follows : 





Suppose we have any rigid body, upon which several forces, P 1 , P 2 , P 3 , 
are acting at several points. Construct the polygon as in the figure at the 
right, the line, 1-2, corresponding to P 1 ; 2-3, to P 2 ; and 3-4, to P 3 . The 
resultant will then be equal to 1-4. Then choose any point, 0, as a pole, 
and draw the rays, S 1 , S' 2 , S 3 , S*. Now, as in the figure on the left, draw a 
line, AT 1 A" 2 , parallel to S 2 , intersecting P 2 prolonged. From AT 2 to K 3 
draw a line parallel to S 3 ; then draw K 1 to A" 4 , parallel to S l ; and AT 3 - 
A' 4 , parallel to £ 4 , and the intersection will give the point, it 4 , through 
which the resultant must pass. 

The position of the pole, 0, does not affect the result, as will be found 
by choosing several poles and observing that the position of the resultant 
is not affected thereby. 

If we have a number of parallel forces acting upon a rigid body, the 
same method may be used, but the diagram becomes simplified. 










A 



Thus, if we have the vertical forces, P 1 , P 2 , P 3 , P 4 , we draw a vertical 
line, as in the diagram on the right, and lay off 1-2 = P 1 , 2-3 = P 2 , 3-4 = P 3 , 
4-5 = P 4 . Taking any pole, 0, and drawing the rays, S 1 , S 2 , S 3 , S 4 , S r °, we 
draw, as in the figure on the left, the line, K l K-, parallel to& 2 ; K 2 K 3 , 
parallel to S* ; 7i' ;5 A' 4 , parallel to SK Then draw K l K^ parallel to S l , and 
K*K'\ parallel to N \ these two lines intersecting at A' 5 . The resultant, 
equal to 1-5, will then pass through A' 5 . 



Statics. 



237 



Funicular Polygons. 

If we have a flexible cord, secured at the ends and having weights sus- 
pended from it at various points, we may use the polar force diagram to 
determine the various forces acting in the combination. 

Thus, if we have a cord suspended from two points, K 1 , K b , and to the 
points K 2 , K 3 , K±, suspend weights, P 1 , P 2 , P 3 , the cord will assume a 
shape similar to that shown in the figure. The combination will be in 
equilibrium, since the flexibility of the cord permits the weights to draw 
it into a position in which the forces balance each other. The various 
parts of the cord will then be subjected to tensions which are to be deter- 
mined. There will also be vertical and horizontal forces at the points K 1 
and K b which are to be found. All of these questions are solved by the 




diagram on the right. First draw the vertical line, 1-4, making 1-2 = P 1 , 
2-3 = P 2 , and 3-4 = P 3 . Then from 1 draw S\ parallel to K l K 2 , and from 
4 draw S 4 , parallel to K±K b , and the intersection of these two rays deter- 
mines the position of pole, 0. The rays S 2 and S 3 are parallel to K 2 K 3 
and K 3 K*. 

The lengths of the various rays, S 1 , S 2 , S s , £ 4 , measured on the same 
scale on which the vertical forces were laid off, will then give the magni- 
tude of the tensions in the parts of the cord to which they are parallel. 
By drawing a horizontal line, OH, through the pole, 0, the vertical will be 
divided at a point, H, and 1H and H 4 will be the vertical reactions at 
K 1 and K b , while the length, OH, will give the horizontal force acting to 
draw the two ends of the cord together. If the butments were removed, 
and a rod extending from K 1 to K b substituted, the length, OH, would 
give the compression on the rod. If the whole diagram be imagined as 
inverted, and the parts of the cord be replaced by rigid struts, the figure 
will represent a framework which will sustain the same weights without 
distortion, the tensions in the various parts of the cord being converted 
into thrusts in the corresponding members of the framework. 

In the preceding example the form taken by the cord is assumed to 
be given, and the only requirement is the determination of the forces. In 
some important cases, to be discussed hereafter, it is desirable to determine 
the form of the curve under various conditions of loading, as well as the 
stresses. Thus, the data given may be the span and the depth of the lowest 
point in the curve ; also, the position and magnitude of the loads ; and it 
may be required to find the form which these conditions give to the cord. 
The importance of these questions will be seen when it is understood that 
the flexibility of the cord permits it to assume a position of equilibrium 



238 



Statics. 



under any loading ; and hence from it can be deduced the stresses which 
are produced in rigid bodies, such as beams and similar constructions. 

It is well known that a cord suspended from two points on the same 
horizontal line, and uniformly loaded, will assume the form of a parabola ; 
but instead of acting on this assumption we may proceed just as if we had 
only to depend upon the methods of graphical statics, and then apply the 
same methods to the cases of unequal loading and unequal distribution.* 



Fig. 1. 




Suppose, Fig. 1, that we have the forces, 1, 2, 3, 4, 5, 6, 7, 8, equal in mag- 
nitude and at equal distances apart, acting vertically, as under the action 
of gravity, and that it is desired to determine the shape of the curve as- 
sumed by a cord sustaining these forces. The points of suspension are 
given at A and K, the forces are given in position, and the sag, ES, of the 
curve is given. The horizontal tension and the vertical reactions at the 
points, A and K, are required, and also the tension in each portion of the 
curve. 

Referring to the force diagram at the right, we draw the horizontal 
line, 40, and draw a perpendicular through 4. We then lay off the spaces, 
A, 1, 2, 3, 4, 5, 6, 7, 8, equal, on any convenient scale, to the forces, and 

choose any point, 0, as a pole, and draw the rays, 10, 20, 30, 80. We 

have taken the point, 0, on the horizontal, because we know that the 
curve is symmetrical, being uniformly loaded, and for reasons which will 
appear hereafter. Now, starting at A, in the diagram to the left, we draw 
Ab, parallel to AO; be, parallel to 10; cd, parallel to 20; and so on until 
we come to iK, parallel to 80. This gives us a curve, A, b, c, d, e,f, g, h, i, K, 
which is a force polygon corresponding to the forces given. The horizontal 
tension at A and at K will then be equal to the distance, 40, measured on 
the same scale as was used for the given forces in making the diagram, 
and the vertical reactions will be equal to A\ and 4-8. 

Now this diagram, while undoubtedly correct, is not the one we want, 
as the sag is too small ; and this is due to the fact that we have taken our 
pole, 0, too far from the point, 4, or, in other words, we have assumed the 
tension too great. Having once obtained one equilibrium curve, however, 
it is easy to transform it into any other one of any desired sag, in the fol- 
lowing manner : 

Draw the horizontal line, PQ, through the lowest point of the curve, 
which we have already obtained. Then prolong the line, Ab, until it 
intersects this horizontal at m; also prolong Ki until it intersects the 
horizontal at n. Draw P'Q' through S, the point of the desired sag, and 
drop perpendiculars from m and n until they intersect P'Q' at m' and n' ; 
join Am' and Kn'. 

Now, in the force diagram at the right of the figure, draw from A a 
line, AO', parallel to Am', and a line, 80', parallel to Kn'. They will in- 
tersect on the horizontal at 0', and this will be a new pole. Using this 

pole by drawing rays to 0' from A, 1, 2, 8, we have a new force 

diagram. Now, starting again at A, we draw Ab', parallel to AO' ; b'c' 



♦The following treatment of the subject of the catenary is substantially the 
same as that given in an article by the author in Engineering-Mechanics, June, 1896. 



Statics. 



239 



parallel to 10', etc., and we get a new catenary, A\ b' , c', i'K, 

which will have just the sag required. The distance, 40', will then be the 
correct horizontal tension, which corresponds to the sag, RS; and the 
tension in any portion of the curve is equal to the length of the corre- 
sponding parallel ray. 

All this is very clear ; "but this is the simple case of uniform loading, 
and might just as well have been solved by drawing a parabola of the 
required span and sag. Suppose now, however, that the loads are not 
uniform. Such an example is shown in Fig. 2. Here the loads are all 
the same, except that at 6, which is as much greater as is indicated by 
the length of the arrow. As before, we know only the magnitude and 
direction of the forces and the span and sag of the curve, and desire to 
find the horizontal tension and other forces. Referring to the force dia- 
gram on the right, we draw a vertical line, .48, making the distances, 



Fig. 2. 



A 




L 


! ! 


I 


1 , 


) 


h 


! 


5 


... 


5l 


c 

C I 


D 

d 


E 1 

. i 


F 

f - 


G y 

! \ 


'H I 


— 


P 


d'^ 


L-.L 

i 


f'_ 




AX, 1-2, 2-3, 7-8, proportional to the various forces, and it will be 

noticed that 5-6 is the large force, the others being equal to each other. 
We now choose any point, 0, for a pole, and draw the rays, A 0, 10, 20, 

80. Then, starting at A in the diagram to the left, we draw Ab, 

parallel to AO; be, parallel to 10; cd, parallel to 20, etc., and get the 
curve, A, b, c, d, e,f, g, h, i, k. We then join k back to A by drawing the 
inclined line, Ak. This gives us a complete polygon, bat it is not the 
one we want, for two reasons : first, it has not the right sag ; and second, 
the points of suspension are not on a horizontal line. We can readily 
bring the points of suspension on a horizontal line, in the following man- 
ner : In the force diagram, draw from the line, 0, a line, OK, parallel to 
kA; then will the distances, R' and R, be the vertical reactions at the 
points of suspension, and OK will be the tension at A and at k, in the 
direction of the line, Ak. If, now, we draw in the force diagram a line, 
KO' , horizontally through K, and place a new pole, 0', on this horizontal 
line vertically under 0, we can draw a new force diagram, as shown in 
the dotted lines, and the polygon drawn from A, with its sides parallel 
to che rays of this new diagram, will give us the dotted curve, which 
has the same sag as the first curve, but has its points of suspension on 
a horizontal line. We thus see that even if the first curve — obtained by 
choosing any pole— does not give us a curve with the required points of 
suspension, that it can readily be transformed into the desired form. If, 
instead of having the points of suspension on a horizontal line, it is desired 
to have them at different elevations, it is only necessary to draw a line 
through K, on the force diagram, parallel to a line joining the desired 
points of suspension, and place the pole on the line so obtained, and the 
desired curve will be found. Now, to obtain the sag which is wanted, 
we have only to proceed as in the first case, Fig. 3. In this figure the dotted 
curve corresponds to the dotted curve of Fig. 2. Draw the horizontal line, 
PQ, through the lowest point, /, of the curve already obtained ; prolong 
Ab to m, and Ki to n. Draw, also, P' Q' horizontally through the desired 
point of lowest sag, and drop perpendiculars from m and n to it at m' and 



240 



Statics. 



n'. Join Am' and Kn', and draw from A, in the force diagram, a line 
parallel to Am', and from 8, a line parallel to Kn f . These two lines will 
intersect at the point, 2 , which will then be the correct pole for the curve 
of the desired sag. Drawing a new set of rays, we have" only to draw the 
new polygon, A, b', c', d', e',f, g f , h f , i' ', K, with its sides parallel to the 
corresponding rays, and the problem is solved. The vertical reactions at 
the point of support are R f and R, and the horizontal tension is equal to 
K0*. 

Fig. 3. 





In actual practice the two operations of bringing the points of support 
to the horizontal (or to any desired inclination), and the adjustment of 
the tension to produce any required sag, may be combined so as to give the 
proper pole at one operation, as shown in Fig. 4, in which, also, the forces 
are all shown as different, so as to show the general nature of the solution. 
We first draw the vertical line of the force diagram on the left, making 
the spaces from A, downward, proportional to the forces of the corre- 
sponding numbers, and then choose a trial pole, 0. Drawing the rays, and 
constructing the polygon, A, b, c, d y e, f, g, h, i, k, and joining kA, we 
have a polygon which has neither the proper position of the points of 

Fig. 4. 



suspension nor the desired sag, but which does express the equilibrium of 
forces, and can therefore be transformed into the form we want. We draw 
PQ. parallel to Ak, and prolong Ab and kt until they intersect PQ at m 
and n. Also draw 1 V Q? horizontally through the desired point of lowest 
Bag, and drop perpendiculars from m and n, intersecting m' and n'. In 
the force diagram draw OK, parallel to Ak, and draw a horizontal line 
through K. Then, by drawing a line from A, parallel to Am' t and from 



Statics. 



241 



8, parallel to Kn r , we find that they intersect at 0' ', on the horizontal line, 
KO', and 0' will at once be the new pole for the final curve, A, b', c', d f , e' , 
f,g',h',i',K. 

As a general idea of the process we may imagine the pole to be con- 
nected to the points, A, 1, 2, 3, 4, 5, 6, 7, 8, by elastic cords, so that they will 
remain taut and straight as is moved about. Then, if we move the pole 
up and down anywhere, always keeping it at a constant distance from the 
line, ^48, we shall obtain diagrams which will give correct polygons for 
the forces under consideration, and of any desired inclination. The hori- 
zontal tension being unchanged, the sag will remain constant in all these 
curves. If we move the pole, 0, to and from the line, AS, we shall obtain 
curves of varying sag and correspondingly varying horizontal tensions, 
and, as we have shown how to obtain the position of the pole for any 
desired sag, we have only to place it there and proceed with the con- 
struction of the curve. If the forces, 1, 2, 3, — , etc., are not spaced 
equally, it is only necessary to draw verticals through their points of 
application and use them in the construction of the curve, instead of the 
lines as given in the figures. By this simple graphical process, therefore, 
all the problems involved in the construction of such curves may be 
rapidly and accurately solved. 

The space which has been given to the variably-loaded catenary in the 
preceding pages will be understood when it is seen that the construction 
of such curves enables the distribution of stresses in a great variety of 
structures to be -readily and accurately determined. The flexible cord, 
being at liberty to assume a position of equilibrium, is free from any bend- 
ing stresses, every portion of the curve being, in fact, a resultant of the 
forces acting upon it, the tension in the various portions of the cord being 
measured by the length of the corresponding ray in the force diagram. 
If, now, we invert the catenary, we have the proper curve for an arch 
subjected to similar forces, the only difference being that the arch is in 
compression, while the catenary is in tension. This will be discussed 
more fully when treating of the arch. 

If, instead of a cord, we have a horizontal beam resting upon two 
supports and loaded in any given manner, we may use the catenary to 
determine the stresses. The beam, unless loaded excessively, will not 
have an appreciable deflection, and so will not place itself in the line of 
the catenary. In consequence, it is subjected to internal stresses of a 
kind differing from the simple tension of the catenary. By drawing the 
catenary and the force diagram we get the data to determine these in- 
ternal forces, and thus are able to proportion the beam to resist the loads 
properly. 

If we have a beam, AG, loaded with parallel forces, Q\ to Q 5 (Fig. 5), 
whose load is to be opposed by reactions, Pi and P 2 , at A and G, we may 
first determine a resultant, Q, of all the forces, and then decompose this 
into values for Pi and P 2 . We also omit the determination of Q altogether, 
and proceed to determine Pi and P 2 
directly, as follows : 

Choose any pole, 0, and form the 
force polygon, Kl . 2 .... 50, and 
construct the cord polygon, making 
its sides parallel to their respective 
rays, and draw ba, parallel to KO, 
and/*?, parallel to 05, their intersec- 
tions with the lines of the forces, Pi 
and P 2 , being a and g. Join ag, 
which will be the closing line of the 
polygon, and its parallel, 06, in the 
force polygon gives P 2 = 5 . 6 and 
p 1 = 6 . 7.' If the sides, db and/#, of 
the cord polygon are prolonged in 
the other direction we obtain a' and 
g', giving, however, the same result, 
since a'g f is parallel to ag. The cord polygon would then be the figure, 
a' , g' , m, b, d, c, e,f, m, a', and m indicates the position of the resultant of 
the forces, Qi to Q5, or of Pi and P 2 . 

The cord polygon, or catenary, therefore, gives the proportion of load 
borne by each of the supports. But it does more, it enables the determina- 
tion of both the shear and the statical moment at any point. 

16 




242 



Statics. 




A Statical Moment is the product of a force by the normal distance 
from the point of resistance against which it acts. Thus, a force of 10 
pounds, hanging from the end of a rod projecting 36 inches from a wall, 
has a statical moment of 36 X 10 = 360 inch-pounds,— moment being merely 
the technical term for leverage. * 

A statical moment is a compound quantity, expressed in terms of force 
and distance, as inch-pounds, foot-tons, kilogramme-metres, etc. In the 
case of a beam resting upon two supports, and having various loads upon 
it, the statical moment at any point is the product of the resultant of all 
the forces acting upon the beam on either side of the point into the dis- 
tance of the line of action of the resultant from the given point for which 

the statical moment is deter- 
mined. 

In order to show how the stati- 
cal moments in any loaded beam 
may be determined graphically, 
take the example shown in Fig. 6. 
After constructing the force 
polygon, ^404, and cord polygon, 
a,b,c, d, e,f, let it be required to 
find the statical moment for any 
point, S, upon the beam. This 
moment is the product of the re- 
sultant of all the forces upon one 
side or the other of the line, SSi, 
into the lever arm, I, of this resultant from SSi. 

The magnitude of this resultant is obtained from the distance, hi == 1 . 5, 
in the force polygon, cut off by the rays, 01 and 05, which are parallel to 
be and fa, and its point of application is determined by prolonging these 
sides until they intersect at g. By drawing the perpendicular, gg , the 
lever arm, I, of the resultant, P = hi, is determined for the force acting at 
the point, S ; and hence we have M = PL 

This multiplication may also be performed graphically. By drawing 
the perpendicular, Ok, in the force polygon, we obtain the altitude of the 
triangle, Ohi, from the base, hi, and this triangle is similar to the triangle, 
gss , whose altitude is I. Call in Ok = H, and ss = t, we have 

P:H=t\l, 
or M=Pl = Ht. 

This proves that the statical moment at any point in a beam is proportional to 
the corresponding ordinate of the cord polygon, parallel to the direction of the 
forces, since H is a constant. By making H equal to unity the moment, 
M, becomes equal to the ordinate, t. It is not necessary to determine the 
position of the point of application, g, of the resultant, since it is the 
relation between the statical moments which is most desirable, whether 
H be chosen as a unit or not. This property of the cord polygon for 
parallel forces is most useful, and an example may be found in the case 
of axles. 

The shearing force in a beam at either support is evidently equal to the 
entire reaction at that support. Thus, in Fig. 6, the shearing force at A is 
equal to Ab on the force diagram. The shearing force at any other point 
in the beam is equal to the distance from 5 to the point on A± corre- 
sponding to that point in the force diagram. Thus, the shearing force at 
B is equal to 1-5, etc. 

Centre of Gravity. 

Every particle of a body is attracted by the force of gravitation to the 
earth, and the sum of all these forces upon the particles constitutes the 
weight of the body. In accordance with the methods already given for 
determining the resultant of a number of parallel forces, the point of J 
application of the resultant of the force of gravity may be found. This 
point is known as the Centre of Gravity of the body. For homogeneous 
bodies the position of the centre of gravity may generally be computed 
from the form of the body. For bodies which are entirely symmetrical 
and homogeneous, the centre of gravity is situated at the centre of figure. 



Statics. 



243 




For bodies which are symmetrical about a given axis, such as a cone, etc., 
the centre of gravity is situated in the axis. Various methods are used for 
determining the position of the centre of gravity of non-symmetrical fig- 
ures, most of them based upon the subdivision of the figure into parts, of 
which the centres of gravity are known. 

The most convenient of these 
is the graphical method. 

This may be done by dividing 
the figure into a number of strips 
of uniform width such that their 
area may be considered as propor- 
tional to their middle ordinate, 
constructing the force and cord 
polygons, and taking the line of 
the resultant as a line of gravity. 
If the figure is not symmetrical, 
it will be necessary to divide the 
figure again in another direction 
and determine another line of gravity, when the position of the centre of 
gravity will be found at the intersection of the two lines. For figures of 
simple form larger determinate sections may be taken instead of strips, 
their area determined in any convenient manner, and the diagram con- 
structed accordingly. 

Suppose, for example, that it is required to determine the position of 
the centre of gravity of the T-shaped section shown in the above cut. The 
figure is symmetrical about the axis, YY, so that the centre of gravity must 
lie somewhere in that line. We may divide the figure into the rectangular 
portions b x c, &i X C\, an( i &2 X g, which we will call respectively the 
areas 1, 2, and 3. 

These three forces are laid off at A, 1, 2, 3, a pole, 0, selected, and K\'K\ 
drawn parallel to OA ; K X K 2 , parallel to 01 ; K 2 K 3 , parallel to 02 ; K 3 K s ' f 
parallel to 03, when the intersection of the sides, K X K X ' and K z K^ r , at M 
gives a point on the line of gravity, MM ', whose intersection, S, with the 
axis, YY, is the centre of gravity of the figure. 

The method of moments may also be used 
Y in determining the position of the centre of 

l_]_^i gravity , as follows : 

! — . : r-J This method is based on the fact that the 

total weight of a body, multiplied by the dis- 
tance of its centre of gravity from any given 
axis,— i.e., its statical moment with regard to 
that axis,— is equal to the sum of the statical 
moments of its various parts. 



r-JLJt 



tively to a, b, and 
axis, XX', will be 



Thus, if we have the section 
here shown,. we see that its figure 
is symmetrical about the axis, 
YY', so the centre of gravity must 
lie in that axis. Taking any con- 
venient axis, XX', we divide the 
i section into the three rectangles, 

A, B, and C, of which the posi- 
tions of the centres of gravity are 

* JT* known, we have their distances 

from the axis, XX', equal respec- 
and their statical moments with reference to the 



Aa, Bb, and Cc. 



The area of the whole figure is equal to A + B + C, which we will call 
M, and the distance of its centre of gravity, x, from the axis, XX', is un- 
known and sought. 

We have 

Aa + Bb + Cc = Mx; 



Aa-\- Bb-\- Cc 
M = 



244 



Centre of Gravity. 



Thus, if A = 4 square inches, I? = 5 square inches, (7=9 square inches, 
and a = 12 inches, b = 9 inches, c = 4 inches, we have 



^4a = 4 X 12 = 48 
ift = 5 X 9 = 45 
Cc = 9 X 4 = 36 

129 

and this, divided by the area of the whole figure, or 18 square inclAa, gives 

129 

-— - = 7.166 for x, the distance of the centre of gravity, S, from the axis, 

18 

XX' The position of the axis, XX', is immaterial, so long as all the 

moments are taken with reference to the same axis. 

When a figure is not symmetrical, the moments must be taken first with 

reference to a vertical axis and then with reference to a horizontal axis, 

and the centre of gravity will be found at the intersection of the two lines 

thus determined. 

In practical work the position of the centre of gravity is often most 

conveniently found by experiment. 

Thus, if a scale drawing of the section be cut out of stiff card-board, or 

better, thin sheet metal, it may be hung up by one corner, a plumb-bob 

made of a fine thread and weight being suspended from the same point, a, 

as in the figure. By marking the point where the thread intersects the 

edge of the section, as atp, the path 
of the vertical across the section 
may be drawn from the supporting 
point. The section is then suspended 
from another point, b, and the point 
p' marked; the intersection of the 
lines ap and bp' gives the position of 
the centre of gravity, s. Care must 
be taken to have the section per- 
fectly free to oscillate about the 
point of suspension, usually a pin, 
and errors due to friction against 
the wall must be avoided. 

Another convenient method is to 
balance the section across a horizon- 
tal knife-edge, in two successive po- 
sitions, marking the intersection of the two positions of the knife-edge. 

This latter method may be conveniently applied by using a draftsman's 

triangular scale as a knife-edge. 

The position of the centre of gravity for some of the more generally 

occurring figures may be obtained from the following diagrams : 




Quadrangle. 

a and b parallel. 




h fb — a\ 
6 I b + a)' 



Triangle. 




Centre of Gravity. 



245 



Half a Circle Plane, or Elliptic 
Plane. 




2 = 0.4244r. 




2cr 
3b ' 



Circle Segment 

a = area. 




Parabola. 





Convex surface, z = %r. 
Solid, z = %r. 



Spherical Sector. 




Solid, z = %(r-^y 



Spherical Segment. 




Convex surface, z = — . 



Cone. 




NH 



Convex surface, z = - 



Solid, 



4* 



246 



Centre of Gravity. 



Conic Frustum. 
h 




Convex h h r E _ r -i 
surfe,, = ---L^ T - r J. 

Solid, « = xl2P + r(U + r)J- 

Pyramidic Frustum. 

.4 and a = area of the two bases. 




Solid, s = A [-^+3a + 2 1 / J ia 1> 
4 L^ + a 




P: W=l:z t 



P ' 



To find the Centre of Gravity of 
Two Bodies, P and Q. 



-a 

— T 



r« 



U — b — >*+-z—*i 



P+Q' 



Pa 
P+Q' 



To find the Centre of Gravity of 
a System of Bodies. 

h — a ^'byzL 



6 = 



Ra 
P-VR' 



Qd 



P + R + Q' 



Half a Circumference of a Circle 
or Ellipse. 




z = 0.4244r. 



Circle Arc, or Elliptic Arc. 




cr __ c{c 2 + 4ft 2 ) 






6 8M> 

For semicircular line, 

2r 
2= — = 0.6366r. 



Cylindric Surface, with a bot- 




: 2(h + r2)' 



Graphical Statics. 247 



Statics of Framed Structures. 

As the distribution of stresses in simple "beams or in suspended cords 
may be determined graphically, so may the stresses in the various mem- 
bers of framed structures be investigated. 

Framed structures are of very general application wherever loads are 
to be supported, and their discussion may be classified as a system by itself, 
while their use extends from the simple trussed beam to the bridge and 
roof truss ; also for walking beams and many other uses. 

The tensile and compressive stresses in these various forms may readily 
be examined by means of the force plan, which consists of both the force 
and cord polygons and their modifications. The subsequent examples 
will serve to illustrate the principal cases. In all of these cases it is 
assumed that at the knots — i.e., at the points where several members meet, 
—a joint is supposed to exist ; or at least no account is taken of the re- 
sistance to bending at the knots. 

In order to form such a plan for any given construction, it is necessary 
first to determine the division and direction of the forces, and then, 
beginning at one of the external forces and laying off its direction and 
magnitude to the next knot, combining it there with the external forces 
at that point, laying off the resultant to the next bend, etc. Upon such 
combinations of force triangles or quadrangles the force plan is constructed. 

If it is desired to determine the directions of the components of a given 
or determined force, the principles laid down in the following rules must 
be borne in mind. 

If one force is to be separated into two or more forces, its direction is to be 
reversed and it is to be made the closing line, S f , in the paths of the other forces. 

If two or more given forces are to be combined with two or more other forces, , 
the force polygon will consist of the given forces and their closing line, S. 

The first rule is only a special case under the second or general rule, 
since the single force may be considered as an unclosed force polygon 
whose closing line passes backward over the same path to the starting 
point. 





In the investigation of each member in a frame without error, it is best 
to assume the member to be cut, and to determine the external forces at 
each section which oppose the internal forces ; the direction of the forces 
may then also be determined with precision. 

I. Simple=Trussed Beams.— The beam, ABC, is supposed to carry at 
B a load equal to 2P, acting in a direction normal to A C, and to be sup- 
ported at A and C. Since AB = BC, the reaction at each support is equal 
to P. It is then required to determine the stresses upon the various 
members from 1 to 5, as marked in the figure. 

Referring to the diagram marked a, let ab be the reaction, P, which 
acts upward at A. We now have to construct a diagram of the internal 
forces acting in AB and AD. To simplify matters we will give these forces 
the same numbers as their corresponding members, drawing 1 parallel to 
AB, and 2 parallel to AD. The direction of the force, P, in the closing 
line of the force triangle determines the direction in the other two sides, 
as shown by the arrows and by the lines 1 and 2. In this case there will 
be compression in AB and tension in AD. 

In order to show this clearly, in all the following strain diagrams the 
I forces acting compressively in struts or posts will be indicated by double 
lines, while all tension members, links, or rods will be shown by single 
lines. 

Following out this idea, we shall, in the following illustrations, show 
all struts or compression members in the construction drawings as having 
a measurable thickness, as if made of wood, while the tension members 



248 



Graphical Statics. 



will be represented by simple lines, although this is not intended to indi- 
cate any limit as to the choice of materials. 

For the knot at B we make abc *= 2P, and, following in the direction 
dac (because the thrust is from A towards B) , join the closing lines 3 and 
4, both of which represent compression. The combination of 2 and 3 
determines 5, which is tension. This gives an entirely symmetrical plan, 




Simple-Trussed Beam. — I. 

which was to be expected from the symmetrical form of the structure, 
and an investigation of one-half is practically sufficient. 

If the load, 2P, is taken as uniformly distributed over the entire dis- 
tance, ABC, instead of being concentrated at B, the reactions at A and B 

p 
will each be equal to ~, and the load at B = P, so that one-half of the 

load on AB and BC is referred to the knots A, B, and C. From these con- 
ditions we obtain the force plan b, which is geometrically similar to the 
other, but only half as large. 

II. Double-Trussed Beam (much used for constructions of all sizes). — 
In this case take vertical forces, P, at B and C, and corresponding vertical 
reactions at A and D. In the first force plan a is drawn equal to P, and 1 
and 2 parallel respectively to AB and AE, thus determining the forces 1 
and 2,-1 being compression, and 2 tension. Lines now drawn parallel to 
BE and EF determine the compression in 3 and the tension in 5, while the 
compression at 4 is the closing line of 3, 1, and P; and the other half of 



1 
A 


P 

1 


> 


P 

B 4 


P P. 

C 1 




2 s 


3 


X 6 


V 




Double-Trussed Beam. — II. 



the diagram is similar. If the vertical forces at A and B are not of the 
same magnitude, which is often the case in practice, the structure should 
be strengthened by the introduction of the diagonals, EC and BF. 

The second diagram shows the construction in this case. Let Pi = a T &i 
be the force acting at A, and P 2 = OoCo be the force acting at B. Draw a 
vertical line from 1 to a horizontal through C\, which gives the length, 3, 
of the vertical force at B, and by drawing the dotted diagonal line their 
resultant is found. If any of the tension members are omitted the frame- 
work will tend to take an inclined position until the various parts are at 
such an angle with each other that both constructions will give the same 
value for 3. For this reason it is best in nearly every case to use the 
diagonal counterbraces. 

III. Triple-Trussed Beam. — The uniformly distributed load upon the 
framework gives the following distribution of forces. The force, 3P = abc, 
is first decomposed in 2 and 1, or ce and ea; then 1 is connected to ab = IP 



Graphical Statics. 



249 



by the line be, and this latter decomposed into 3 and 4, or ef and/b ; 2 and 
3 are now joined by/c, and the components at 5 and 6, or fg and gc, found. 
Since 6 and 10 are equal to each other, we may draw ch parallel to GH, 
and equal to eg, which gives gh = 7 ; the rest of the force plan is similar 
to the first half. 




Triple-Trussed Beam. — III. 

IV. Another form of Triple- Trussed Beam is shown below. — The space 
between B and is twice as great as between A and B, and the uniformly 
distributed load is equal to 12P, acting at the various knots, as shown in 
the figure. 





Triple-Trussed Beam.— IY. 

In the force plan make abc == 5P, and draw parallel to 1 and 2 the lines 
ae and ec; then join 1 with 3p (for the knot at B), and decompose into 3 
and 4, or e/and fb. Now combine 2 with 3, giving cf, and draw 5 and 6 
parallel to FC and FG, respectively. This case differs from the preceding, 
in that 5 is now compression instead of tension. The equality of the 
forces 6 and 10 gives gh — 7, and the similar half of the diagram need not 
be drawn. 



7P 



2P 2P 2P 2P 2P 2P 2P 




h -"~----- 






IS 



^ 



^2 8J 



^^ 



2. 



Multiple-Trussed Beam. — V. 



250 



Graphical Statics. 



V. Multiple»Trussed Beam.— The beam, A J, is divided into eight 
equal parts, which are represented as being uniformly loaded, the load at 
each knot being shown in the figure. In constructing the force plan we 
make ae = 7P, and by drawing the lines parallel to 1 and 2 we obtain of 
and fe; then lay off ab = 2P, and join the resultant, bf. This decomposes 
into 3 and 4, oifg and gb. The forces 2 and 3 combine to give the resultant, 
ge, which, by drawing lines parallel to KC and KL, gives gh and he for the 
values of 5 and 6. We now find that to proceed further we have three 
forces of given direction only, and, since this is indeterminate, we must 
obtain one magnitude as well. This, for example, may be done for the 
force 7, as follows : the strut, CL, sustains the vertical components of 5 
and 9, as well as its own direct load, 2P. Now 5 and 9 are equal to each 
other, since they are placed symmetrically, and carry equal loads from the 
struts, BK and KM; hence, in the force plan, we may make hi, which 
represents the force 7, equal to twice the projection of 5 upon the vertical 
+ 2P. This we can now combine with 6 = he, giving ie, which in turn 
decomposes into im and me, or 10 and 11. Returning to the knot, C, we 
may now take the line, hi, and by drawing parallels to CL, CM, and CD, 
obtain the figure, hike, which determines the forces 8 and 9. In the same 
manner proceed from 12 to 15, which will complete the half plan. It may 
be noted that the principal beam, A J, is subjected to a uniform compres- 
sion throughout its entire length. 

The force plan will, of course, be modified by various distributions of 
the load, as in the case of simple beams. 

Roof trusses furnish many and varied examples of framework. In the 
following examples a uniformly distributed vertical load is assumed, so 





that the burden upon any portion of a rafter may be considered as pro- 
portional to the length of that portion. 

I. Roof with Simple Principals.— A uniform load, 2P, upon each 
half gives as the external forces P, 2P, and P at A, B, and C. Lay off in 
the force plan ab = P, and draw ac and be parallel to AB and AC, deter- 
mining the forces 1 and 2,-1 being compression, and 2 tension. Then draw 
the vertical, ce, and also draw be parallel to CD, thus giving both 3 and 5, 
and the diagram is completed by drawing de. 

II. Roof with Single-Trussed Principals.— This form is similar to 
the preceding, with the addition of the struts, Ci? and CF. The distance, 
A E, is to EB as 3 is to 2 ; and the loads upon the respective portions are 
6P and 47 , 1 which give the forces at the various knots, as shown in the 
figure. Make ac in the force plan equal to 7P, and by drawing lines 
parallel to AE and AC obtain the forces 1 and 2, or ad and de; then com- 
bine 1 with 5P ab, and decompose the dotted resultant into de and eb, 
respectively parallel to EC and EB, giving the forces 3 and 4, both being 
compression. By repeating 2 and 3, in drawing 7 and 8, we obtain the 
figure, edefg, in which eg gives 5. This latter force might also have been 



Graphical Statics. 



251 



found by combining 4 and 4P, and decomposing the resultant by lines 
parallel to .BCand BF, an illustration of the various methods in which the 
force plan may be used. 
H III. Another form, with Single-Trussed Principals.— This roof is similar 
to the preceding, except that the struts, EC and CF, are placed horizon- 
tally. In this case AE = EB, and the external forces at A and D are both 





Single-Trussed Roof.— II. 

equal to 3P. The forces from a to c in the force plan are determined as 
before, giving da and cd for the forces 1 and 2, and the combination of 1 
with 2P gives the resultant, db y from which the thrusts 3 and 4, or de and 
eb, are obtained. The value of 1 is the same as 3, and 8 is the same as 2, 
while 5 is the closing line of cdedf or of cdf. The force 5 must also be the 
combination of the equal forces 4 and 6 with 2P, which the diagram shows 
to be the case. If the rod, CB, is omitted, as is frequently done, the strut, 
ECF, if there is no joint at C, will oppose its resistance to bending to the 




Single-Trussed Roof.— III. 



force 5 ; but there will be a tendency to rise at the apex, B, if the fastening 
be not made sufficiently strong. 

IV. Third Roof with Single-Trussed Principals.— In this form of truss, 
frequently known as the Belgian or French truss, the single vertical rod of 
the "preceding form is replaced by a triangle, BCD. The struts are placed 



252 



Graphical Statics. 



in the middle of the rafters and the external forces are distributed as 
shown in the figure. In the force plan abc — 3P, and 1 and 2 are deter- 
mined as before. By the decomposition of the resultant of 1 and 2P we 
obtain the forces 3 and 4, or de and be, and from the resultant, ec, of the 
forces 2 and 3 we get the tensions 5 and 6, in cf and ef. The second half 
of the diagram is the symmetrical counterpart of the first. 





Single-Trussed Roof— IV. 

V. Roof Truss with Double=Trussed Principals.— This construc- 
tion does not differ greatly from the preceding, except that the struts 
employed to strengthen the rafters are divided into two. The spaces are 
equal to each other and the load uniformly distributed. As shown in the 
figure this gives a reaction of 5P, or A and D. In the force plan ad = 5P, 
and lines parallel to AE and A C drawn, determining the forces 1 and 2, or 
de and ea. We then combine ea with ab = 2P, and decompose the dotted 
resultant, eb, into the thrusts, ef and /&, or 3 and 4, by drawing these lines 





Double-Trussed Roof.— V. 



parallel to EC and EF. Again, we take the resultant of the forces 4 and 
2P and decompose it into 5 and 6, or/<7 and gc, which brings us to the 
middle of the symmetrical figure. The force 7 is the resultant of 6 and its 
counterpart, 8, and the load 2P, and the half of this force is therefore 
equal to the projection of 6 upon the vertical, less P, or, in the diagram, 
Xjodh. 



Graphical Statics. 



253 



VI. English Roof Truss, with MuItiple=Trussed Principals.— Here 

we have inclined struts, with vertical tie-rods. The load is again uni- 
formly distributed, each space bearing the load of 2P. The reactions at A 
1 -,and D are each = IP. In the force plan we have ab + be -f cd + de = 3 X 
2P + P = 7P, which gives the length of ae. The forces 1 and 2 are found 
by drawing fa and ef parallel to AE and AL. Now consider 1 as combined 





Multiple-Trussed Roof.— VI. 



with ab = 2P, and the resultant, fb, decomposed into fg and gb, giving the 
forces 3 and 4. Again, combine 2 and 3, and then decompose the resultant, 
ge, into 5 and 6, or gh and he, by drawing these latter parallel to LF and 
LM. In this manner we continue until we reach 12, or Id, which we then 
project upon the vertical. Now, taking from dm one-half the load P = de, 
we have me for one-half the stress on the middle rod, BC. The remaining 
half of the force plan is similar. 



D 12 e 




Polygonal Roof Truss.— VII. 



VII. Polygonal= or Sickel-Shaped Roof Truss.— This roof may be 
considered as a modification of the preceding form, and is used for higher 
and wider spans. It is hardly proper to assume that the load is here 
uniformly distributed, even if the spaces are equal, for in the case of 
snow much less weight would be carried by the steep portions, AB or GH, 
than by the natter surfaces, CD or DE. We must therefore estimate the 
forces Pi, Po, P 3 , acting as B, C, D, E, F } G, and make the reactions at A 
and B equal to Q = Pi + P 2 + P 3 . 



254 



Graphical Statics. 



In the force plan ab = Pi, be = P 2 , cd = P 3 , and ad = Q, which is first 
decomposed into 1 and 2 by drawing ea and de parallel to AB and AJ; 
then, combining 1 with P lt and decomposing the resultant, as before, we 
get 3 and 4, or ef and fb. Having 2 and 3, we get in like manner 5 and 6# 
or gf and dg ; then combining 4 and 5 with P 2 , and decomposing with 
parallels to CK and CD, we obtain the forces 8 and 9, and so proceed until 
we reach 12, which is the middle of the symmetrical figure. The members 
KL, DL, EL, and ML are all subject to tension. 



WIND STRESSES. 

In designing large and important roof trusses it is important to investi- 
gate the stresses due to wind pressure, as well as those due to the weight 
of the roof and of snow ; and, indeed, in some cases the resistance to wind 
is the most important of all. 

As an illustration of the applicability of the graphical method to the 
determination of wind stresses, we will take the English roof truss, whose 




Wind Stresses. 



conditions under a vertical load have already been examined, and con- 
sider it as also subjected to a wind stress, W. 

We have first to determine the forces, Qi and Q 2 , acting at the points, A 
and D. The wind pressure will be taken as acting on the surface of the 
roof from A to B. Let W be the resultant of the entire wind pressure 
acting normal to AB, and let P be the total vertical load upon that half 
of the truss. By combining these two forces we obtain the direction, FS, 
of their resultant, and also its magnitude, which we then lay off on the 
force plan at cc\. Upon the other half of the truss we have only the 
vertical load, which may be considered as acting at J, and equal in mag- 
nitude to P. By prolonging its direction until it intersects the previously 
determined line at S, we have at S a point in the resultant of the entire 
load upon the roof, including wind pressure. By making r^, in the force 
plan equal to P, we have ac for the direction of this resultant, which may 
then be laid off at ST in the drawing. In order to determine the forces 



Graphical Statics. 255 



Qi and Q 2 we must recollect that when we have two closing forces to 
determine" we must also have at least two conditions given. In this case, 
then, we must first find the direction of Qi and Q 2 . 

The wind pressure produces a horizontal thrust which must be met by 
the stability of the walls or columns upon which the roof rests. In each 
case it must be determined whether this horizontal thrust is borne equally 
or unequally by both supports, and in what proportion it is divided. To 
this end we first find the proportion of the vertical component of the 
force ac, which comes upon each support (as found by the intersection of 
ST, prolonged with AD), and then combine these vertical forces with their 
respective horizontal components. It often happens that all the horizontal 
thrust is borne by one of the supports, which it must of course be pre- 
pared to resist. This often occurs in the case of railway stations, and 
under such circumstances the direction of each force must be determined 
separately. First prolong the vertical at D downward until it intersects 
ST, and join the intersection with A (the lines are only indicated in the 
figure). This gives the direction of the force at A. We have now both 
the direction of the reaction at D and the direction of that at A. We must 
also consider the distribution of the forces at the various knots between A 
and B and between B and D. We have for the points between A and B 
the resultants between the proportional parts of P and W, while from B to 
D we have simply the proportional parts of P. This gives at A the force 
Pi ; at E, P, and G, the force Po ; at the peak, the force P 3 ; at JET, J, and K, 

P P 

the force P 4 = — - ; and at D, the vertJeal force P 5 = — . 

Returning now to the force plan, we make cd = Pi,de = ef = fg = P 2 , gh 
= P 3 , hi = ik = kl = P 4 , and la = P 5 . We now have finally the length, bl, 
for the value of the reaction, Q 2 , at the point, D, and a line (not shown) 
from b to d gives the magnitude bf the force, Q lf acting at A. 

The determination of the stresses in the various members can now 
readily be made. The decomposition of bd by drawing bm and md parallel 
respectively to AE and AL gives the forces 1 and 2. We thus proceed until 
we reach the rod, PC, or No. 13, for which we get the tension, rs = 13, by 
drawing the vertical, rs from r, until it intersects the line, ns, drawn 
parallel to BD. We then continue to determine the forces from 15 to 25, 
as already shown. The force plan shows that under these conditions 
similarly placed struts are subjected to dissimilar stresses. The determina- 
tion of the stresses might have been made in the reverse order, beginning 
with the triangle, xbl, which should give the same results, and which may 
be used to prove the accuracy of the work. A proof is also made by the 
accuracy with which the line, wx, drawn from w, parallel to KO, intersects 
the point, x, which was first determined by the intersection of bx and Ix. 
As a matter of fact, it will be found to require careful drawing in order to 
insure the closing of the diagram. 

By comparing the last force plan with that found for the same roof 
truss, under vertical loads only, it will be seen how greatly the wind 
stresses affect the structure. In order to complete the calculation, a second 
plan should be drawn, assuming the wind to act also upon BD. 

FRAMED BEAMS. 

Beams of various forms are often framed in various shapes and made 
both of wrought and cast iron, and have many applications, such as 
walking beams for steam engines, for cranes, arms, etc. A few examples 
will show the method of investigation for such cases. 

I. Cantilevers with Straight Members.— The load, P, acts at A in 
a direction normal to the axis of the frame, which is supported at B and 
C. The force plan is constructed as follows : Draw ab = P, and from its 
extremities draw ac and be parallel to 1 and 2, which gives the forces in 
those members. Each of these is then decomposed into two other forces, 
— 1 into 3 and 4, 2 into 5 and 6, giving the triangles, bee and adc. 

The forces 3 and 5 are then combined and the resultant decomposed 
into 7 and 8. To do this we transfer 5 = dc tofe, and join the resultant, fb, 
which can readily be separated into 7 and 8. We proceed in this manner 
for the remaining members, and as the frame is symmetrical about the 
axis, gc, only one-half of the diagram need be completed. The lines, ga 
and bg, which are the final resultants of 15 with 17, and 16 with 18, are 



256 



Graphical Statics. 



also the external forces at B and C, the points of attachment, provided 
that their direction be permitted to remain the same. 








Cantilever. — I. 

II. Double-Loaded Frame.— In this case we have the force, Pi, acting 
downwards at A, and a force, P 2 , acting upwards at D, while the points of 
attachment remain at B and C, as before. The members, AB and AC, are 
polygonal-formed. The force plan is drawn just as before, until the force 
13 is reached. At D the members are attached to each other at their inter- 
section, so that the force, P 2 , acts upon both 15 and 16. At this same point 
we have the action of the forces 12 and 13. Now join the extremities of 
12 and 13 by the dotted line shown, and mark off the length of the force, 
P 2 , which is subtracted, because its action is upward, thus obtaining the 
resultant of the three forces. We can then draw 15 and 16 and proceed 
without interruption to 20. Finally, we draw bf and ea, the external forces 
at Qi and Q 2 , which hold the entire frame in equilibrium. 




Cantilever. — II. 



III. Framed Boom.— This figure is a portion of a framed arch which 
may be used for the projecting boom of a large crane. At A and D we 
have the forces, Pi and P 2 , and at B and C the external forces, Qi and Q 2 . 
The force plan is now required to determine the internal forces acting on 
the various members of the structure. Before this can be done we must 
first determine the as yet unknown direction of the force, Q*. Prolong Pi 
and P 2 to their intersection at E, and by drawing in the force plan the 
triangle, o6c, determine the direction, FE, of their resultant; then prolong 
Q] until it intersects EF at G, and join CG, which will be the required 
direction of the force, Q 2 . Completing the figure in the force plan, we 
have cd = = Q\ and da = § 2 - We now proceed from Pi = ab and lay off the 
forces 1 and 2. decomposing 2 into 3 and 4 ; combine 3 and 1 and decompose 
their resultant, obtaining 5 and 6. We thus proceed until we reach 12, 
which we obtain by combining 9 and 8 and decomposing the resultant into 
11 and 12. We now have to combine 10 and 11 with P 2 , and decompose the 



Gkaphical Statics. 



257 



resultant into 13 and 14. We first transfer the force 11 to e, making it equal 
to ef, in order to avoid the confusion of lines, which would occur if the 
construction were made at a. Now, drawing the path 11. 10, P 2 , we have 
the closing line, cf, which decomposes into 13 and 14. We then have 15 




Framed Boom. — III. 

and 16 from the resultant of 13 and 12, and finally, 17, as the line joining 
15 and 16 with d, since 16 and 17 must have the resultant, ad = Q 2 . If the 
work is correctly done, we will find 17 falls parallel to BC, which affords a 
convenient and valuable proof for the whole work. 

BRIDGE TRUSSES 

may be examined in a similar manner 
to roof trusses. 

I. Simple Truss.— In the case of 
a truss of four panels, with vertical 
struts and diagonal tie-rods, as in the 
figure, we have on each pillar a load, 
P, except at the ends, where it is equal 

p 
to — , this giving a total load of 4P, or 

a vertical reaction of 2P on each pier. 
The diagram shown is constructed for 
one-half of the truss, the forces in the 
\ other half being identical. In the dia- 
gram we make ad = 2P. Since %P is 
supported directly upon the pier at A, 
we make ab = JP« Then draw 2, paral- 
lel to BD, and 3, parallel to BC, the 
lengths of these lines giving the stresses 
in the corresponding members. From 
c draw 5, parallel to CD, and from b 
draw 6, parallel to CE. Combine 5 and 
2 with P for a resultant, ed, and draw 
7, parallel to BE, and 8, parallel to DF. Each member will then have its 
load given by the lengths of the lines in the diagram, the double lines rep- 
resenting compression and the single lines tension, as before. The middle 
strut, FE, bears a compression equal to its top load, P. 

17 






° s \ e 


■■ 
I 

L- 

i 
[. 


\ 


^^-"\ 


I 

Sim i 


I G b 
i 

)le Truss. — '. 



258 



Graphical Statics. 



II. Simple Truss.— In the case of a truss with diagonal struts and 
vertical tie-rods, as in the figure, we have similar loading, and the diagram 

is given below. The tension on EF 
is zero, and there is no compression 
on BD. 

III. Combined Truss.— By com- 
bining the two simple trusses the 
combination is formed in which all 
the loads may be doubled for the 
same stresses as shown in the previous 
diagrams, except for those members which 




Ag 



r i 


" i 


" i 


2P 


x 


X 


xl 


X 



Simple Truss. — II. 



C E 

Combined Truss. 



C 
-III. 



coincide. We thus have loads of 2P on the vertical struts, except the 
middle one, while the loads on the diagonals remain unchanged. 



Leverage. 

The statical moment of a force, as already explained, is the leverage of 
that force,— that is, the magnitude of the force, multiplied bv the perpen- 
dicular distance from the centre about which it acts. If two or more 
forces are in equilibrium, so that motion does not take place, their statical 
t moments must be equal. This is only a general 

~ a » i « n "I statement of what may be called the principle 

S of the lever. 

f Thus, the statical moment of the force, P, is 

©I the force multiplied bv the distance, a, from the 
(P) fulcrum,/, or = Pa. In like manner the stati- 
cal moment of P' is equal to P r a f , and, if the 
beam remains stationary, Pa = P'a'. 
This is true no matter how the lever arms may be disguised by the form 
or material which may include them. Thus, the force may act at the 
perimeter of a wheel, the radius of the wheel then becoming the lever 
arm, or it may be included in some other form ; but the forces themselves 
must always be considered as acting upon lever arms of a length equal to 
the perpendicular distance from the lines of action of the forces to the 
fulcrum. 

in the case of a force acting at any point to overturn a mass, the resist- 
ance must be considered as the weight of the body acting at the centre of 
gravity. 




Thus, in the case of either wall shown in the illustration, the force, P, 
acting to overturn the wall about the corner, A, is opposed by a force, O, 
equal to the weight of the wall, acting at a lever arm, AM, equal to the 



Motion. 



259 



distance of the corner, A, from the centre of gravity, S, measured at right 
angles to the line of the force, G. This gives for the moment of stability of 
the wall 
- AM X S. 



MOTION. 
Falling Bodies. 

According to the law of gravitation enunciated by Sir Isaac Newton, 
every particle of matter in the universe attracts every other particle of matter 
with a force which varies directly as the mass, and inversely as the square of the 
distance. 

In accordance with this law any body above the surface of the earth, 
when permitted to fall freely, does so with an accelerated velocity. 

The unit or measure of force of gravity is assumed to be the velocity a 
falling body has attained at the end of the first second of descent. This 
unit is commonly denoted by the letter g ; its value at the level of the sea 
in New York is g = 32.17 feet per second, in vacuum, g is called the accel- 
eration of gravity. The space fallen through in the first second is Y^g = 
16.085 feet. 

This value increases with the latitude, and decreases with the elevation 
above the level of the sea. 

I = latitude, h = height in feet above the level of the sea, and r — radius 
I of the earth in feet, at the given latitude, I. 

r = 208 87510 (1 + 0.00164 cos 21), 

g = 32.16954 (1 — 0.00284 COS 21) (l — —\. 

Notation. 
j S = the space in feet which the falling body passes through in the time T. 
\ u = the space in feet which the body falls in the Tth second. 

V = velocity in feet per second of the falling body at the end of the time T. 
I T = time in seconds the body is falling. 

In the metric system the value of g is given in metres per second, and 
is taken as equal to 9.81 metres at latitude 45° and at the level of the sea. 

Formulas for Accelerated Motion. 

Velocity, V* in Feet per Second. 



1. 

2. 



V = 

V = 



S = 



s = 



10. 



113. 



!F = 



gT. 

2S 
T ' 

gT* 

2 * 
VT 
2 * 

g' 

2S 



3. 



4. 



V=l/2gS. 
F=8.02i/& 



4a. 



F=4.429l/& 
(Metric.) 



Space, £> Fallen through in Feet. 

V 2 

V 2 




8a. 



S = 



V 2 



12a. 



19.62 ' 
(Metric.) 



V~s 



2.04 
(Metric) 



Space Fallen through in the Tth Second 



g + 2- 






'260 Motion. 



Example 1. What velocity has a body attained after having fallen freely 
for a time of T = 2% seconds ? 

Velocity, V= 32.17 X 2.5 = 80.2 feet per second. 

Example U. A body is dropped from a height of 5= 98 feet. What*i 
velocity will it have on reaching the ground, and what time is required 
for its fall ? _ 

Formula 4. Velocity, V= 802 -|/98 = 79.3939 feet per second. 

-./ c 1/98 

Formula 12. Time, T = \ M = \ M = 2.46 seconds. 
4.01 4.01 

Example 5. A body was dropped at the opening of a hole in the rock, 
and reached the bottom in T= 3.5 seconds. Required the depth of the 
hole? 

Formula 5. Depth, 5 = fl4p = * = 196.98 feet. 

Example 8. What space must a body fall through in order to acquire a 
velocity V = 369 feet per second ? 

V 2 369 2 

Space ' s =ii^ = CT = 2m6feet 

Example 10. What time is required for a body to fall £ = 2116.6 feet, 
when the final velocity V— 369 feet per second? 

m 2S 2X2116.6 „ AtM , 

Time, T = -=7- — ^^ = 11.472 seconds. 

V ob9 

Example 13. A body falls freely for a time of T= 4% seconds. How 
much will it fall in the last second ? 

Formula 13. u = g{ T— %) = 32.17 (4.5 — 0.5) = 128.68 feet. 

Retarded Motion. 

A body thrown up vertically will obtain inversely the same motion 
as when it falls down, because it is the same force that acts upon it, 
causing retarded motion when it ascends, and accelerated motion when it 
descends. 

V = the velocity at which the body starts to ascend. 

v = velocity at the end of the time t. 

T = time in seconds in which the body will ascend. 

t = any time less than T. 

S — height in feet to which the body will ascend. 

s = the space it ascends in the time t. 

Velocity in Feet per Second at the End of the Time t. 

15. v =V—gt. I 16. V = -J — "Y" 

Height of Ascension in the Time t. 
8 =t(^V-g~y I 18. •-<(« + ?£-). 

Starting Velocity in Feet per Second. 

19. y=v + gt. I 20. F=j+#y. 

Time of Ascension in Seconds. 



17 



9 I 9 \g* 9 it$ { 

Starting and Ending Velocities. 

23. v=i/T'S — 2gi. I 24. V= \/v 2 + 2^8. 

Formulas for Tand S are the same as for accelerated motion. 



Gravity. 



261 



Example 22. A ball starts to ascend with a velocity of 135 feet per 
second. At what velocity will it strike an object 60 feet above ? 
Find the time t by the Formula 22. 



t = 



135 



/ 1352 
" \ 32.16 



2X1 



- 0.41 seconds, 



32.16 \ 32.16 32.16 

mtil it strikes ; and from Formula 15 we have 

v = 135 — 32.16 X 0.41 = 121.83 feet per second. 

Example 2U. With what velocity must a body start to ascend in order to 
strike an object s = 15 feet above with a velocity v — 10 feet per second ? 



Velocity, V= i/lO 2 + 2 X 32.17 X 15 -= 32.63 feet per second. 

Force of Gravity. 




V-= gTsin. x = \/2gS sin x, 
gT* V* 



S = 



2 sin x 2(7 sin z' 



gsinx \ g 




A body will fall from o the dis- 
ances a, 6, c, and d, in equal times. 




A body will fall from a to b, via c, 
l the shortest time, if the curve is 
fj cycloid. 

| S = 4d, the length of the cycloid, 



h^S-Afe 




T = 



VW 
gF 

VW 

gT 



-4 



2SW 
gF ' 

2S W 

g T 2 ' 



W= P + Q, and F= P— i 



262 



Falling Bodies. 



Palling Bodies. 

English Units. 

V= velocity in feet per second at the end of fall. 
T= time in seconds of the fall. 
S = space fallen through in feet. 



V 


T 


S 


V 


T 


S 


V 


T 


8 


0.1 


0.0031 


.00015 


5.1 


0.1585 


0.4042 


11 


0.3419 


1.8804 


0.2 


0.0062 


.00031 


5.2 


0.1616 


0.4202 


12 


0.3730 


2.2380 


0.3 


0.0093 


0.0014 


5.3 


0.1647 


0.4364 


13 


0.4041 


2.6266 


0.4 


0.0124 


0.0025 


5.4 


0.1678 


0.4530 


14 


0.4352 


3.0464 


0.5 


0.0155 


0.0039 


5.5 


0.1709 


0.4700 


15 


0.4663 


3.4975 


0.6 


0.0186 


0.0055 


5.6 


0.1740 


0.4872 


16 


0.4973 


3.9784 


0.7 


0.0217 


0.0076 


5.7 


0.1771 


0.5047 


17 


0.5284 


4.4914 


0.8 


0.0248 


0.0099 


5.8 


0.1802 


0.5226 


18 


0.5595 


5.0355 


0.9 


0.0279 


0.0125 


5.9 


0.1833 


0.5407 


19 


0.5906 


5.6107 


1. 


0.0311 


0.0155 


6. 


0.1865 


0.5595 


20 


0.6217 


6.2170 


1.1 


0.0342 


0.0188 


6.1 


0.1896 


0.5782 


21 


0.6527 


6.8502 


1.2 


0.0373 


0.0224 


6.2 


0.1927 


0.5973 


22 


0.6838 


7.5218 


1.3 


0.0404 


0.0262 


6.3 


0.1958 


0.6168 


23 


0.7149 


8.2213 


1.4 


0.0435 


0.0304 


6.4 


0.1989 


0.6365 


24 


0.7460 


8.9520 


1.5 


0.0466 


0.0335 


6.5 


0.2020 


0.6565 


25 


0.7771 


9.7125 


1.6 


0.0497 


0.0381 


6.6 


0.2051 


0.6768 


26 


0.8082 


10.566 


1.7 


0.0528 


0.0432 


6.7 


0.2082 


0.6975 


27 


0.8393 


11.330 


1.8 


0.0559 


0.0485 


6.8 


0.2113 


0.7184 


28 


0.8704 


12.185 


1.9 


0.0590 


0.0551 


6.9 


0.2144 


0.7397 


29 


0.9015 


13.072 


2. 


0.0622 


0.0622 


7. 


0.2176 


0.7616 


30 


0.9325 


13.987 


2.1 


0.0653 


0.0685 


7.1 


0.2207 


0.7835 


31 


0.9636 


14.936 


2.2 


0.0684 


0.0756 


7.2 


0.2238 


0.8057 


32 


0.9947 


15.915 


2.3 


0.0715 


0.0822 


7.3 


0.2269 


0.8282 


33 


1.0258 


16.926 


2.4 


0.0746 


0.0895 


7.4 


0.2300 


0.8510 


34 


1.0569 


17.967 


2.5 


0.0777 


0.0971 


7.5 


0.2331 


0.8741 


35 


1.0879 


19.038 


2.6 


0.0808 


0.1050 


7.6 


0.2362 


0.8975 


36 


1.1190 


20.142 


2.7 


0.0839 


0.1135 


7.7 


0.2393 


0.9213 


37 


1.1501 


21.277 


2.8 


0.0870 


0.1218 


7.8 


0.2424 


0.9453 


38 


1.1812 


22.443 


2.9 


0.0901 


0.1305 


7.9 


0.2455 


0.9697 


39 


1.2123 


23.640 


3. 


0.0932 


0.1398 


8. 


0.2487 


0.9948 


40 


1.2434 


24.868 


3.1' 


0.0963 


0.1492 


8.1 


0.2518 


1.0168 


41 


1.2745 


26.127 


3.2 


0.0994 


0.1590 


8.2 


0.2549 


1.0451 


42 


1.3056 


27.417 


3.3 


0.1025 


0.1691 


8.3 


0.2580 


1.0707 


43 


1.3367 


28.739 


3.4 


0.1056 


0.1795 


8.4 


0.2611 


1.0966 


44 


1.3678 


29.407 


3.5 


0.1087 


0.1886 


8.5 


0.2642 


1.1228 


45 


1.3989 


31.475 


3.6 


0.1118 


0.2012 


8.6 


0.2673 


1.1494 


46 


1.4300 


32.890 


3.7 


0.1149 


0.2125 


8.7 


0.2704 


1.1762 


47 


1.4611 


34.336 


3.8 


0.1170 


0.2223 


8.8 


0.2735 


1.2034 


48 


1.4922 


35.813 


3.9 


0.1201 


0.2:555 


8.9 


0.2766 


1.2259 


49 


1.5233 


37.321 


4. 


0.1243 


0.2486 


9. 


0.2797 


1.2586 


50 


1.5544 


38.830 


4.1 


0.1274 


0.261] 


9.1 


0.2828 


1.2867 


51 


1.5854 


40.413 


4.2 


0.1305 


0.2740 


9.2 


0.2859 


1.3151 


52 


1.6165 


42.029 


4.3 


0.1336 


0.2872 


9.3 


0.2<890 


1.3438 


53 


1.6475 


43.659 


4.4 


0.1367 


0.2939 


9,1 


0.2921 


1.3729 


54 


1.6786 


45.322 


4.5 


0.1398 


0.3145 


9.5 


0.2952 


1.4022 


55 


1.7097 


47.017 


4.6 


0.1429 


0.3286 


9.6 


0.2983 


1.4318 


56 


1.7407 


48.740 


4.7 


0.1460 


1 i:;i 


9.7 


0.3014 


1.4618 


57 


1.7718 


50.396 


4.8 


0.1491 


0.3578 


9.8 


0.3045 


1.4920 


58 


1.8029 


52.284 


4.9 


0.1522 


l).:;7'J ( J 


9.9 


0.3076 


1.5226 


59 


1.8340 


54.103 


5. 


0.1554 


0.3885 


10. 


0.3108 


1.5540 


60 


1.8651 


55.953 



Falling Bodies. 



263 



Falling Bodies. 

English Units. 






J2S 



s = - 



V 


T 


S 


V 


T 


5 


V 


T 


5 


65 


2.0206 


65.669 


530 


16.478 


4366.6 


1030 


32.027 


16494 


70 


2.1769 


76.260 


540 


16.788 


4452.8 


1040 


32.338 


16815 


75 


2.3314 


87.427 


550 


17.099 


4701.7 


1050 


32.649 


17141 


80 


2.4868 


97.472 


560 


17.409 


4874.5 


1060 


32.950 


17463 


85 


2.6422 


112.29 


570 


17.720 


5050.2 


1070 


33.261 


17794 


90 


2.7976 


125.89 


580 


18.030 


5228.7 


1080 


33.572 


18129 


95 


2.9530 


140.27 


590 


18.341 


5410.6 


1090 


33.883 


18446 


100 


3.1085 


155.42 


600 


18.651 


5595.3 


1100 


34.194 


18806 


I 110 


3.4194 


188.07 


610 


18.961 


5783.1 


1110 


34.504 


19149 


i 120 


3.7302 


223.81 


620 


19.271 


5974.0 


1120 


34.815 


19496 


130 


4.0411 


262.67 


630 


19.582 


6168.3 


1130 


35.126 


19846 


140 


4.3519 


304.63 


640 


19.893 


6365.7 


1140 


35.436 


20198 


150 


4.6627 


349.70 


650 


20.204 


6566.3 


1150 


35.747 


20504 


! 160 


4.9736 


397.88 


660 


20.515 


6770.0 


1160 


36.058 


20913 


170 


5.2844 


449.18 


670 


20.826 


6976.7 


1170 


36.369 


21275 


180 


5.5953 


503.36 


680 


21.137 


7186.6 


1180 


36.680 


21641 


190 


5.9061 


561.08 


690 


21.448 


7399.5 


1190 


36.991 


22009 


200 


6.2170 


621.70 


700 


21.759 


7615.6 


1200 


37.302 


22381 


210 


6.5279 


689.43 


710 


22.070 


7834.8 


1210 


37.613 


22755 


220 


6.8387 


752.26 


720 


22.380 


8056.8 


1220 


37.924 


23133 


! 230 


7.1496 


822.20 


730 


22.691 


8282.2 


1230 


38.235 


23514 


240 


7.4604 


895.25 


740 


23.002 


8510.7 


1240 


38.546 


23898 


250 


7.7713 


971.41 


750 


23.313 


8742.4 


1250 


38.857 


24285 


260 


8.0821 


1050.6 


760 


23.623 


8976.7 


1260 


39.168 


24676 


270 


8.3930 


1133.1 


770 


23.934 


9214.6 


1270 


39.479 


25069 


280 


8.7038 


1218.5 


780 


24.245 


9455.5 


1280 


39.780 


25459 


290 


9.0147 


1308.2 


790 


24.556 


9699.6 


1290 


40.090 


25855 


300 


9.3255 


1398.8 


800 


24.868 


9947.2 


1300 


40.411 


26267 


310 


9.6363 


1493.7 


810 


25.179 


10197 


1310 


40.722 


26673 


1 320 


9.9472 


1591.6 


820 


25.490 


10451 


1320 


41.033 


27081 


1 330 


10.258 


1690.6 


830 


25.801 


10707 


1330 


41.343 


27493 


340 


10.569 


1791.7 


840 


26.112 


10967 


1340 


41.654 


27908 


1 350 


10.879 


1903.8 


850 


26.423 


11230 


1350 


41.965 


28326 


I 360 


11.190 


2014.2 


860 


26.733 


11495 


1360 


42.276 


28747 


1 370 


11.501 


2127.7 


870 


27.044 


11764 


1370 


42.587 


29172 


: 380 


11.812 


2244.3 


880 


27.354 


12035 


1380 


42.897 


29599 


390 


12.123 


2364.0 


890 


27.665 


12311 


1390 


43.208 


30029 


400 


12.434 


2486.8 


900 


27.976 


12589 


1400 


43.519 


30463 


410 


12.745 


2612.7 


910 


28.287 


12871 


1410 


43.820 


30893 


420 


13.055 


2741.5 


920 


28.598 


13155 


1420 


44.131 


31333 


430 


13.366 


2873.7 


930 


28.908 


13442 


1430 


44.442 


31776 


440 


13.677 


3008.9 


940 


29.219 


13733 


1440 


44.753 


32222 


450 


13.989 


3144.8 


950 


29.530 


14027 


1450 


45.064 


32671 


460 


14.300 


3289.0 


960 


29.841 


14323 


1460 


45.375 


33123 


j 470 


14.611 


3433.6 


970 


30.152 


14623 


1470 


45.686 


33579 


480 


14.922 


3581.3 


980 


30.463 


14927 


1480 


45.997 


34037 


! 490 


15.233 


3732.1 


990 


30.774 


15233 


1490 


46.308 


34499 


| 500 


15.545 


3886.2 


1000 


31.085 


15542 


1500 


46.619 


34964 


510 


15.856 


4043.3 


1010 


31.396 


15855 


1510 


46.930 


35432 


520 


16.167 


4203.4 


1020 


31.707 


16179 


1520 


47.241 


35853 



264 



Falling Bodies. 



Palling Bodies. 

Metric System. 
Space, s, for terminal velocity, v, in metres. 



2<7' 



V 


S 


V 


S 


. v 


S 


0.0 


0.0000 


4.0 


0.8157 


8.0 


3.2627 


1 


0.0005 


1 


0.8570 


1 


3.3447 


2 


0.0020 


2 


0.8993 


2 


3.4278 


3 


0.0046 


3 


0.9426 


3 


3.5120 


4 


0.0082 


4 


0.9869 


4 


3.5971 


5 


0.0127 


5 


1.0323 


5 


3.6832 


6 


0.0184 


6 


1.0787 


6 


3.7704 


7 


0.0250 


7 


1.1261 


7 


3.8586 


8 


0.0326 


8 


1.1746 


8 


3.9478 


9 


0.0413 


9 


1.2240 


9 


4.0381 


1.0 


0.0510 


5.0 


1.2745 


9.0 


4.1293 


1 


0.0617 


1 


1.3260 


1 


4.2216 


2 


0.0734 


2 


1.3785 


2 


4.3149 


3 


0.0862 


3 


1.4320 


3 


4.4092 


4 


0.0999 


4 


1.4866 


4 


4.5045 


5 


0.1147 


5 


1.5421 


5 


4.6009 


6 


0.1305 


6 


1.5987 


6 


4.6982 


7 


0.1473 


7 


1.6563 


7 


4.7966 


8 


0.1652 


8 


1.7149 


8 


4.8960 


9 


0.1840 


9 


1.7746 


9 


4.9965 


2.0 


0.2039 


6.0 


1.8352 


10.0 


5.0979 


1 


0.2248 


1 


1.8969 


1 


5.2004 


2 


0.2467 


2 


1.9596 


2 


5.3039 


3 


0.2697 


3 


2.0234 


3 


5.4084 


4 


0.2936 


4 


2.0881 


4 


5.5139 


5 


0.3186 


5 


2.1539 


5 


5.6204 


6 


0.3446 


6 


2.2207 


6 


5.7280 


7 


0.3716 


7 


2.2885 


7 


5.8366 


8 


0.3997 


8 


2.3573 


8 


5.9462 


9 


0.4287 


9 


2.4271 


9 


6.0568 


3.0 


0.4588 


7.0 


2.4980 


11.0 


6.1685 


1 


0.4899 


1 


2.5699 


12.0 


7.3410 


2 


0.5220 


2 


2.6428 


13.0 


8.6155 


3 


0.5552 


3 


2.7167 


14.0 


9.9919 


4 


0.5893 


4 


2.7916 


15.0 


11.4703 


5 


0.6245 


5 


2.8676 


16.0 


13.0507 


6 


0.6607 


6 


2.9446 


17.0 


14.7330 


7 


0.6979 


7 


3.0226 


18.0 


16.5172 


8 


0.7361 


8 


3.1016 


19.0 


18.4035 


9 


0.7754 


9 


3.1816 


20.0 


20.3916 



Falling Bodies. 



265 



Falling Bodies. 

Metric System. 

Terminal velocity, v, for space, s, in metres. 

v = \/2gs. 



s 


V 


S 


V 


S 


V 


0.0 


0.0000 


4.0 


8.8580 


8.0 


12.5271 


1 


1.4006 


1 


8.9681 


1 


12.6052 


2 


1.9807 


2 


9.0767 


2 


12.6827 


3 


2.4259 


3 


9.1842 


3 


12.7598 


4 


2.8012 


4 


9.2904 


4 


12.8365 


5 


3.1318 


5 


9.3953 


5 


12.9127 


6 


3.4307 


6 


9.4991 


6 


12.9884 


7 


3.7056 


7 


9.6019 


7 


13.0637 


8 


3.9614 


8 


9.7035 


8 


13.1385 


9 


4.2017 


9 


9.8040 


9 


13.2130 


1.0 


4.4290 


5.0 


9.9036 


9.0 


13.2870 


1 


4.6452 


1 


10.0021 


1 


13.3606 


2 


4.8517 


2 


10.0997 


2 


13.4338 


3 


5.0499 


3 


10.1963 


3 


13.5066 


4 


5.2405 


4 ' 


10.2921 


4 


13.5790 


5 


5.4244 


5 


10.3869 


5 


13.6511 


6 


5.6023 


6 


10.4809 


6 


13.7228 


7 


5.7747 


7 


10.5740 


7 


13.7940 


8 


5.9421 


8 


10.6664 


8 


13.8650 


9 


6.1049 


9 


10.7580 . 


9 


13.9355 


2.0 


6.2635 


6.0 


10.8488 


10.0 


14.0057 


1 


6.4182 


1 


10.9388 


1 


14.0756 


2 


6.5693 


2 


11.0281 


2 


14.1451 


3 


6.7169 


3 


11.1167 


3 


14.2143 


4 


6.8614 


4 


11.2046 


4 


14.2831 


5 


7.0029 


5 


11.2918 


5 


14.3516 


6 


7.1415 


6 


11.3783 


6 


14.4198 


7 


7.2776 


7 


11.4642 


7 


14.4877 


8 


7.4111 


8 


11.5495 


8 


14.5552 


9 


7.5423 


9 


11.6340 


9 


14.6224 


3.0 


7.6712 


7.0 


11.7180 


11.0 


14.6893 


1 


7.7981 


1 


11.8014 


12.0 


15.3425 


2 


7.9228 


2 


11.8842 


13.0 


15.9692 


3 


8.0457 


3 


11.9665 


14.0 


16.5720 


4 


8.1667 


4 ' 


12.0482 


15.0 


17.1535 


5 


8.2859 


5 


12.1293 


16.0 


17.7160 


6 


8.4035 


6 


12.2099 


17.0 


18.2612 


7 


8.5194 


7 


12.2900 


18.0 


18.7907 


8 


8.6337 


8 


12.3695 


19.0 


19.3056 


9 


8.7466 


9 


12.4485 


20.0 


19.8071 



266 



Falling Bodies. 



Falling Bodies. 

Metric System. 

Space, s, in metres for time, t, from 0.1 to 10 seconds. 

s = %gt 2 . 



t 


S 


t 


S 


t 


S 


0.0 


0.0000 


3.0 


44.1362 


6.0 


176.5446 


1 


0.0490 


1 


47.1276 


1 


182.4785 


2 


0.1962 


2 


50.2171 


2 


188.5104 


3 


0.4414 


3 


53.4047 


3 


194.6404 


4 


0.7846 


4 


56.6904 


4 


200.8685 


5 


1.2260 


5 


60.0742 


5 


207.1947 


6 


1.7654 


6 


63.5561 


6 


213.6190 


7 


2.4030 


7 


67.1360 


7 


220.1413 


8 


3.1386 


8 


70.8140 


8 


226.7618 


9 


3.9723 


9 


74.5901 


9 


233.4802 


1.0 


4.9040 


4.0 


78.4643 


7.0 


240.2968 


1 


5.9339 


1 


82.4365 


1 


247.2115 


2 


7.0618 


2 


86.5069 


2 


254.2243 


3 


8.2878 


3 


90.6753 


3 


261.3351 


4 


9.6119 


4 


94.9418 


4 


268.5440 


5 


11.0340 


5 


99.3063 


5 


275.8510 


6 


12.5543 


6 


103.7690 


6 


283.2560 


7 


14.1726 


7 


108.3297 


7 


290.7592 


8 


15.8890 


8 


112.9886 


8 


298.3604 


9 


17.7035 


9 


117.7455 


9 


306.0597 


2.0 


19.6161 


5.0 


122.6004 


8.0 


313.8571 


1 


21.6267 


1 


127.5535 


1 


321.7526 


2 


23.7354 


2 


132.6046 


2 


329.7461 


3 


25.9422 


3 


137.7538 


3 


337.8377 


4 


28.2471 


4 


143.0011 


4 


346.0274 


5 


30.6501 


5 


148.3465 


8.5 


354.3153 


6 


33.1512 


6 


153.7900 


9.0 


397.2254 


7 


35.7503 


7 


159.3315 


9.5 


442.5875 


8 


38.4475 


8 


164.9711 


10.0 


490.4017 


9 


41.2428 


9 


170.7088 


11.0 


593.3911 



Statics. 



267 



M-4 



1st 



Leverage. 

Static Moments. 

<tf + a 'f + a"/" = & r + V r ' + & //,r " 



.F: W=l 

1. J--^ 

2. TF=^-. 


£, 


1 ^ a 


, r ^a 
4 ' X== TF+^' 




JP: W=l:L, FL = Wl. 



5. .F = 



6. TF = 



Wl 
L ' 

Ik 

I ' 



7. L = 



8. J = 



Wa 
W—F' 

Fa 
W—F' 




F: W=l 


L, 




jPX = Wl. 


9. F=^. 




11. 


r Wa 
L ~ F-W 


• 10. w= FL . 




1?, 


!-*_. 



I ' 



F— W' 




If the sum of the moments that 
act to move the "body in one direc- 
tion are equal to the sum of the 
moments that act opposite, the act- 
ing forces will be in equilibrium ; c 
being the centre or fulcrum. 









*• 


x * 






> 



P Q B 

To find the fulcrum, c, when three 
forces act on the lever, 

Ex = Q{a — b — x) + F(a — x), 

_ Qa + Fa— Qb 




Q = weight of the lever, x = dis- 
tance from the centre of gravity of 
the lever to the fulcrum. Balance 
the lever over a sharp edge, and the 
centre of gravity is found. 



Wl— Qx 



W-- 



FL + Qx 
I 



268 Dynamics. 



DYNAMICS. 

We have already referred to the fact that in practical engineering work 
there are but three elementary quantities: Force, expressed as a weight; 
Space, expressed as a lineal distance; and Time, expressed in hours, min- 
utes, or seconds. 

In the problems of Dynamics, or the study of force and motion, we 
have the following relations, in which 

P = force, 5 = space, T— time, M— mass, 

S' PSf 

V = -=■ = velocity, P = —=- = FV = power, 

K = FS = work, K = %MV 2 = work. 



Dynamical Formulas. 

Force or Pressure, in Pounds. 



V 


I 2. F= m * P . 1 3. , * 1 4. 


f=JL. 

VT 


Velocity, in Feet, per Second. For Uniform Motion. 


T 


I 6 ' F -^p' 1 7 ' * -p"~"" 1 8 - 


PP 


Time 


of Action, in Seconds. For Uniform Motion. 


*~f 


1 FS 1 FS 1 

I 10 - r = P - I 11 - y =55o/.P- I 12 - 

Power, in Foot-Pounds, per Second. 


PF 


P = FV. 


J 14. P = ~L. | 15. P=550 7P. | 16. 


P=* 



13. 



Space passed through in the Time T. 

17. 5= VT. I 18. S=-=^-. I 19. g = 550 ^ 1P .l 20. £=i 
I PI p I P 

Horse-power. 

21. JB^-^L I 22. IP=~. I 23. IP=-^-. I 24. J3P = - * 



550* I ' 550' I ! 550P' I 550 T * 

Work, in Foot-Pounds. 

25. A' = PFPin time 71 I 27. K = FS. 

26. 7T = PPin time T I 28. K = 550 JP Tin time T. 

It will be observed in the preceding formulas that an element is never 
divided by an element ; but a function is divided by an element only when 
that function contains that element. 

Power divided by velocity gives force, because power contains the ele- 
ments force and velocity; but power cannot be divided by time, because 
time is not a constituent element of power. 

Work can be divided by either one or two of its three constituent fac- 
tors. When work is divided by either two of its elements, the product 
will be the third element. 

Different elements or functions cannot be added to or subtracted from 
one another. Power or space cannot be added to or subtracted from work. 
Force, velocity, or time cannot be added to or subtracted from space. 



Dynamics. 269 



In the metric system, force is given in Kilogrammes and space in 
Metres. Work is given in Kilogrammetres ; Power in Kilogrammetres per 
second. 

The metric Horse-power is 75 kilogrammetres per second, = 32,547 foot- 
pounds per minute, or about 1.4 per cent, less than the British horse-power 
of 33,000 foot-pounds per minute. 

Work (K= FS). 

Work is the product obtained by multiplying together the elements 
force, F, and space, S. 

Work may also be expressed by K — PT, or the product of power and 
time. 

The work of a steam-engine operating with a constant power will be 
directly as the time of operation, and so with all labor, whether it be 
mechanical or manual. 

Moment of a Force (Fl). 

The moment of a force is its lever arm at right angles to its direction of 
action multiplied by its intensity in pounds or tons. 

Momentum (MV). 

The momentum of a moving body is the intensity of that constant force 
which, resisting its movement, will bring it to rest in one second. 

weight 



M = 



32.2 * 
velocity in feet per second. 



Moment of Inertia (MVr). 

The moment of inertia of a rotating body is the moment of its momen- 
tum, and is equal to its momentum, MV, multiplied by its radius of oscil- 
lation, r. 

Radius of Oscillation. 

The radius of oscillation is the mean lever-arm of the momentum of a 
revolving body. It is equal to the moment of inertia divided by the 
momentum of the revolving body. 

Radius of Gyration. 

The square of the radius of gyration of an oscillating body is equal to 
the product of the radius of oscillation and of the distance of the centre 
of gravity of the suspended body from its point of suspension. 

The intensity of the force of momentum is proportional to the distance 
of the centre of gravity from the axis of suspension, and the mean lever- 
age of the momentum is the radius of oscillation. The square of the 
"radius of gyration," then, is a convenient product of these two quan- 
tities, as including both, and therefore giving them in a convenient 
mathematical form. If a straight rod be balanced at its middle, we are 
obliged to consider each half separately and add them together. 

Units of Work. 

The usual unit of work in the British notation is the foot-pound, equal 
to one pound raised through a space of one foot. For large measurements, 
where this unit is too small, the foot-ton is used, this being the usual unit 
in ordnance computations. 

Units of Power. 

The unit of power most generally used in England and America is the 
horse-power. In rating the early steam-engines Watt made a number of 
experiments with powerful draught horses, and arrived at the value of 



270 Dynamics. 



22,000 foot-pounds per minute as the horse-power. In order to allow liberal 
measure in proportioning his steam-engines, he increased this by 50 per 
cent., and called the steam horse-power 33,000 foot-pounds per minute, or 
550 foot-pounds per second. 

The unit of power generally used in connection with electrical work is 
the watt, and for most mechanical purposes the kilowatt = 1000 watts is 
used. One English horse-power = 746 watts, or one kilowatt = 1.34 horse- 
power. The metric horse-power = 736 watts. Since electric generators are 
usually rated in kilowatts, and are frequently coupled directly to steam- 
engines, or even built into combined generating sets with them, the power 
of steam-engines is sometimes rated in kilowatts, and this practice is 
probably destined to become more and more general as electric driving 
is introduced. For all practical purposes the kilowatt may be taken as 
three-quarters of a horse-power. 

Formulas for Rotary Motion. 



F = force. 

P = power, in foot-pounds, per sec- 
ond. 
V = velocity, in feet, per second. 
S = space, in feet. 
T = time, in seconds. 



K= work, in foot-pounds. 
R = radius from centre of rotation. 
n — revolutions per minute. 
N = total number of revolutions in 
a time T. 



Force, ~F> acting in the Direction of the Tangent. 

29. F= ™ I 30. ,_«££ I 31. *-*"* I 32. ,-«»» 

2-rrRn I Rn I RnT I Rn 

Circumferential Velocity and Revolutions per Minute. 
33. V-*** I 34. F-0J0fl2»J 35. n = °§I. I 36. — 5g£ 

Time of Operation, in Seconds. 
37. r-?££ I 38. T^-i^. I 39. T= ™ U. T=- FRN 



Rn ' \ ' FRn' I ' 9.55P' I " 87.5/P' 

Radius of Revolution. 

41. B = !^Z. I 42. * = 9 -^. I 43. K = ^. I 44. B = ~»* 

n \ Fii \ Fn \ FnT 

Power Generated, in Foot-pounds, per Second. 

p== <^RnF I p= FR ± I FRN I ^ = 9.55Pr 

60 I °" 9.55 ' I ' r 9.55 T I FR ' 

Space Generated, in Feet. 

Horse-power Generated. 

__ _„ PPn I PPi^ I 87.5 iPTI M Ar S 

"'■ U ^-5252-- I 54 ' JP = 8T5T I 55 ' *--jaH 56 ' ^ = 2^ 

Work Accomplished, in Foot-pounds, in Time T. 



Dynamics. 271 



Force, Power, and Work in Moving Bodies. 

It requires force, power, and work to change the state of motion or rest 
of a body. 

In the dynamic expression M V= FT we have 



MV 
Force, F = —=-. 

F ' 



FT 

3. M=-^-. 

4. V=™. 

M 



The force, F, required to set a mass, if, in motion with velocity, V, 
depends inversely on the time, T, of action. The more time the less need 
the force be for a certain velocity, and therefore it cannot be determined 
what force has set a mass in motion without knowing its time of action ; 
but when the mass and its velocity are given, then we can determine the 
exact amount of work bestowed on the motion. 

Multiply the dynamic momentum by the velocity, V, and we have 

MV 2 = FVT 

V 
Here we recognize the work, -~-FT, which is that bestowed on the mass, 

M, in giving it the velocity, V, or the mass multiplied by one-half the 
square of its velocity is the work stored in it. 

Vis=viva. — The term M V 2 has formerly been called vis-viva, but that 
term is now seldom used. 

The real work in foot-pounds is %MV 2 = %FVT The space, S, in which 
the mass was set in motion is £ = % VT, which inserted in the formula 
gives the 

Work, K= %MV 2 =-- FS. 

Dynamical Formulas for Accelerated or Retarded 
Motion. 

Constant Force, in Pounds, acting on a Body free to move. 



GW = W = 2WS = WV 2 PT_ = 2PW 2K_ = K^ 

~ g ~ gT ~ gT 2 ~ 2gS ~ S ~ \ gT ~ OT 2 ~ S' 

Final Velocity in the Time, T, or Uniform Velocity of a moving Body. 

2gK 
W ' 



?-F-HrA-^-^~ I £-^-4 



Time, in Seconds, in which the Force acts on the Body free to move. 



V = wv 

G gF ' 



J 2WS _ l_2S_ _ 
"M gF ~\G 



Constant Acceleration of the Force, .F, in Feet per Second. 

r== 9^_ = ^S =z Ji == V 2 __ gPT FV 2 _ gK 2K 
W~T 2 ~T~2S~ WS ~ FT ~ WS ~ FT 2 ' 

Space, in Feet, in which the Force acts on the Body free to move. 

__ GT 2 _ VT_ _ V*_ _ gFT 2 _ PT^ _ gPT 2 _ gK _ _2T 
"" 2 ~2~20^2W~F~ WV ~ GW ~ F' 

Weight, in Pounds, of the moving Body. 

gF[ = gFT 2 2gFS _ gFT _ gPT^ gF 2 T = 2gK = gT 2 K 

G 2S ~~ V 2 " V ~ 2S 2 ~ 2P ~ V 2 ~~ 2S 2 ' 



272 Revolving Bodies. 



Mean Power in Effects during the Time, T, or in the Space, 5. 

FS _ gF°-T _ 2WS* _ WV 2 _ 2K = TK = VK = FV* 
T ~~ 2W " gT* ~ 2gT ~ T ~ 2S ~ S ~" GT ' 

Work, in Foot=pounds, concentrated in a moving Body. 

jr— ro- wy2 _ FVT _ GWVT = FGT* gF*T 2 ^2SP 

K — JTb— 2g - 2 - 2g 2 - 2W - T - Fl. 



REVOLVING BODIES. 
Centre of Gyration. 

The Centre of Gyration is a point in a revolving body in which, if 
all the revolving matter were there collected, it would obtain equal angu- 
lar velocity from and sustain equal resistance to the force that gives it the 
rotary motion. The distance of the centre of gyration from the axis of 
rotation for different shapes in practical work will be found in the dia- 
grams on pages 273 and 274. 



Formulas for Accelerated Circular Motion. 






Force, .F, in Pounds, acting on the Lever or Radius, r, to rotate 
the Body. 

Wx*n WxW = 60iT = K 

' ~ 307.49 Tr ~ 2.562 TV ~~ nrnT ~ 2nrN' 

Final Revolutions per Minute in the Time T. 



_ 120 N _ 307.49 FTr _ 60 K _ i bS72.2K 
T Wx' 2 ~ 7rrTF~^y W& * 

Total Number of Revolutions in the Time T. 

_ Tn_ _ 2.562 FT*r _ K T \K_ 

~ 120 — Wx* ~ 2irrF *~ lMbx\W 

Time of Acceleration, in Seconds, from the Start of Change of 
Motion. 



T = 



V WxW 
2.562 F 



Wx^n j Wx*N _ 60 K _ x\/WK 



307.49 Fr \ 2.562 Fr nrnF 4.09 Fr' 
Radius of Gyration, in Feet, of the revolving Body. 



V 307.49 FrT _ I 2.562 FrT* 
Wn ~ \ WN 



KT 334.9 K 



3.$N\/WNFr n\/WnTFr' 



W = 



Weight, in Pounds, of the revolving Body. 
807.49 TFr 2.562 T*Fr 5872.2 K KT 2 



x^N n?0? 2.452 a&ZV a ' 



Work, in Foot-pounds, concentrated in a revolving Body. 

H> ,/•- 2.452 W&N* irrnFT „ „«, 
A 587*2 — T> = —GO - ■ = 2nrNF ' 



Radius of Gyration. 



273 



Radius of Gyration. 



A Line or Bar. 

I 



-x- 



rT 



\s 


7 -1 


1 1 


-I 


\±-x-* 





x = 0.5773Z, 



x = 0.2887Z. 



A Circumference around its Di= 

ameter. 
A Disk around its Centre. 
A Cylinder around its Axis. 




x = 0.7072r. 



A Disk around its Diameter. 




x = 0.5r. 



A Sphere around its Diameter. 




Spherical shell, x = 0.8165r, 
Solid, x = 0.6324r. 



18 



Parallelopipedon. 



■ft 



"^Mi 



- a "^ 



^-7> 



x = \ 



12 ' 



* = ^/— 12— + <* + <*. 



Cylinder. 






■4 
-4 




X = 


Z2 + 3r2 
12 ' 




2 +3r 2 




12 * 



Cone. 




274 



Central Forces. 




x =^ro 



10 \ B 2 -}- Rr + r2 J 



3 / JJ5 



\i2 3 — r*/ 



20\i2 3 



Cylinder and Sphere. 



a; = y'a? + %r 2 , 




Wedge and Ring. 




r = internal radius of ring, i? = ex- 
ternal radius of ring. 

x = 0.204 V 12J2 + #> + & 2 , 



/2P + - 



Ply-wheel. 




^(? : TT^ = a* : 42. 
Fly-wheel with Arms. 




0/TT _ , . Trr i£* + r 2 , 4r 2 + i 
z 2 ( TF + to) = W g h w- 



12 



— I 6 W(R* + r 2 ) + io(4r2 + fr 2 ) 
*~\ 12(TT+iy) 



CENTRAL FORCES. 



Central Forces are of two kinds, centrifugal and centripetal. 

Centrifugal Force is the resistance which a revolving body offers to 
being moved in the arc of a circle. 

Centripetal Force is that by which a revolving body is attracted or 
attached to its centre of motion. 

The centrifugal and centripetal forces are opposites to each other, and 
when equal the body revolves in a circle ; but when they differ the body 
will revolve in other curved lines, as the ellipse, the parabola, etc., ac- 
cording to the nature of the difference in the forces. If the centrifugal 
force is o while the other is acting, the body will move straight to the 
centre of motion ; and if the centripetal force is o while the other is act 



Centrifugal Forces. 



275 



ing, the body will depart from the circle in a straight line, tangent to the 
circle in the point where the centripetal force ceased to act. The central 
forces are distinct from the force that has set the body in motion. 

If the centrifugal force be made use of to produce an effect, such effect 
f will be at the expense of the one producing the rotary motion. 

Notation. 
F = centrifugal force, in pounds. 
W= the weight of the revolving body, in pounds. 
v = velocity of the revolving body, in feet, per second. 
R — radius of the circle in which the body revolves, in feet. 
n = number of revolutions per minute. 



Wtfi 
gR 



Wv* 
Z2.2R' 



F = 



W= 



4WRttW = WRn* 
60*0 ~~ 2933 
FgR _ 2933-P 



= 0.00034 WRn?, 



Rn* 



Wv 2 
Fg '' 



2933F 
WW ' 



n =\ 



'2933JP 
WR ' 



■V 



FRg 
W ' 




Thus, if we have a weight of 63 pounds at a radius of 4 feet 4 inches, 
making 163 revolutions per minute, we have 

W= 63, R = 4.333, n = 163, 

and the centrifugal force, or tension produced on the radial arm, will be 

0.00034 WRn? = 0.00034 X 63 X 4.333 X 163 2 = 2466 pounds. 



Centrifugal Force of a Ring. 




; r = internal radius, R ■- 
radius. 



external 



F = 



Wn 2 (R — r) 
tt4150 * 



Centrifugal Force of a Grinding 
Stone, Thin Disk, or Cylinder 

rotating around its centre. 




WRn* 
7r4150* 



Centrifugal Force of a Cylinder 

rotating around the diameter of its 
base. 




WnH 

5867 ' 



Centrifugal Force of a Ball. 




F = 



Wn?R 
2933 * 



276 



Pendulum. 




2tt 

2933 

n 2 ' 
2933 

nH '' 



Governor. 

54.16 



54.16 




n 2 cos x 



'= \/P — h?, 



x = angle made by arm with the vertical 
axis. 

For a weighted governor of the Porter type, 
in which W = the axial weight and B = the 
weight of the ball, the height, h, will be equal 
to the height of a simple governor multiplied 



PENDULUM. 

Simple Pendulum is a material point under the action of gravitation, 
and suspended at a fixed point by a line of no weight. 

Compound Pendulum is a suspended rod and body of sensible mag- 
nitude, fixed as the simple pendulum. 

Centre of Oscillation is a point at which if all the matter in the com- 
pound pendulum were there collected, it would make a simple pendulum 
oscillate in the same periods. 

Angle of Oscillation is the space a pendulum describes when in 
motion. 

The velocity of an oscillating body through the vertical position is 
equal to the velocity a body would obtain by falling vertically the dis- 
tance versed sine of half the angle of oscillation. 

Notation. 
I = length of the simple pendulum, or the distance between the centre of 

suspension and centre of oscillation, in inches. 
t = time, in seconds, for n oscillations. 
n — number of single oscillations in the time t. 

Example. Required the length of a pendulum that will vibrate seconds ? 
Here n = 1 and t = 1". 

t 2 
I = 39.10—5- = 39.10 inches, the length of a pendulum for seconds. 

Example. Required the length of a pendulum that will make 180 vibra- 
tions per minute? Here t = 60" and n = 180. 



1 = 



39.10*2 39.10 X 60 2 



= 4.344 inches. 



1802 

Example. How many vibrations will a pendulum of 25 inches length 
make in 8 seconds ? 

6.254« 6.254 X 8 iA .. ., 

n = — t=- = 7= — = 10 vibrations. 

y l V25 

A pendulum is 137.67 inches long and makes 8 vibrations in 



Example. 
15 seconds. 



Required the acceleration of gravity, g f 

0.8225^2 0.8225 X 137.67 
9 = 



-^- =32.209. 

A compound pendulum of two iron balls, P and Q, having ^ 



Example. 
the centre of suspension between themselves, as shown in the illustrations 
on the opposite page. P = 38 pounds, Q = 12 pounds, a = 25 inches, and 
b = 18 inches. How long is the simple pendulum, and how many vibra- 
tions will the pendulum make in 10 seconds? 



Pendulum. 



277 



aP—bQ 25X38 — 18X12 



I = 



P+ Q 
a?P + WQ 252 



38 + 12 
C38+18 2 ) 



12 



14.68 inches. 
37.68 inches, 



x(P + Q) 14.68(38 + 12) 

the length of the single pendulum. 

6.254* 6.254X10 „ A ™ „ . 1n 

n = — = ==^ = 10.193 vibrations in 10 seconds. 

V 1 y 37.68 

If a compound pendulum is hung up at its centre of oscillation, the 
former centre of suspension will be the centre of oscillation and the pen- 
dulum will oscillate the same time. 

Simple Pendulum. 



1 = 



t = 



12g& _ 39.1ff 

7l 2 ' 




£-#-# 



A = centre of gravity, 
B = centre of gyra- 
tion, 
C = centre of oscilla- 
tion. 

a : b = b : I, 

b = ]/al = 1.414a, 
I = l%a. 




Compound Pendulum. 





r = radius of cylinder, 

. _ 16a 2 + 3r 2 
"" 12a ' 

= 4a r 2 
3 + 4a* 




1 = 



aPP+WQ 
aP + bQ' 



P and Q expressed in 
pounds or cubic con- 
tents. 

Connecting wire 
neglected. 



centre of suspen- 
sion, 




Connecting wire 
neglected. 



278 Impact. 

Length of Pendulum Vibrating Seconds at Sea=IeveI.* 

Latitude. Metres. Inches. 

At Equator 0° 00' 0.99092 39.012 

At Washington, D. C 38° 53' 0.99299 39.094 

At New York 40° 43' 0.99316 39.101 

At Latitude 45° 45° 00' 0.99355 39.116 

At London, Eng 51° 31' 0.99414 39.139 

At Stockholm 59° 21' 0.99481 39.166 



IMPACT. 
Impact of Moving Bodies. 

When bodies in motion come in collision with each other, the sum of 
their concentrated momentum will be the same after the collision as 
before, but their velocities and sometimes their directions of motion will 
differ. 

In the illustrations on page 279 the bodies are supposed to move in the 
same straight line, and the formula illustrates the consequences after col- 
lision. 

Notation. 

M and m = weight of the bodies, in pounds. 

V and v = their respective velocities, in feet, per second. 

V and v r = respective velocities of the bodies after impact. 

K and k = coefficient of elasticity, which for perfectly hard bodies k = 
and for perfectly elastic bodies k = 1 ; therefore the elastic 
coefficient will always be between and 1. When the 
bodies are perfectly hard their velocities after impact will 
be common. 

For if, £" = 



M{V—V) ' ' m{v — vy 

Example 1. The non-elastic body weighs if = 25 pounds, and moves at 
a velocity V = 12 feet per second ; m = 16 pounds and v = 9 feet per sec- 
ond. Required the bodies' common velocities v = ? after impact, both 
bodies moving in the same direction. 

, MV + mv 25X12 + 16X9 _, A DO . . , 

v' = „ = oc , .,„ — = 10.83 feet per second. 

M + m 25 + 16 ^ 

Example 2. The perfect elastic body if = 84 pounds, V = 18 feet per 
second, m = 48 pounds, and v = 27 feet per second. Required the velocity 
V f = ? after impact with the body m, the bodies moving in opposite direc- 
tions. 

18(84-48) -2X48X27 
V = ~ 8T+48 = ~ 23 ' 64 - 

The negative sign denotes that the body will return after the collision 
with a velocity of 23.63 feet per second. 

Example 8. The partly elastic body if = 38 pounds and V = 79 feet per 
second will strike the body in rest m = 24 pounds. What will be the 
velocity v = ? of the body m, its elasticity being k' = 0.6. 

„ = 79X38(1 + 0. 6). . 70 . 6 feet per second . 
oo + 2A. 

When a moving body strikes a stationary elastic plane its course of 
departure from the plane will be equal to its course of incidence. 

* Authority of T. C. Mendenhall, Superintendent United States Coast and 
Geodetic Survey, January, 1894. 



Impact. 



279 



The Bodies Perfectly Hard. 



The bodies move in the same direction. 

v'{M + m) = M V + mv, 

MV + mv 

v = . 

M -\-m 




The bodies move in opposite directions. 
v f (M + m) = MV — mv, 
t _ MV — mv 
M + m 







Only one body in motion. 







The Bodies Elastic. 



The bodies move in the same direction. 



V = 



V(M— Km) + vm(l + K) 

M -\- m 
MV(1 + k) ± v(m + kM) 
M -\- m 




A-> 



The bodies move in opposite directions. 
F(if — Zm) — vm(l + J6T) 



V' = 



v' 



M -\- m 
if F(l + k) — v(m — kM) 
M + m 




^(S 



Only one body in motion. 
F(3f— Km) 



V? = 



if -f m 

V3f(l + fe) 

if + w 




280 Friction. 



FRICTION. 

The resistance to motion which is experienced when one body is moved 
upon another is expressed by the general term Friction. Theoretically, it 
is assumed to be due to the interlocking of the roughness and inequalities 
which exist to a greater or less degree in the surfaces of all solids. The 
term friction, however, is applied to the resistance encountered by air or 
gases flowing through pipes or flues, or by water in pipes and channels, 
and in all cases of motion it is an element to be considered. 

The first modern study of the subject of friction was that made in 
France by General Morin about 1831, and for a long time the laws enunci- 
ated by him as the result of his experiments, and the coefficients of friction 
given by him, were generally accepted and extensively reprinted. It is 
now generally understood, however, that these laws and results were true 
only for the conditions under which they were made, and that modern 
operative conditions require them to be modified. At the same time, the 
general results of Morin 's experiments may here be referred to as forming 
a basis for the more recent data. 

Morin' s experiments were made by measuring the force required to 
cause one body to slide upon another. The ratio between this force and 
the pressure upon the sliding body is called the coefficient of friction, so 
that the coefficient of friction is the proportion which the resistance of 
friction bears to the pressure upon the sliding body. The pressure upon 
the sliding body is always taken as acting normal to the sliding surfaces. 
As a result of his experiments, Morin announced : 

1. Friction-is directly proportional to the pressure ; the coefficient being 
thus constant at all pressures. 

2. Friction, both in amount and coefficient, is independent of the areas 
in contact ; the pressure remaining the same. 

3. The coefficient of friction is independent of the velocity of the rub- 
bing surfaces. This is understood to refer to the friction of motion, since 
it takes a greater force to overcome the friction of rest than to maintain 
the surfaces in motion thereafter. 

The second law is a natural consequence of the first, since any increase 
in area for the same total pressure reduces the pressure per unit of area in 
the same proportion. If the area be doubled, the pressure per square inch 
will be halved, but there will be twice as many square inches, and the 
frictional resistance will be unchanged. 

The principal modifications which have to be made in these laws, in 
the light of modern practice, are in the expansion of an expression made 
by Morin himself in connection with the experiments,— namely, that the 
condition of the surfaces must be taken into consideration. It is now possi- 
ble to produce surfaces, both plane and cylindrical, so far superior to those 
with which Morin experimented that the coefficients deduced by him, and 
tabulated in many reference books since, are now considered far too great 
in nearly every case. The improvement in lubricants and the influence 
of temperature also enter as factors, and the number of variables thus 
introduced make it impossible to do more than furnish general data for 
preliminary use; and in all undertakings of importance experimental 
determinations should be made with the given materials, as nearly under 
the actual conditions as possible. 

For plane-sliding friction, in which the speed of the movement is mod- 
erate and the pressures not excessive, Morin's laws and coefficients are 
fairly correct, although the latter are somewhat higher than are found 
with highly-polished surfaces, well lubricated. 

Morin's coefficients of friction, given by him in detail under numerous 
varying conditions, may be taken in general as follows : 

Material. Coefficient. 

Wood on wood, dry 0.25 

Metal on metal, drv 0.15 

well lubricated 0.07 to 0.0S 

Any attempt to use closer refinements when the exact conditions are 
not known is both useless and deceptive. 



Friction. 



281 



Journal Friction. 

Recent experiments have shown that Morin's laws do not hold for 
revolving journals at high speeds and under heavy pressures. 

The experiments of Mr. Beauchamp Tower* showed that the coefficient 
of friction, /, increased as the square root of the linear velocity, and 
diminished directly with the increase in pressure. 

If v = the linear velocity in feet per second, and p = the pressure in 

v v 
pounds per square inch, the coefficient/ = c- — , in which c is a constant, 

dependent upon the lubricant. 

The following values of c may be used with pressures of 100 to 600 
pounds per square inch : 



Lubricant. c 

Olive oil 0.289 

Lard oil 0.281 

Mineral grease 0.431 



Lubricant. c 

Sperm oil 0.194 

Rape oil 0.212 

Mineral oil 0.276 



With olive oil lubrication, at a velocity of 3% feet per second, and 
pressures ranging from 520 pounds down to 100 pounds per square inch, 
the coefficient of friction, /, varied from 0.001 to 0.0055. 

In Mr. Tower's experiments the pressure at which the journal seized 
varied from 520 to 625 pounds per square inch of projected area, — that is, 
the length multiplied by the diameter. 

On collar bearings, such as are used for the thrust bearings of screw- 
propeller shafts, the coefficient of friction is found to be independent of 
the speed, and for pressures between 45 and 75 pounds per square inch it 
ranges between 0.040 and 0.035. Good practice allows a pressure of 50 
pounds per square inch, at which a coefficient of 0.036 may be used. 

In computing the resistance of friction a clear distinction must be made 
between the frictional resistance itself and the work of friction. The work 
is measured by the product of the frictional resistance and the lineal speed 
of the rubbing surfaces, this giving the power absorbed in foot-pounds per 
minute. 

The question of temperature is often an important element in frictional 
resistance, the work of friction appearing as heat, which, if not carried 
away, produces a rapid rise in temperature. The increased temperature 
reduces the viscosity of the lubricant, which is then forced out by the 
pressure, and the bearing runs dry and seizes. 

Pillow-blocks and similar bearings should contain sufficient mass of 
metal to permit the heat to be conducted away freely, and attempts to 
economize in metal by coring out to the limit of mere strength may reduce 
the thermal conductivity of a bearing to such an extent as to render it 
liable to heat. 

The thrust bearings of marine engines, and other bearings in which it 
is of great importance that motion should not be interrupted, are made 
with passages for the circulation of cooling water, which is to be turned 
on promptly in case of an abnormal increase in temperature. 

The maximum pressures which are permitted depend upon the nature 
of the motion. When the pressure is exerted continuously in the same 
direction, as in the case of heavy shafting, etc., there is not the same 
opportunity for the lubricant to flow in, as in the case of alternating or 
intermittent pressure. In the first case the pressure should not exceed 450 
pounds per square inch at a maximum, and should be kept down to 250 or 
300 pounds, when practicable. For the second case, such as crank pins, in 
which the pressure acts in alternate directions, pressures from 500 to 900 
pounds per square inch are used in stationary practice and 1200 to 1800 
pounds per square inch in locomotive engines. 



* Proceedings of the Institution of Mechanical Engineers, 1883. 



282 Materials of Engineering. 



MATERIALS OF ENGINEERING. 

Martens divides the materials of engineering into two main classes : 

1. Materials of Construction. Being those which constitute the com- 
pleted structures. To this class belong the metals, woods, stone, cement, 
etc. 

2. Materials of Consumption. Being those which are consumed or 
transformed while being used. These include such substances as coal, 
water, oil, etc. 

While these distinctions are not rigid or absolute, they may serve as a 
convenient classification. 

Materials of Construction may be considered according to their 
physical or their chemical properties, or both. 

. For engineering purposes the chemical properties are not so generally 
considered as are the physical properties, although in some respects the 
ultimate composition, as well as the manner of combination, must be 
taken into account. Materials must generally be defined according to 
their chemical nomenclature, after which their physical properties de- 
mand the most attention. No attempt will be made here to discuss any 
but the materials in general use, the rarer elements and their combina- 
tions forming properly the subjects for treatises on chemistry and physics. 

Apart from their chemical composition, the principal properties of 
importance to the engineer are : 

Density, represented by specific gravity. 
Resistance, or capacity to oppose stresses. 
Hardness, or opposition to penetration. 
Toughness, or capacity for elongation under tension. 
Brittleness, the opposite of toughness. 

Besides these there are many other physical properties, such as behavior 
during heating or cooling, fusing, or working in innumerable ways, but 
these must be considered in connection with the operations in which they 
appear. 

The Specific Gravity, or relative density of substances, is the ratio of 
the weight of a given volume of the substance to the same volume of 
water. For gases, the unit of comparison is an equal volume of air. The 
water unit in specific gravity determinations is assumed to be pure and at 
its temperature of greatest density. 

Since, in the metric system, the units of weight are derived from the 
units of volume in terms of the weight of water, the specific gravity of 
any substance is also its weight in metric units. Thus, if the specific 
gravity of a certain iron is 7, a cubic centimetre of it will weigh 7 
grammes, or a cubic decimetre will weigh 7 kilogrammes. In English 
units there is no such integral relation between the units of weight and 
volume of water, and hence the weight of a cubic inch or cubic foot of 
any substance must be given in pounds in addition to the specific gravity, 
or it can be computed from the latter by multiplying it by the weight of 
the given volume of water. 

Since a submerged body is buoyed up by a force equal to the weight of 
an equal volume of water, the specific gravity of any solid substance may 
be found by the following methods : 

To Find the Specific Gravity. 

W= weight of a body in the air. 

w = weight of the body (heavier than water) immersed in water. 

S = specific gravity of the body. Then 

1. TF— w: W=1:S. S= nr W . 

W—w 



Specific Gravity. 283 



Required the specific gravity of a piece of iron ore weighing 6.345 
pounds in the air and 4.935 pounds in water, £ = ? 



4.5, the specific gravity. 



6.345 — 4.935 

When the body is lighter than water, attach to it a heavier body that is 
able to sink the lighter one. 

5 = specific gravity of the heavier attached body. 
s = specific gravity of the lighter body. 
W= weight of the two bodies in air. 
w = weight of the two bodies in water. 
V= weight of the heavier body in air. 
v = weight of the lighter body in air. 



o 

To a piece of wood, which weighs v = 14 pounds in the air, is fastened 
a piece of cast-iron, V = 28 pounds ; the two bodies together weigh w = 11.7 
pounds in water. Required the specific gravity of the wood? 

W= F+v = 28 + 14 = 42 pounds. 
S = 7.2, the specific gravity of cast-iron. 

14 

Formula 2. S = — = 0.529, the specific gravity of the 

40 ii 7 _ &_ wood ( Spanish white poplar). 

7.2 

A simple way to obtain the specific gravity of wood is to make it into a 
rod and place it vertically in water; then, when in equilibrium, the im- 
mersed end is to the whole rod as the specific gravity is to 1. 

A cylinder of wood is 6 feet 3 inches long. When immersed vertically 
in water it will sink 3 feet 9 inches by its own weight. Required its spe- 
cific gravity ? 

3.75 : 6.25 = S : 1. £ = ~£- = 0.600. 
6.25 



To Find the Percentage of Alloy in Metals, or to Find the Propor- 
tions of Two Ingredients in a Compound. 

3 W-s(W-w) 

1— i 
S 

A metal compounded of silver and gold weighs W = 6 pounds in the 
air, and in water w = 5.636 pounds. Required the proportions of silver and 
gold? 

S = 19.36, the specific gravity of gold. 
s = 10.51, the specific gravity of silver. 

«. . u . „ 6 — 10.51(6 — 5.636) . „„ , . .. , 

Weight V= TnT\ = 4 * 755 pounds of gold and 

1 10 - 51 1.245 pounds of silver. 

19.36 



284 



Specific Gravity. 



NameB of substances. 



Metals. 

Platinum, rolled 

" wire 

11 hammered. 

1 ' purified 

" crude, grs. . 

Gold, hammered 

" pure cast 

" 22 carats' fine.. . 

11 20 " " ... 

Mercury, solid at — 40° 

at +32° F. . . 

" 60° F. .. 

11 212° F. .. 

Lead, pure 

' ' hammered 

Silver, hammered 

11 pure 

Bismuth 

Red lead 

Cinnabar 

Manganese 

Copper, wire & rolled . 

" pure 

Bronze, gun metal 

Brass, common 

Steel, cast-steel 

•' common soft .. . 
" hard'ed & temp. 

Iron, pure 

" wrought & rol'd. 

1 ' hammered 

" cast-iron 

Tin, from Bohmen. . . 

" English 

Zinc, rolled 

" cast 

Antimony 

Aluminium .' . 

Arsenic 

Stones and Earths 

Topaz, Oriental 

Emery 

Diamond 

Limestone, green 

11 white — 

Asbestos, starry 

Glass, flint 

11 white 

11 bottle 

" green 

Marble, Parian 

11 African 

" Egyptian 

Mica 

Hone, white razor. . . 

Chalk 

Porphyry 

Spar, green 

" blue 

Alabaster, white 





4- .2 


ja fit • 


§ > 


.a>g-s 


CO bO 






Lb. 


22.669 


.798 


21.042 


.761 


20.337 


.736 


19.500 


.706 


15.602 


.565 


19.361 


.700 


19.258 


.697 


17.486 


.733 


15.702 


.568 


15.632 


.566 


13.619 


.493 


13.580 


.491 


13.375 


.484 


11.330 


.410 


11.388 


.412 


10.511 


.381 


10.474 


.379 


9.823 


.355 


8.940 


.324 


8.098 


.293 


8.030 


.290 


8.878 


.321 


8.788 


.318 


8.700 


.315 


7.820 


.282 


7.919 


.286 


7.833 


.283 


7.818 


.283 


7.768 


.281 


7.780 


.282 


7.789 


.282 


7.207 


.261 


7.312 


.265 


7.291 


.264 


7.191 


.260 


6.861 


.248 


6.712 


.244 


2.500 


.090 


5.763 


.208 


4.011 


.145 


4.000 


.144 


3.521 


.127 


3.180 


.115 


3.156 


.114 


3.073 


.111 


2.933 


.106 


2.892 


.104 


2.732 


.0987 


2.642 


.0954 


2.838 


.1030 


2.708 


.0978 


2.668 


.0964 


2.800 


.1000 


2.838 


.1040 


2.784 


.1000 


2.765 


.0999 


2.704 


.0976 


2.693 


.0971 


2.730 


.0987 



Names of substances. 



Stones.— Continued. 

Alabaster, yellow 

Coral, red 

Granite, Susquehanna 

" Quincy 

" Patapsco 

11 Scotch 

Marble, white Italian 

" common 

Talc, black 

Quartz 

Slate 

Pearl, Oriental 

Shale 

Flint, white 

11 black 

Stone, common 

" Bristol 

mill 

" paving 

Gypsum, opaque 

Grindstone 

Salt, common 

Saltpetre 

Sulphur, native 

Common soil 

Rotten stone 

Clay 

Brick 

Nitre 



Plaster of Paris \ 

Ivory 

Sand 

Phosphorus 

Borax 

Coal, anthracite j 

11 Maryland 

" Scotch 

" New Castle 

11 bituminous 

Charcoal, triturated . . 

Earth, loose 

Amber 

Pimstone 

Lime, quick 

Charcoal 



Woods (Dry). 

Alder 

Apple-tree 

Ash, the trunk 

Bay-tree 

Beech 

Box, French 

H Dutch 

" Brazilian red. 
Cedar, wild 

" Palestine... 

11 Indian 

11 American... 



2.6 
2.700 
2.704 
2.652 
2.640 
2.625 
2.708 
2.686 
2.900 
2.660 
2.672 
2.650 
2.600 
2.594 
2.582 
2.520 
2.510 
2.484 
2.416 
2.168 
2.143 
2.130 
2.090 
2.033 
1.984 
1.981 
1.930 
1.900 
1.900 
1.872 
2.473 
1.822 
1.800 
1.770 
1.714 
1.640 
1.436 
1.355 
1.300 
1.270 
1.270 
1.31 
1.500 
1.078 
1.647 
.804 
.441 



.800 

.793 

.845 

.822 

.852 

.912 

1.328 

1.031 

.596 

.613 

1.315 

.561 



Specific Gravity. 



285 



Names of substances. 



Woods.— Continued. 

Citron 

Cocoa- wood 

Cherry-tree 

Cork 

Cypress, Spanish 

Ebony, American 

11 Indian 

Elder-tree 

Elm, trunk of 

Filbert-tree 

Fir, male 

M female 

Hazel 

Jasmine, Spanish 

Juniper-tree 

Lemon-tree 

Lignum- vitae 

Linden-tree 

Log- wood 

Mastic-tree 

Mahogany 

Maple 

Medlar 

Mulberry 

Oak, heart of, 60 old. 

Orange-tree 

Pear-tree 

Pomegranate-tree 

Poplar 

" white Spanish 

Plum-tree 

Quince-tree 

Sassafras 

Spruce 

" old 

Pine, yellow 

" white 

Vine 

Walnut 

Yew, Dutch 

" Spanish 

Liquids. 

Acid, acetic 

" nitric 

" sulphuric 

11 muriatic 

" fluoric. 

1 ' phosphoric 

Alcohol, commercial 

" pure 

Ammoniac, liquid . . . 

Beer, lager 

Champagne 

Cider 

Egg 

Ether, sulphuric 

Honey 

Human blood 

Milk 

Oil, linseed 



O ^ 


*»& 




a& • 


§"► 


.SPg-S 


0° So 






Lb. 


.726 


.0263 


1.040 


.0376 


.715 


.0259 


.240 


.0087 


.644 


.0233 


1.331 


.0481 


1.209 


.0437 


.695 


.0252 


.671 


.0243 


.600 


.0217 


.550 


.0199 


.498 


.0180 


.600 


.0217 


.770 


.0279 


.556 


.0201 


.703 


.0254 


1.333 


.0482 


.604 


.0219 


.913 


.0331 


.849 


.0307 


1.063 


.0385 


.750 


.0271 


.944 


.0342 


.897 


.0324 


1.170 


.0423 


.705 


.T)255 


.661 


.0239 


1.354 


.0490 


.383 


.0138 


.529 


.0191 


.785 


.0284 | 


.705 


.0255 


.482 


.0174 


.500 


.0181 


.460 


.0166 


.660 


.0239 


.554 


.0200 


1.327 


.0480 


.671 


.0243 


.788 


.0285 


.807 


.0292 


1.062 


.0384 


1.217 


.0440 


1.841 


.0666 


1.200 


.0434 


1.500 


.0542 


1.558 


.0563 


.833 


.0301 


.792 


.0287 


.897 


.0324 


1.034 


.0374 


9.970 


.0360 


1.018 


.0361 


1.090 


.0394 


.739 


.0267 


1.450 


.0524 


1.054 


.0381 


1.032 


.0373 


.940 


.0340 



Names of substances. 



Liquids. — Continued, 

Oil, olive 

" turpentine 

" whale 

Proof spirit 

Vinegar 

Water, distilled 

" sea 

" Dead Sea 

Wine 

" port 

Miscellaneous. 

Asphaltum -j 

Atmospheric air 

Beeswax 

Butter 

Camphor 

India rubber 

Fat of beef 

" hogs 

1 ' mutton 

Gamboge 

Gunpowder, loose — 
" shaken.. 

11 solid... | 

Gum Arabic 

Indigo 

Lard 

Mastic — 

Spermaceti 

Sugar 

Tallow, sheep 

calf 

11 ox 



Oases, Vapors. 

Acetylene 

Ammonia gas 

Atmospheric air 

Carbonic acid 

Carbonic oxid 

Carburetted hydrog.. 

Chlorine 

Ethylene 

Hydrogen 

Methane 

Nitrogen 

Oxygen 

Smoke of bitum. coal 

" " wood 

Steam at 212° 

Sulphuretted hydrog. 

Sulphurous acid 

Vapor of alcohol 

" " turp. spir. .. 

" " water 






.915 

.870 

.932 

.925 

1.080 

1.000 

1.030 

1.240 

.992 

.997 



.905 
1.650 
.0012 
.965 
.942 



.923 

.936 

.923 

1.222 

.900 

1.000 

1.550 

1.800 

1.452 

1.009 

.947 

1.074 

.943 

1.605 

.924 

.934 

.923 



.910 

.590 

1.000 

1.527 

.972 

.972 

2.500 

.984 

.069 

.566 

.972 

1.104 

.102 

.900 

.488 

1.777 

2.222 

1.613 

5.013 

.623 



286 



Weight of Iron. 



Weight of Flat Rolled Iron per Lineal Foot. 

For Thicknesses from T ^ inch to 2 inches, and Widths from 1 inch to 3% 

inches. 







Iron weighing 480 pounds per cubic foot. 








Width, in inches. 


1 


« 


IX 


1% 


2 


2K 


2V 2 


2% 


3 


3% 


3^ 


3% 




Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


a 


.208 


.260 


.313 


.365 


.417 


.469 


.521 


.573 


.625 


.677 


.729 


.781 


Vs 


.417 


.521 


.625 


.729 


.833 


.938 


1.04 


1.15 


1.25 


1.35 


1.46 


1.56 


a 


.625 


.781 


.938 


1.09 


1.25 


1.41 


1.56 


1.72 


1.88 


2.03 


2.19 


2.34 


m 


.833 


1.04 


1.25 


1.46 


1.67 


1.88 


2.08 


2.29 


2.50 


2.71 


2.92 


3.13 


A 


1.04 


1.30 


1.56 


1.82 


2.08 


2.34 


2.60 


2.86 


3.13 


3.39 


3.65 


3.91 


% 


1.25 


1.56 


1.88 


2.19 


2.50 


2.81 


3.13 


3.44 


3.75 


4.06 


4.38 


4.69 


& 


1.46 


1.82 


2.19 


2.55 


2.92 


3.28 


3.65 


4.01 


4.38 


4.74 


5.10 


5.47 


X 


1.67 


2.08 


2.50 


2.92 


3.33 


3.75 


4.17 


4.58 


5.00 


5.42 


5.83 


6.25 


A 


1.88 


2.34 


2.81 


3.28 


3.75 


4.22 


4.69 


5.16 


5.63 


6.09 


6.56 


7.03 


K 


2.08 


2.60 


3.13 


3.65 


4.17 


4.69 


5.21 


5.73 


6.25 


6.77 


7.29 


7.81 


tt 


2.29 


2.86 


3.44 


4.01 


4.58 


5.16 


5.73 


6.30 


6.88 


7.45 


8.02 


8.59 


% 


2.50 


3.13 


3.75 


4.38 


5.00 


5.63 


6.25 


6.88 


7.50 


8.13 


8.75 


9.38 


If 


2.71 


3.39 


4.06 


4.74 


5.42 


6.09 


6.77 


7.45 


8.13 


8.80 


9.48 


10.16 


% 


2.92 


3.65 


4.38 


5.10 


5.83 


6.56 


7.29 


8.02 


8.75 


9.48 


10.21 


10.94 


H 


3.13 


3.91 


4.69 


5.47 


6.25 


7.03 


7.81 


8.59 


9.38 


10.16 


10.94 


11.72 


l 


3.33 


4.17 


5.00 


5.83 


6.67 


7.50 


8.33 


9.17 


10.00 


10.83 


11.67 


12.50 


A 


3.54 


4.43 


5.31 


6.20 


7.08 


7.97 


8.85 


9.74 


10.63 


11.51 


12.40 


13.28 


3^ 


3.75 


4.69 


5.63 


6.56 


7.50 


8.44 


9.38 


10.31 


11.25 


12.19 


13.13 


14.06 


A 


3.96 


4.95 


5.94 


6.93 


7.92 


8.91 


9.90 


10.89 


11.88 


12.86 


13.85 


14.84 


K 


4.17 


5.21 


6.25 


7.29 


8.33 


9.38 


10.42 


11.46 


12.50 


13.54 


14.58 


15.63 


T°S 


4.37 


5.47 


6.56 


7.66 


8.75 


9.84 


10.94 


12.03 


13.13 


14.22 


15.31 


16.41 


% 


4.58 


5.73 


6.88 


8.02 


9.17 


10.31 


11.46 


12.60 


13.75 


14.90 


16.04 


17.19 


A 


4.79 


5.99 


7.19 


8.39 


9.58 


10.78 


11.98 


13.18 


14.38 


15.57 


16.77 


17.97 


% 


5.00 


6.25 


7.50 


8.75 


10.00 


11.25 


12.50 


13.75 


15.00 


16.25 


17.50 


18.75 


A 


5.21 


6.51 


7.81 


9.11 


10.42 


11.72 


13.02 


14.32 


15.63 


16.93 


18.23 


19.53 


% 


5.42 


6.77 


8.13 


9.48 


10.83 


12.19 


13.54 


14.90 


16.25 


17.60 


18.96 


20.31 


H 


5.63 


7.03 


8.44 


9.84 


11.25 


12.66 


14.06 


15.47 


16.88 


18.28 


19.69 


21.09 


% 


5.83 


7.29 


8.75 


10.21 


11.67 


13.13 


14.58 


16.04 


17.50 


18.96 


20.42 


21.88 


H 


6.04 


7.55 


9.06 


10.57 


12.08 


13.59 


15.10 


16.61 


18.13 


19.64 


21.15 


22.66 


% 


6.25 


7.81 


9.38 


10.94 


12.50 


14.06 


15.63 


17.19 


18.75 


20.31 


21.88 


23.44 


M 


6.46 


8.07 


9.69 


11.30 


12.92 


14.53 


16.15 


17.76 


19.38 


20.99 


22.60 


24.22 


2 


6.67 


8.33 


10.00 


11.67 


13.33 


15.00 


16.67 


18.33 


20.00 


21.67 


23.33 


25.00 



For Steel add 2 per cent. 



Weight of Iron. 



287 



Weight of Flat Rolled Iron per Lineal Foot. 

For Thicknesses from ^ inch to 2 inches, and Widths from 4 inches to 

6% inches. 







Iron weighing 480 


pounds per cubic foot. 






00 . 

DD DO 


Width, in inches. 


la 


4 


4^ 


43^ 


4% 


5 


534 


5% 


5% 


6 


6% 


6V 2 


6% 




Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


A 


.833 


.885 


.938 


.990 


1.04 


1.09 


1.15 


1.20 


1.25 


1.30 


1.35 


1.41 


y 8 


1.67 


1.77 


1.88 


1.98 


2.08 


2.19 


2.29 


2.40 


2.50 


2.60 


2.71 


2.81 


& 


2.50 


2.66 


2.81 


2.97 


3.13 


3.28 


3.44 


3.59 


3.75 


3.91 


4.06 


4.22 


% 


3.33 


3.54 


3.75 


3.96 


4.17 


4.38 


4.58 


4.79 


5.00 


5.21 


5.42 


5.63 


& 


4.17 


4.43 


4.69 


4.95 


5.21 


5.47 


5.73 


5.99 


6.25 


6.51 


6.77 


7.03 


% 


5.00 


5.31 


5.63 


5.94 


6.25 


6.56 


6.88 


7.19 


7.50 


7.81 


8.13 


8.44 


& 


5.83 


6.20 


6.56 


6.93 


7.29 


7.66 


8.02 


8.39 


8.75 


9.11 


9.48 


9.84 


% 


6.67 


7.08 


7.50 


7.92 


8.33 


8.75 


9.17 


9.58 


10.00 


10.42 


10.83 


11.25 


& 


7.50 


7.97 


8.44 


8.91 


9.38 


9.84 


10.31 


10.78 


11.25 


11.72 


12.19 


12.66 


% 


8.33 


8.85 


9.38 


9.90 


10.42 


10.94 


11.46 


11.98 


12.50 


13.02 


13.54 


14.06 


n 


9.17 


9.74 


10.31 


10.89 


11.46 


12.03 


12.60 


13.18 


13.75 


14.32 


14.90 


15.47 


% 


10.00 


10.63 


11.25 


11.88 


12.50 


13.13 


13.75 


14.38 


15.00 


15.63 


16.25 


16.88 


it 


10.83 


11.51 


12.19 


12.86 


13.54 


14.22 


14.90 


15.57 


16.25 


16.93 


17.60 


18.28 


Vs 


11.67 


12.40 


13.13 


13.85 


14.58 


15.31 


16.04 


16.77 


17.50 


18.23 


18.96 


19.69 


h 


12.50 


13.28 


14.06 


14.84 


15.63 


16.41 


17.19 


17.97 


18.75 


19.53 


20.31 


21.09 


i 


13.33 


14.17 


15.00 


15.83 


16.67 


17.50 


18.33 


19.17 


20.00 


20.83 


21.67 


22.50 


& 


14.17 


15.05 


15.94 


16.82 


17.71 


18.59 


19.48 


20.36 


21.25 


22.14 


23.02 


23.91 


Vs 


15.00 


15.94 


16.88 


17.81 


18.75 


19.69 


20.63 


21.56 


22.50 


23.44 


24.38 


25.31 


A 


15.83 


16.82 


17.81 


18.80 


19.79 


20.78 


21.77 


22.76 


23.75 


24.74 


25.73 


26.72 


M 


16.67 


17.71 


18.75 


19.79 


20.83 


21.88 


22.92 


23.96 


25.00 


26.04 


27.08 


28.13 


A 


17.50 


18.59 


19.69 


20.78 


21.88 


22.97 


24.06 


25.16 


26.25 


27.34 


28.44 


29.53 


% 


18.33 


19.48 


20.63 


21.77 


22.92 


24,06 


25.21 


26.35 


27.50 


28.65 


29.79 


30.94 


& 


19.17 


20.36 


21.56 


22.76 


23.96 


25.16 


26.35 


27.55 


28.75 


29.95 


31.15 


32.34 


% 


20.00 


21.25 


22.50 


23.75 


25.00 


26.25 


27.50 


28.75 


30.00 


31.25 


32.50 


33.75 


& 


20.83 


22.14 


23.44 


24.74 


26.04 


27.34 


28.65 


29.95 


31.25 


32.55 


33.85 


35.16 


% 


21.67 


23.02 


24.38 


25.73 


27.08 


28.44 


29.79 


31.15 


32.50 


33.85 


35.21 


36.56 


ii 


22.50 


23.91 


25.31 


26.72 


28.13 


29.53 


30.94 


32.34 


33.75 


35.16 


36.56 


37.97 


% 


23.33 


24.79 


26.25 


27.71 


29.17 


30.63 


32.08 


33.54 


35.00 


36.46 


37.92 


39.38 


H 


24.17 


25.68 


27.19 


28.70 


30.21 


31.72 


33.23 


34.74 


36.25 


37.76 


39.27 


40.78 


Vs 


25.00 


26.56 


28.13 


29.69 


31.25 


32.81 


34.38 


35.94 


37.50 


39.06 


40.63 


42.19 


H 


25.83 


27.45 


29.06 


30.68 


32.29 


33.91 


35.52 


37.14 


38.75 


40.36 


41.98 


43.59 


2 


26.67 


28.33 


30.00 


31.67 


33.33 


35.00 


36.67 


38.33 


40.00 


41.67 


43.33 


45.00 



For Steel add 2 per cent. 



288 



Weight of Iron. 



Weight of Flat Rolled Iron per Lineal Foot. 

For Thicknesses from ^ inch to 2 inches, and Widths from 7 inches to 

9% inches. 

Iron weighing 480 pounds per cubic foot. * ; 



oB aJ 
g,d 


Width, in inches. 


is 


7 


7% 


7% 


7% 


8 


8M 


sy 2 


8% 


9 


*v± 


<>V* 


9% 




Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


& 


1.46 


1.51 


1.56 


1.61 


1.67 


1.72 


1.77 


1.82 


1.88 


1.93 


1.98 


2.03 


Vs 


2.92 


3.02 


3.13 


3.23 


3.33 


3.44 


3.54 


3.65 


3.75 


3.85 


3.96 


4.06 


A 


4.38 


4.53 


4.69 


4.84 


5.00 


5.16 


5.31 


5.47 


5.63 


5.78 


5.94 


6.09 


v± 


5.83 


6.04 


6.25 


6.46 


6.67 


6.88 


7.08 


7.29 


7.50 


7.71 


7.92 


8.13 


A 


7.29 


7.55 


7.81 


8.07 


8,33 


8.59 


8.85 


9.11 


9.38 


9.64 


9.90 


10.16 


Vs 


8.75 


9.06 


9.38 


9.69 


10.00 


10.31 


10.63 


10.94 


11.25 


11.56 


11.88 


12.19 


A 


10.21 


10.57 


10.94 


11.30 


11.67 


12.03 


12.40 


12.76 


13.13 


13.49 


13.85 


14.22 


% 


11.67 


12.08 


12.50 


12.92 


13.33 


13.75 


14.17 


14.58 


15.00 


15.42 


15.83 


16.25 


A 


13.13 


13.59 


14.06 


14.53 


15.00 


15.47 


15.94 


16.41 


16.88 


17.34 


17.81 


18.28 


% 


14.58 


15.10 


15.63 


16.15 


16.67 


17.19 


17.71 


18.23 


18.75 


19.27 


19.79 


20.31 


H 


16.04 


16.61 


17.19 


17.76 


18.33 


18.91 


19.48 


20.05 


20.63 


21.20 


21.77 


22.34 


% 


17.50 


18.13 


18.75 


19.38 


20.00 


20.63 


21.25 


21.88 


22.50 


23.13 


23.75 


24.38 


*i 


18.96 


19.64 


20.31 


20.99 


21.67 


22.34 


23.02 


23.70 


24.38 


25.05 


25.73 


26.41 


% 


20.42 


21.15 


21.88 


22.60 


23.33 


24.06 


24.79 


25.52 


26.25 


26.98 


27.71 


28.44 


if 


21.88 


22.66 


23.44 


24.22 


25.00 


25.78 


26.56 


27.34 


28.13 


28.91 


29.69 


30.47 


l 


23.33 


24.17 


25.00 


25.83 


26.67 


27.50 


28.33 


29.17 


30.00 


30.83 


31.67 


32.50 


A 


24.79 


25.68 


26.56 


27.45 


28.33 


29.22 


30.10 


30.99 


31.88 


32.76 


33.65 


34.53 


% 


26.25 


27.19 


28.13 


29.06 


30.00 


30.94 


31.88 


32.81 


33.75 


34.69 


35.63 


36.56 


A 


27.71 


28.70 


29.69 


30.68 


31.67 


32.66 


33.65 


34.64 


35.63 


36.61 


37.60 


38.59 


% 


29.17 


30.21 


31.25 


32.29 


33.33 


34.38 


35.42 


36.46 


37.50 


38.54 


39.58 


40.63 


ft 


30.62 


31.72 


32.81 


33.91 


35.00 


36.09 


37.19 


38.28 


39.38 


40.47 


41.56 


42.66 


% 


32.08 


33.23 


34.38 


35.52 


36.67 


37.81 


38.96 


40.10 


41.25 


42.40 


43.54 


44.69 


A 


33.54 


34.74 


35.94 


37.14 


38.33 


39.53 


40.73 


41.93 


43.13 


44.32 


45.52 


46.72 


% 


35.00 


36.25 


37.50 


38.75 


40.00 


41.25 


42.50 


43.75 


45.00 


46.25 


47.50 


48.75 


A 


36.46 


37.76 


39.06 


40.36 


41.67 


42.97 


44.27 


45.57 


46.88 


48.18 


49.48 


50.78 


% 


37.92 


39.27 


40.63 


41.98 


43.33 


44.69 


46.04 


47.40 


48.75 


50.10 


51.46 


52.81 


« 


39.38 


40.78 


42.19 


43.59 


45.00 


46.41 


47.81 


49.22 


50.63 


52.03 


53.44 


54.84 


% 


40.83 


42.29 


43.75 


45.21 


46.67 


48.13 


49.58 


51.04 


52.50 


53.96 


55.42 


56.88 


H 


42.29 


43.80 


45.31 


46.82 


48.33 


49.84 


51.35 


52.86 


54.38 


55.89 


57.40 


58.91 


% 


43.75 


45.31 


46.88 


48.44 


50.00 


51.56 


53.13 


54.69 


56.25 


57.81 


59.38 


60.94 


H 


45.21 


46.82 


48.44 


50.05 


51.67 


53.28 


54.90 


56.51 


58.13 


59.74 


61.35 


62.97 


2 


46.67 


48.33 


50.00 


51.67 


53.33 


55.00 


56.67 


58.33 


60.00 


61.67 


63.33 


65.00 



For Steel add 2 per cent. 



Weight of Iron. 



289 



Weight of Flat Rolled Iron per Lineal Foot. 

For Thicknesses from ^ inch to 2 inches, and Widths from 10 inches to 
12% inches. 







Iron weighing 480 pounds pei 


cubic foot 










Width, in inches. 


la 


10 


10% 


10% 


10% 


11 


"M 


11% 


11% 


12 


12^ 


12% 


12% 




Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


& 


2.08 


2.14 


2.19 


2.24 


2.29 


2.34 


2.40 


2.45 


2.50 


2.55 


2.60 


2.66 


Vs 


4.17 


4.27 


4.38 


4.48 


4.58 


4.69 


4.79 


4.90 


5.00 


5.10 


5.21 


5.31 


& 


6.25 


6.41 


6.56 


6.72 


6.88 


7.03 


7.19 


7.34 


7.50 


7.66 


7.81 


7.97 


M 


8.33 


8.54 


8.75 


8.96 


9.17 


9.38 


9.58 


9.79 


10.00 


10.21 


10.42 


10.63 


T 5 * 


10.42 


10.68 


10.94 


11.20 


11.46 


11.72 


11.98 


12.24 


12.50 


12.76 


13.02 


13.28 


% 


12.50 


12.81 


13.13 


13.44 


13.75 


14.06 


14.38 


14.69 


15.00 


15.31 


15.63 


15.94 


A 


14.58 


14.95 


15.31 


15.68 


16.04 


16.41 


16.77 


17.14 


17.50 


17.86 


18.23 


18.59 


% 


16.67 


17.08 


17.50 


17.92 


18.33 


18.75 


19.17 


19.58 


20.00 


20.42 


20.83 


21.25 


A 


18.75 


19.22 


19.69 


20.16 


20.63 


21.09 


21.56 


22.03 


22.50 


22.97 


23.44 


23.91 


% 


20.83 


21.35 


21.88 


22.40 


22.92 


23.44 


23.96 


24.48 


25.00 


25.52 


26.04 


26.56 


H 


22.92 


23.49 


24.06 


24.64 


25.21 


25.78 


26.35 


26.93 


27.50 


28.07 


28.65 


29.22 


3 A 


25.00 


25.62 


26.25 


26.88 


27.50 


28.13 


28.75 


29.38 


30.00 


30.63 


31.25 


31.88 


H 


27.08 


27.76 


28.44 


29.11 


29.79 


30.47 


31.15 


31.82 


32.50 


33.18 


33.85 


34.53 


Va 


29.17 


29.90 


30.63 


31.35 


32.08 


32.81 


33.54 


34.27 


35.00 


35.73 


36.46 


37.19 


H 


31.25 


32.03 


32.81 


33.59 


34.38 


35.16 


35.94 


36.72 


37.50 


38.28 


39.06 


39.84 


i 


33.33 


34.17 


35.00 


35.83 


36.67 


37.50 


38.33 


39.17 


40.00 


40.83 


41.67 


42.50 


& 


35.42 


36.30 


37.19 


38.07 


38.96 


39.84 


40.73 


41.61 


42.50 


43.39 


44.27 


45.16 


% 


37.50 


38.44 


39.38 


40.31 


41.25 


42.19 


43.13 


44.06 


45.00 


45.94 


46.88 


47.81 


A 


39.58 


40.57 


41.56 


42.55 


43.54 


44.53 


45.52 


46.51 


47.50 


48.49 


49.48 


50.47 


X 


41.67 


42.71 


43.75 


44.79 


45.83 


46.88 


47.92 


48.96 


50.00 


51.04 


52.08 


53.13 


& 


43.75 


44.84 


45.94 


47.03 


48.13 


49.22 


50.31 


51.41 


52.50 


53.59 


54.69 


55.78 


% 


45.83 


46.98 


48.13 


49.27 


50.42 


51.56 


52.71 


53.85 


55.00 


56.15 


57.29 


58.44 


& 


47.92 


49.11 


50.31 


51.51 


52.71 


53.91 


55.10 


56.30 


57.50 


58.70 


59.90 


61.09 


« 


50.00 


51.25 


52.50 


53.75 


55.00 


56.25 


57.50 


58.75 


60.00 


61.25 


62.50 


63.75 


& 


52.08 


53.39 


54.69 


55.99 


57.29 


58.59 


59.90 


61.20 


62.50 


63.80 


65.10 


66.41 


% 


54.17 


55.52 


56.88 


58.23 


59.58 


60.94 


62.29 


63.65 


65.00 


66.35 


67.71 


69.06 


ti 


56.25 


57.66 


59.06 


60.47 


61.88 


63.28 


64.69 


66.09 


67.50 


68.91 


70.31 


71.72 


% 


58.33 


59.79 


61.25 


62.71 


64.17 


65.63 


67.08 


68.54 


70.00 


71.46 


72.92 


74.38 


if 


60.42 


61.93 


63.44 


64.95 


66.46 


67.97 


69.48 


70.99 


72.50 


74.01 


75.52 


77.03 


Vs 


62.50 


64.06 


65.63 


67.19 


68.75 


70.31 


71.88 


73.44 


75.00 


76.56 


78.13 


79.69 


15 
T5 


64.58 


66.20 


67.81 


69.43 


71.04 


72.66 


74.27 


75.89 


77.50 


79.11 


80.73 


82.34 


2 ' 


66.67 


68.33 


70.00 


71.67 


73.33 


75.00 


76.67 


78.33 


80.00 


81.67 


83.33 


85.00 



For Steel add 2 per cent. 
19 



290 



Weight of Iron. 



Weight of Square and Round Wrought=iron per 
Lineal Foot. 



© tTd 

w CD ® 

.si- 3 

2-o.S 

H 


Weight of 
square bar, 
in pounds. 


Weight of 
round bar, 
in pounds. 


O -aj 

as 4S rt 
05 © o 

2 * a 

H 


Weight of 
square bar, 
n pounds. 


Weight of 
round bar, 
in pounds. 


m ® % 

2 a .s 


Weight of 
square bar, 
in pounds. 


Weight of 
round bar, 
in pounds. 









2 


13.33 


10.47 


4 


53.33 


41.89 


A 


.013 


.010 


A* 


14.18 


11.14 


A 


55.01 


43.21 


% 


.052 


.041 


Vs 


15.05 


11.82 


Vs 


56.72 


44.55 


A 


.117 


.092 


A 


15.95 


12.53 


T% 


58.45 


45.91 


M 


.208 


.164 


H 


16.88 


13.25 


X 


60.21 


47.29 


T 5 S 


.326 


.256 


5 
TS 


17.83 


14.00 


5 
TS 


61.99 


48.69 


% 


.469 


.368 


% 


18.80 


14.77 


% 


63.80 


50.11 


A 


.638 


.501 


A 


19.80 


15.55 


T5 


65.64 


51.55 


X 


.833 


.654 


% 


20.83 


16.36 


% 


67.50 


53.01 


A 


1.055 


.828 


A 


21.89 


17.19 


A 


69.39 


54.50 


% 


1.302 


1.023 


K 


22.97 


18.04 


% 


71.30 


56.00 


ti 


1.576 


1.237 


tt 


24.08 


18.91 


H 


73.24 


57.52 


% 


1.875 


1.473 


% 


25.21 


19.80 


% 


75.21 


59.07 


u 


2.201 


1.728 


ii 


26.37 


20.71 


ii 


77.20 


60.63 


% 


2.552 


2.004 


% 


27.55 


21.64 


Vs 


79.22 


62.22 


if 


2.930 


2.301 


if 


28.76 


22.59 


ii 


81.26 


63.82 


l 


3.333 


2.618 


3 


30.00 


23.56 


5 


83.33 


65.45 


A 


3.763 


2.955 


A 


31.26 


24.55 


A 


85.43 


67.10 


% 


4.219 


3.313 


X 


32.55 


25.57 


Vs 


87.55 


68.76 


A 


4.701 


3.692 


A 


33.87 


26.60 


A 


89.70 


70.45 


X 


5.208 


4.091 


M 


35.21 


27.65 


% 


91.88 


72.16 


A 


5.742 


4.510 


A 


36.58 


28.73 


t\ 


94.08 


73.89 


% 


6.302 


4.950 


% 


37.97 


29.82 


% 


96.30 


75.64 


A 


6.888 


5.410 


7 


39.39 


30.94 


A 


98.55 


77.40 


^ 


7.500 


5.890 


X 


40.83 


32.07 


X 


100.8 


79.19 


9 
IB 


8.138 


6.392 


A 


42.30 


33.23 


A 


103.1 


81.00 


^ 


8.802 


6.913 


Vs 


43.80 


34.40 


% 


105.5 


82.83 


ii 


9.492 


7.455 


H 


45.33 


35.60 


U 


107.8 


84.69 


^ 


10.21 


8.018 


3 A 


46.88 


36.82 


% 


110.2 


86.56 


H 


10.95 


8.601 


ii 


48.45 


38.05 


ii 


112.6 


88.45 


Vs 


11.72 


9.204 


Vs 


50.05 


39.31 


Vs 


115.1 


90.36 


H 


12.51 


9.828 


ii 


51.68 


40.59 


H 


117.5 


92.29 



For Steel add 2 per cent. 



Weight of Iron. 



291 



Weight of Square and Round Wrought=iron per 
Lineal Foot. 



° if °o 

OD 3 ® 

a a a 

H 


Weight of 
square bar, 
in pounds. 


III 

£2.2 


CE £ <» 
<D CD O 


Weight of 
square bar, 
in pounds. 


Weight of 
round bar, 
in pounds. 


° u* 

CD © ® 
® CD o 

o 2 -. 
H 


iT « 

O CD S 

^ 5 & 


Weight of 
round bar, 
in pounds. 


6 


120.0 


94.25 


8 


213.3 


167.6 


10 


333.3 


261.8 


A 


122.5 


96.22 


A 


216.7 


170.2 


i 


337.5 


265.1 


% 


125.1 


98.22 


% 


220.1 


172.8 


% 


341.7 


268.4 


A 


127.6 


100.2 


A 


223.5 


175.5 


A 


346.0 


271.7 


k 


130.2 


102.3 


M 


226.9 


178.2 


v± 


350.2 


275.1 


5 
T6 


132.8 


104.3 


A 


230.3 


180.9 


A 


354.5 


278.4 


K 


135.5 


106.4 


% 


233.8 


183.6 


% 


358.8 


281.8 


7 


138.1 


108.5 


7 
1^ 


237.3 


186.4 


7 


363.1 


285.2 


% 


140.8 


110.6 


% 


240.8 


189.2 


3^ 


367.5 


288.6 


A 


113.6 


112.7 


A 


244.4 


191.9 


A 


371.9 


292.1 


« 


146.3 


114.9 


% 


248.0 


194.8 


% 


376.3 


295.5 


tt 


149.1 


117.1 


tt 


251.6 


197.6 


H 


380.7 


299.0 


% 


151.9 


119.3 


% 


255.2 


200.4 


% 


385.2 


302.5 


H 


154.7 


121.5 


13 

1^ 


258.9 


203.3 


if 


389.7 


306.1 


% 


157.6 


123.7 


^ 


262.6 


206.2 


% 


394.2 


309.6 


it 


160.4 


126.0 


If 


266.3 


209.1 


15 


398.8 


313.2 


7 


163.3 


128.3 


9 


270.0 


212.1 


11 


403.3 


316.8 


A 


166.3 


130.6 


A 


273.8 


215.0 


A 


407.9 


320.4 


% 


169.2 


132.9 


% 


277.6 


218.0 


Vs 


412.6 


324.0 


A 


172.2 


135.2 


A 


281.4 


221.0 


A 


417.2 


327.7 


H 


175.2 


137.6 


M 


285.2 


224.0 


& 


421.9 


331.3 


A 


178.2 


140.0 


A 


289.1 


227.0 


A 


426.6 


335.0 


% 


181.3 


142.4 


% 


293.0 


230.1 


% 


431.3 


338.7 


7 


184.4 


144.8 


A 


296.9 


233.2 


A 


436.1 


342.5 


K 


187.5 


147.3 


X 


300.8 


236.3 


% 


440.8 


346.2 


9 


190.6 


149.7 


A 


304.8 


239.4 


A 


445.6 


350.0 


% 


193.8 


152.2 


% 


308.8 


242.5 


% 


450.5 


353.8 


tt 


197.0 


154.7 


H 


312.8 


245.7 


tt 


455.3 


357.6 


% 


200.2 


157.2 


% 


316.9 


248.9 


% 


460.2 


361.4 


i% 


203.5 


159.8 


« 


321.0 


252.1 


if 


465.1 


365.3 


Vs 


206.7 


162.4 


% 


325.1 


255.3 


% 


470.1 


369.2 


it 


210.0 


164.9 


it 


329.2 


258.5 


iS 


475.0 


373.1 



For Steel add 2 per cent. 



292 



Weight of Iron. 



Weight of Flat Rolled Iron, in Kilogrammes, per 
Lineal Metre. 





Thickness. 


mm 


2 mm 


4 mm 


6 mm 


8 mm 


10 mm 


12 mm 


15 mm 


20 mm 


25 mm 


30 mm 


35 mm 


40 mm 


5 


0.078 


0.155 


0.233 


310 


388 


0.465 


0.581 


0.775 


0.969 


1.163 


1.356 


1.550 


10 


0.155 


0.310 


0465 


0620 


0.775 


0.930 


1.163 


1.550 


1.93S 


2.325 


2.713 


3.100 


15 


0.233 


0.465 


0.69S 


0.930 


1.163 


1.395 


1.744 


2.325 


2.906 


3.4S8 


4.069 


4.650 


20 


0.310 


0.620 


0.930 


1.210 


1.550 


1.860 


2.325 


3.100 


3.875 


4.650 


5.425 


6.200 


25 


0.388 


0.775 


1.163 


1.550 


1.938 


2.325 


2.906 


3.875 


4.844 


5.813 


6.781 


7.750 


30 


0.465 


0.930 


1.395 


1.860 


2 325 


2.790 


3.488 


4.650 


5.813 


6.975 


8.138 


9.300 


35 


0.543 


1.085 


1.628 


2.170 


2.713 


3.255 


4.069 


5.425 


6.781 


8.138 


9.494 


10.850 


40 


0.620 


1.240 


1.860 


2.480 


3.100 


3.720 


4.650 


6.200 


7.750 


9.300 


10.850 


12.400 


45 


0.698 


1.395 


2.093 


2.790 


3.488 


4.185 


5.231 


6.975 


8.719 


10.463 


12.206 


13.950 


50 


0.775 


1.550 


2.325 


3100 


3.875 


4.650 


5.813 


7.750 


9.688 


11.625 


13.563 


15.500 


55 


0.853 


1.705 


2.558 


3.410 


4.263 


5.115 


6.394 


8.525 


10.656 


12.788 


14.919 


17.050 


60 


0.930 


1.860 


2.790 


3.720 


4.650 


5.5S0 


6.975 


9.300 


11.625 


13.950 


16.275 


18.600 


65 


1.008 


2 015 


3 023 


4.030 


5.038 


6 045 


7.556 


10.075 


12.594 


15.113 


17.631 


20.150 


70 


1.085 


2.170 


3.255 


4.340 


5.425 


6.510 


8 138 


10.850 


13.563 


16.275 


18.988 


21.700 


75 


1.163 


2.325 


3. 488 


4.650 


5.813 


6.975 


8.719 


11.625 


14.531 


17.438 


20.344 


23.250 


80 


1.240 


2.480 


3.720 


4.960 


6.200 


7.440 


9.300 


12.400 


15.500 


18.600 


21.700 


24.800 


85 


1,318 


2.635 


3953 


5.270 


6.5S8 


7.905 


9.862 


13.175 


16.469 


19.763 


23.056 


26.350 


90 


1.395 


2.790 


4.185 


5.580 


6.975 


8 370 


10.463 


13.950 


17.43S 


20.925 


24.413 


27.900 


95 


1.473 


2.945 


4.41S 


5.890 


7.363 


8.835 


11.041 


14.725 


1S.406 


22.088 


25.769 


29.450 


100 


1.550 


3.100 


4.650 


6.200 


7.750 


9 300 


11.625 


15.500 


19.375 


23.250 


27.125 


31.000 


110 


1.705 


3.410 


5.115 


6.820 


8.525 


10.230 


12.789 


17.050 


21 314 


25 575 


29.838 


34.100 


120 


1.860 


3.720 


5.580 


7.410 


9.300 


11.160 


13.950 


18.600 


23.250 


27.900 


32.550 


37.200 


130 


2.015 


4.030 


6.045 


8.060 


10 075 


12.090 


15.113 


20.150 


25.188 


30.225 


35.263 


40.300 


140 


2.170 


4.340 


6.510 


8.680 


10.850 


13 020 


16 275 


21.700 


27.125 


32.550 


37.975 


43.400 


150 


2.325 


4.650 


6.975 


9.300 


11.625 


13.950 


17.438 


23.250 


29.062 


34.875 


40.688 


46.500 


160 


2.480 


4.960 


7.440 


9.920 


12.400 


14.880 


18.600 


24.800 


31.000 


37.200 


43.400 


49.600 


170 


2.635 


5.270 


7.905 


10.540 


13.175 


15.810 


19.763 


26.350 


32.938 


39.525 


46.113 


52.700 


180 


2.790 


5.580 


8.370 


11.160 


13.950 


16.740 


20.925 


27.900 


34.875 


41.850 


48 825 


55.800 


190 


2.945 


5.890 


8.835 


11.780 


14 725 


17.670 


22.088 


29.450 


36.813 


44.175 


51.538 


58.900 


200 


3.100 


6.200 


9.300 


12.400 


15.500 


18.600 


23.250 


31.000 


38.750 


46.500 


54.250 


62.000 



For Steel add 2 per cent. 



Weight of Iron. 



293 



Weight of Square and Round Wrought=iron, in 
Kilogrammes, per Lineal Metre. 



Thickness or 
diameter. 


Square. 


Round. 


Thickness or 
diameter. 


Square. 


Round. 


Mm. 


Kg. 


Kg. 


Mm. 


Kg. 


Kg. 


5 


0.195 


0.152 


50 


19.450 


15.215 


6 


0.280 


0.219 


52 


21.009 


16.459 


7 


0.381 


0.298 


54 


22.686 


17.749 


8 


0.498 


0.390 


56 


24.398 


19.088 


9 


0.630 


0.493 


58 


26.172 


20.476 


10 


0.778 


0.609 


60 


28.080 


21.913 


11 


0.941 


0.737 


62 


29.906 


23 398 


12 


1.120 


0.877 


64 


32.147 


24.930 


13 


1.315 


1.028 


66 


33.890 


26.514 


14 


1.525 


1.193 


68 


35.975 


28.146 


15 


1.751 


1.370 


70 


38.122 


29.825 


16 


1.992 


1.558 


72 


39.743 


31.554 


17 


2.248 


1.759 


74 


42.603 


33.333 


18 


2.520 


1.972 


76 


44.937 


35.158 


19 


2.809 


2.197 


78 


47.334 


37.032 


20 


3.112 


2.435 


80 


49.792 


38.953 


21 


3.431 


2.684 


85 


56.195 


43.977 


22 


3.765 


2.946 


90 


63.018 


49.303 


23 


4.116 


3.220 


95 


70.215 


54.934 


24 


4.481 


3.506 


100 


77.800 


60.860 


25 


4.863 


3.804 


105 


85.775 


67.107 


26 


5.259 


4.115 


110 


94.138 


73.651 


27 


5.672 


4.437 


115 


102.891 


80.500 


28 


6.100 


4.772 


120 


112.000 


87.650 


29 


6.543 


5.119 


125 


121.563 


95.107 


30 


7.002 


5.478 


130 


131.500 


102.867 


31 


7.477 


5.849 


135 


141.791 


110.933 


32 


7.967 


6.232 


140 


152.500 


119.302 


33 


8.472 


6.629 


145 


163.575 


127.976 


34 


9.009 


7,036 


150 


175.100 


136.954 


35 


9.531 


7.456 


155 


186.915 


146.236 


36 


10.083 


7.889 


160 


199.200 


155.812 


37 


10.651 


8.333 


165 


211.811 


165.714 


38 


11.234 


8.789 


170 


224.842 


175.910 


39 


11.833 


9.258 


175 


238.263 


186.410 


40 


12.448 


9.738 


180 


252.000 


197.213 


41 


13.078 


10.212 


185 


266.271 


208.322 


42 


13.724 


10.737 


190 


280.900 


219.735 


43 


14.385 


11.255 


195 


295.835 


231.452 


44 


15.062 


11,784 


200 


311.200 


243.473 


45 


15.755 


12.326 


205 


326.920 


256.790 


46 


16.462 


12.880 


210 


343.090 


269.465 


47 


17.187 


13.446 


215 


359.600 


282.453 


48 


17.925 


14.024 


220 


376.550 


295.744 


49 


18.680 


14.614 


225 


393.860 


309.340 



For Steel add 2 per cent. 



294 



Weight of Sheet-metal. 



Weight of Sheet=metal. 

British Units. 
Weight of Iron, Copper, Lead, and Zinc per Square Foot. 



Thickness, 
in inches. 


Cast-iron. 


Wrought- or 
sheet-iron. 


Sheet- 
copper. 


Sheet-lead. 


Sheet-zinc. 




Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


A 


2.346 


2.517 


2.890 


3.694 


2.320 


Vs 


4.693 


5.035 


5.781 


7.382 


4.642 


& 


7.039 


7.552 


8.672 


11.074 


6.961 


34 


9.386 


10.070 


11.562 


14.765 


9.275 


& 


11.733 


12.588 


14.453 


18.456 


11.61 


Vs 


14.079 


15.106 


17.344 


22.148 


13.93 


& 


16.426 


17.623 


20.234 


25.839 


16.23 


y* 


18.773 


20.141 


23.125 


29.530 


18.55 


j% 


21.119 


22.659 


26.016 


33.222 


20.87 


% 


23.466 


25.176 


28.906 


36.913 


23.19 


n 


25.812 


27.694 


31.797 


40.604 


25.53 


% 


28.159 


30.211 


34.688 


44.296 


27.85 


i§ 


30.505 


32.729 


37.578 


47.987 


30.17 


Vs 


32.852 


35.247 


40.469 


51.678 


32.47 


h 


35.199 


37.764 


43.359 


55.370 


34.81 


i 


37.545 


40.282 


46.250 


59.061 


37.13 


Vs 


42.238 


45.317 


52.031 


66.444 


41.78 


M 


46.931 


50.352 


57.813 


73.826 


46.42 


% 


51.625 


55.387 


63.594 


81.210 


51.04 


v* 


56.317 


60.422 


69.375 


88.592 


55.48 


% 


61.011 


65.458 


75.156 


95.975 


60.35 


% 


65.704 


70.493 


80.938 


103358 


65.00 


% 


70.397 


75.528 


86.719 


110.740 


69.61 


2 


75 090 


80.563 


92.500 


118.128 


74.25 



Weight of Sheet-metal. 



295 



Weight of Sheet=metal. 

Metric Units. 
Weight, in Kilogrammes, per Square Metre. 



Thickness. 


Cast-iron. 


Wrought-iron. 


Steel. 


Copper. 


Mm. 


Kg- 


Kg. 


Kg. 


Kg. 


0.5 


3.625 


3.89 


3.925 


4.45 


1 


7.25 


7.78 


7.85 


8.90 


2 


14.50 


15.56 


15.70 


17.80 


3 


21.75 


23.34 


23.55 


26.70 


4 


29.00 


31.12 


31.40 


35.60 


5 


36.25 


38.90 


39.25 


44.50 


6 


43.50 


46.68 


47.10 


53.40 


7 


50.75 


54.46 


54.95 


62.30 


8 


58.00 


62.24 


62.80 


71.20 


9 


65.25 


70.02 


70.65 


80.10 


10 


72.50 


77.80 


78.50 


89.00 


11 


79.75 


85.58 


86.35 


97.90 


12 


87.00 


93.36 


94.20 


106.80 


13 


94.25 


101.14 


102.05 


115.70 


14 


101.50 


108.92 


109.90 


124.60 


15 


108.75 


116.70 


117.75 


133.50 


16 


116.00 


124.48 


125.60 


142.40 


17 


123.25 


132.26 


133.45 


151.30 


18 


130.50 


140.04 


141.30 


160.20 


19 


137.75 


147.82 


149.15 


169.10 


20 


145.00 


155.60 


157.00 


178.00 


21 


152.25 


163.38 


164.85 


186.90 


22 


159.50 


171.17 


172.70 


195.80 


23 


166.75 


178.94 


180.55 


204.70 


24 


174.00 


186.72 


188.40 


213.60 


25 


181.25 


194.50 


196.25 


222.50 


26 


188.50 


202.28 


204.10 


231.40 


27 


195.75 


210.06 


211.95 


240.30 


28 


203.00 


217.84 


219.80 


249.20 


29 " 


210.25 


225.62 


227.65 


258.10 


30 


217.50 


233.40 


235.50 


267.00 



296 



Weight of Materials. 



Weight of Rolled Sheets of Wrought=iron and Steel. 

British Units. 
Specific Gravity of Iron, 7.70 ; of Steel, 7.85. f & 





Birmingham Wire 


Gauge. 


American 


(B. & S.) Wire Gauge. 


No. of 
gauge. 




Weight, in pounds, 




Weight, in pounds, 


Thickness, 


per square foot. 


Thickness, 


per square foot. 




in inches. 






in inches. 








Iron. 


Steel. 


Iron. 


Steel. 


0000 


.454 


18.16 


18.52 


.4600 


18.40 


18.76 


000 


.425 


17.00 


17.34 


.4096 


16.39 


16.72 


00 


.380 


15.20 


15.50 


.3648 


14.59 


14.88 





.340 


13.60 


13.87 


.3249 


13.00 


13.26 


1 


.300 


12.00 


12.24 


.2893 


11.57 


11.80 


2 


.284 


11.36 


11.59 


.2576 


10.31 


10.52 


3 


.259 


10.35 


10.56 


.2294 


9.18 


9.36 


4 


.238 


9.52 


9.71 


.2043 


8.17 


8.33 


5 


.220 


8.80 


8.98 


.1819 


7.27 


7.42 


6 


.203 


8.12 


8.28 


.1620 


6.48 


6.61 


7 


.180 


7.19 


7.34 


.1443 


5.77 


5.88 


8 


.165 


6.60 


6.73 


.1285 


5.14 


5.24 


9 


.148 


5.92 


6.04 


.1144 


4.57 


4.66 


10 


.134 


5.36 


5.47 


.1019 


4.07 


4.15 


11 


.120 


4.80 


4.89 


.0907 


3.63 


3.70 


12 


.109 


4.35 


4.44 


.0808 


3.23 


3.29 


13 


.095 


3.80 


3.87 


.0720 


2.88 


2.93 


14 


.083 


3.32 


3.38 


.0641 


2.56 


2.61 


15 


.072 


2.88 


2.94 


.0571 


2.28 


2.32 


16 


.065 


2.60 


2.65 


.0508 


2.03 


2.07 


17 


.058 


2.32 


2.37 


.0453 


1.81 


1.84 


18 


.049 


1.96 


1.99 


.0403 


1.61 


1.64 


19 


.042 


1.68 


1.71 


.0359 


1.43 


1.46 


20 


.035 


1.39 


1.42 


.0320 


1.27 


1.30 


21 


.032 


1.27 


1.30 


.0285 


1.13 


1.16 


22 


.028 


1.11 


1.14 


.0253 


1.01 


1.03 


23 


.025 


.997 


1.02 


.0226 


.903 


.921 


24 


.022 


.880 


.898 


.0201 


.805 


.821 


25 


.020 


.800 


.816 


.0179 


.715 


.729 


26 


.018 


.719 


.734 


.0159 


.638 


.651 


27 


.016 


.640 


.653 


.0142 


.570 


.581 


28 


.014 


.560 


.571 


.0126 


.505 


.515 


29 


.013 


.520 


.531 


.0113 


.450 


.459 


30 


.012 


.480 


.489 


.0100 


.400 


.409 


31 


.010 


.399 


.408 


.0089 


.357 


.364 


32 


.009 


.359 


.367 


.0080 


.318 


.324 


33 


.008 


.320 


.326 


.0071 


.283 


.288 


34 


.007 


.280 


.286 


.0063 


.252 


.257 


35 


.005 


.200 


.204 


.0056 


.224 


.228 


36 


.004 


.159 


.162 


.0050 


.200 


.204 



WE1GH1 01 M 41 i.i'i ■•• L ■:. 



'l'.)l 



Dimensions and Weights of Spheres. 
British [ i" r 

/r,f;-, in poUIldS. 



Di&mefc r 


Hurf.-i/ ' 


'■;.;,• 


Iron. 


Lead. 


Water. 


[nch. 


Sq, Inch. 


Cub. inch. 


Lb. 


Lb. 


Lb. 


LOOO 


8.1416 


.8236 


.1366 


.2147 


.0188 


1.126 


8.0760 


.7155 


.101/ 


..',002 


.0/01 


L.280 


4.9087 


1.02/0 


.207/ 


.4200 


.0368 


L.375 


5 9395 


1.501 1 


,3560 


.5570 


.0100 


L.600 


7.0686 


1.7071 


.1007 


.7/18 


.00/0 


1.626 




2.2107 


.5801 


.0227 


.0609 


1.700 


0/11 


2.8061 


.7828 


1. 1 528 


,1050 


ISlh 


11.014 


3.451 1 


.9000 


1.1150 


.1/4/ 


2.000 


12.866 


4.1888 


1.00/0 


1.7180 


.1608 


2.126 


14. 186 




1.3124 


2.0631 


.1800 


2.280 


15.001 


5.0010 




2.4482 


.2147 


2.875 


17.720 


7.0143 


L.8334 


2.8811 


.2525 


2.800 


L9.635 


8.1812 




8.5551 


.2015 


2.628 


21.047 


9.4708 


2.4725 




,3410 


2.780 




10.889 


2.8100 


1.1028 


.80/0 


2.875 


25.007 


I J.442 


5.25 1 2 


5.1050 


.1170 


3.000 


28.274 


J 1.1/7 






.8089 


8.128 


80.660 


15.979 


4.1721 


6 5568 


.5752 


8.280 


83.188 


17.071 


4.6835 


7.3623 


.0171 




88 : 


20.129 


5.2612 


8.2521 


.7/10 


8.800 


184 


22.440 




0.207/, 


,8081 


8 628 


41.282 


24.941 


0.5080 


1 0.2/ 1 




8.780 


44.170 


27.012 


7.2135 


I I 323 


9941 




47.17/ 


80.466 


7.0550 


12.500 


1.000/ 


4.00 


60.265 


83.510 


8.7361 


13.744 


1/001 


4.28 


50.745 


40.101 


10.510 


10.482 


1.1170 


4.. 7; 


63.617 


47.715 




10.500 


1.7177 


4.75 


70 


50.115 


11.000 


28.085 


2.0202 


5.00 


It 640 


05.150 


17.00/, 


20.81/, 


2.3562 


5.25 


80.500 


75.700 


10.810 


81.089 


2.7270 


6.80 


05.0.',; 


87.114 


22.720 


i 720 


5.1/01 


5.75 


103.87 


00 511 


26.000 


40.850 




6.0 


11/. 10 


115.10 




40. .',85 


4.0710 


0.5 


L32.73 


143.79 


87.458 


570 


5.1705 


7.0 


163.94 


170.50 


40.8/0 


78.669 


6.4683 


7.5 


170.71 


220 


57.587 


00.50/ 


7.05/0 


8.0 


201.00 


208.08 


m 


100.55 


0.0500 


6.8 




821.55 


8&S39 


131.38 


11.570 


9.0 


254.47 


381.70 


00.51 


150. 55 


13.741 






118.02 


117.0.', 


18-4.12 


10.101 


10.0 


814.16 


! ; 00 


180.. 7) 


211.75 


1 8 . 850 


11.0 


L13 


000.01 


181.70 


285 83 


26.289 


12.0 


462 


001.78 


235.87 


.",71.00 


82.572 


13.0 


630.92 


1150.3 


•-02 


471 ,80 


41.111 


11.0 


015.72 


11/0.7 


874.86 


580.27 


51 7/1 


15.0 


706.84 


1 707. 1 


400.00 


7/1 • 


68.616 


16.0 




2144.6 


0.11 


870.01 


77.200 


17.0 


i . 96 


2572.4 


070.71 


1055.0 


92.607 


18.0 


1017.8 


505/.. 


700.08 


1252.4 


100.08 


10.0 


11/1.1 


3591.3 


080.27 


1472.9 


120.20 


20.0 


1286.6 


4188.8 


1092.00 


1718.0 


J 50.80 



298 



Weight of Materials. 



Dimensions and Weights of Cast=iron Spheres. 

Metric Units. 

Sizes in centimetres, weights in kilogrammes. 

For Lead, multiply by 1.575. 



Diameter. 


Volume. 


Weight. 


Diameter. 


Yolume. 


Weight. 


Cm. 


Cub. cm. 


Kg. 


Cm. 


Cub. cm. 


Kg. 


1.0 


.524 


.004 


21.0 


4849.05 


35.16 


1.5 


1.767 


.013 


21.5 


5203.72 


37.73 


2.0 


4.189 


.030 


22.0 


5575.28 


40.42 


2.5 


8.181 


.059 


22.5 


5964.12 


43.24 


3.0 


14.137 


.102 


23.0 


6370.63 


46.19 


3.5 


22.449 


.165 


23.5 


6795.20 


49.27 


4.0 


33.510 


.243 


24.0 


7238.23 


52.48 


4.5 


47.713 


.346 


24.5 


7700.11 


55.83 


5.0 


65.45 


.475 


25.0 


8181.23 


59.31 


5.5 


87.11 


.632 


25.5 


8681.98 


62.94 


6.0 


113.10 


.820 


26.0 


9202.77 


66.72 


6.5 


143.79 


1.043 


26.5 


9744.08 


70.64 


7.0 


179.59 


1.302 


27.0 


10305.99 


74.72 


7.5 


220.89 


1.601 


27.5 


10889.22 


78.95 


8.0 


268.08 


1.944 


28.0 


11494.04 


83.33 


8.5 


321.56 


2.331 


28.5 


12120.85 


87.88 


9.0 


381.70 


2.767 


29.0 


12770.08 


92.58 


9.5 


448.92 


3.255 


29.5 


13442.02 


97.45 


10.0 


523.60 


3.796 


30.0 


14137.17 


102.49 


10.5 


606.13 


4.394 


31.0 


15598.53 


113.09 


11.0 


696.91 


5.053 


32.0 


17157.28 


124.39 


11.5 


796.33 


5.773 


33.0 


18816.57 


136.42 


12.0 


904.78 


6.560 


34.0 


20579.53 


149.20 


12.5 


1022.64 


7.414 


35.0 


22449.30 


162.76 


13.0 


1150.35 


8.340 


36.0 


24429.02 


177.11 


13.5 


1288.25 


9.340 


37.0 


26521.95 


192.28 


14.0 


1436.76 


10.416 


38.0 


28730.91 


208.30 


14.5 


1596.26 


11.573 


39.0 


31059.35 


225.18 


15.0 


1767.15 


12.812 


40.0 


33510.32 


242.95 


15.5 


1949.82 


14.14 


41.0 


36086.96 


261.63 


16.0 


2144.66 


15.55 


42.0 


38792.39 


281.24 


16.5 


2352.07 


17.05 


43.0 


41629.77 


301.82 


17.0 


2572.44 


18.65 


44.0 


44602.24 


323.37 


17.5 


2806.16 


20.34 


45.0 


47712.94 


345.91 


18.0 


3053.63 


22.14 


46.0 


50965.01 


369.50 


18.5 


3315.24 


24.04 


47.0 


54361.60 


394.12 


19.0 


3591.36 


26.04 


48.0 


57905.58 


419.82 


19.5 


3882.42 


28.15 


49.0 


61600.87 


446.61 


20.0 


4188.79 


30.37 


50.0 


65449.85 


474.51 


20.5 


4510.87 


32.70 


100.0 


523598.80 


3796.09 



Weight of Materials. 



299 



Weight of Cast=iron Pipe per Foot in Length. 

British Units. 

For WroughtHbron multiply by 1.067 ; for Lead, by 1.575 ; for Copper, by 

1.23 ; for Brass, by 1.16. 









Thickness of 


pipe, 


in inches. 


3~ 

M 13 


u 


% 


% 


% 


% 


Vs 


1 


iVs 


1^ 


tVs 


IK 


i% 


2 


Ins. 


Lb. 


Lb. 


Lb 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


1 


3.07 


























H 


3.69 


























% 


4.30 


























4.92 


























2 


5.53 


8.76 


























6.15 
6.76 


9.69 
10.6 
























% 


7.37 


11.5 
























3 


7.98 


12.5 


17. 


1 22.3 




















£ 


8.60 


13.4 


18., 


) 23.8 




















9.21 


14.3 


19/ 


1 25.4 




















/4 


9.83 


15.2 


20. < 


) 26.9 




















4 


10.3 


16.1 


22. 


I 28.5 


35.1 


42.0 
















x 4 


11.1 


17.1 


23.' 


t 30.0 


36.9 


44.1 
















H 


11.7 


18.0 


24. ( 


5 31.5 


38.8 


46.3 
















12.3 


18.9 


25.5 


5 33.1 


40.6 


48,5 
















5 


12.9 
13 5 


19.8 
20.8 


27.] 
28.,' 


34.6 
$ 36.1 


42.5 
44.3 


50.6 

52 8 


59.1 
61.5 


67.8 
70.6 












M 












A 


14 ?, 


21.7 


29, 


) 37.7 


46.1 


54.9 


64.0 


73.3 












148 


22.6 


3(U 


J 39.2 


48.0 


57.1 


66.4 


76.1 












6 


15.4 
16.6 


23.5 
25.4 


32.( 
34.* 


) 40.8 
> 43.8 


49.8 
53 5 


59.2 
63 5 


68.9 
73 8 


78.9 
84.4 


89.2 
95 3 


99.8 
107.0 








* 








7 


17.8 


27.2 


36.< 


) 46.9 


57.2 


67.8 


78.7 


89.4 


102.0 


113.0 


126 


151 


177 


y 9 


19.1 


29.1 


39.^ 


: 50.0 


60.9 


72.1 




95.5 




120 


133 






8 


20.3 


30.9 


41.5 


5 53.1 


64.6 


76.4 


88.6 


101.0 


114.0 


127.0 


140 


168 


197 


K 


21.5 


32.8 


44.; 


; 56.1 


68.3 


80,7 


93 5 












207 


9 


22.8 


34.6 


46.J 


I 59.2 


72.0 


85.1 


98.4 


112.0 


126.0 


140.0 


155 


185 


217 


« 


24.0 


36.4 


49.5 


I 62.3 


75.7 






118.0 


132.0 


147.0 


163 






10 


25.1 


38.3 


51/ 


I 65.3 


79.4 


93.6 


108.0 


123.0 


138.0 


154.0 


170 


202 


235 


V, 


26.4 


40.1 


54.] 


. 68.4 


83.0 


97 9 






145 


161.0 




211 




11 


27.6 


42.0 


56.( 


> 71.5 


86.7 


102.0 


118.0 


134.0 


151.0 


168.0 


185 


220 


255 


fc 


28.8 


43.8 


59.: 


74.6 


90.4 


107.0 


123.0 


140.0 


157.0 


174 


192 


228 


265 


12 


30.0 


45.7 


61.* 


) 77.7 


94.1 


111.0 


128.0 


145.0 


163.0 


181.0 


199 


237 


275 


13 






66.^ 


L 83.8 


102.0 


120.0 


138.0 


156.0 


175.0 


195.0 


214 


254 


294 


14 






71.' 


t 89.4 


109.0 


128.0 


148.0 


168.0 


188.0 


208.0 


229 


271 


314 


15 






76.^ 


\ 96.1 


116.0 


137.0 


158.0 


179.0 


200.0 










16 






81.5 


I 102.0 


124.0 


145.0 


167.0 


190.0 


212.0 


235.0 


258 


306 


353 


17 






86.1 


L 108.0 


131.0 


154.0 


177.0 


201.0 


225.0 






323 




18 






91.( 


) 115.0 


139.0 


163.0 


187.0 


212.0 


237.0 


262.0 


288 


340 


393 


19 






96.( 


) 121.0 


146.0 


171.0 


197.0 


223.0 


249.0 


276.0 


303 


357 


412 


20 






101.( 


) 127.0 


153.0 


180.0 


207.0 


234.0 


261.0 


289.0 


317 


375 


432 


21 








133.0 


161.0 


188.0 


217.0 


245.0 


274.0 


303.0 


332 


392 


452 


22 








139.0 


168.0 


196.0 


227.0 


256.0 


286.0 


316.0 


347 


409 


471 


23 








145.0 


175.0 


206.0 


236.0 


267.0 


298.0 


330.0 


362 


426 


491 


24 








152.0 


183.0 


214.0 


246.0 


278.0 


311.0 


343.0 


375 


444 


511 


25 










190.0 


223.0 


256.0 


289.0 


323.0 


357.0 


391 


461 


531 


26 










198.0 


231.0 


266.0 


300.0 


335.0 


370.0 


406 


478 


550 


27 










205.0 


240.0 


276.0 


311.0 


348.0 


384.0 


421 


495 


570 


28 










212.0 


249.0 


286.0 


323.0 


360.0 


397.0 


436 


512 


590 


30 










227.0 


266.0 


305.0 


345.0 


384.0 


424.0 


465 


547 


629 


32 










242.0 


283.0 


325.0 


367.0 


409.0 


451.0 


495 


581 


668 


34 










257.0 


300.0 


345.0 


389.0 


434.0 


479.0 


524 


616 


708 


36 










271.0 


318.0 


364.0 


411.0 


458.0 


506.0 


554 


650 


746 


42 










315.0 


370.0 


423.0 


478.0 


532.0 


588.0 


644 


753 


864 


48 










359.0 


422.0 


482.0 


544.0 


605.0 


669.0 


733 


856 


982 



The weight of a spigot and faucet joint may be taken as equal to 8 
inches of straight pipe, and the weight of two flanges as equal to 12 inches 
of straight pipe. 



300 



Weight of- Mateeials. 



Weight of Cast=iron Pipe. 

Metric Units. 

Weight, in Kilogrammes, per Metre of Length. 

For Wrought-iron multiply by 1.067 ; for Lead, by 1.575 ; for Copper, by 
1.23 ; for Brass, by 1.16. 



Inside 




Thickness, in millimetres. 




diameter. 


10 


15 


20 


25 


30 


40 


Mm. 


Kg. 


Kg. 


Kg. 


Kg. 


Kg. 


Kg. 


25 


8.0 
9.1 


13.7 
15.4 


20.5 
22.8 


28.5 
31.3 






30 


41.0 




35 


10.2 


17.1 


25.1 


34.4 


44.4 


68.3 


40 


11.4 


18.8 


27.3 


37.0 


47.8 


72.9 


45 


12.5 


20.5 


29.6 


39.9 


51.2 


77.4 


50 


13.7 


22.2 


31.8 


42.8 


54.7 


82.0 


60 


15.9 


25.6 


36.5 


48.4 


61.5 . 


91.1 


70 


18.2 


29.0 


41.0 


54.1 


68.3 


100.2 


80 


20.5 


32.5 


45.6 


59.8 


75.2 


109.3 


90 


22.8 


36.0 


50.1 


65.5 


82.0 


118.4 


100 


25.1 


39.3 


54.7 


71.1 


88.8 


129.8 


110 


27.3 


42.8 


59.2 


76.9 


95.7 


136.7 


120 


29.6 


46.1 


63.8 


82.5 


102.5 


144.9 


130 


31.8 


49.5 


68.3 


88.3 


109.3 


154.9 


140 


\ 34.4 


52.8 


72.9 


94.0 


116.2 


164.0 


150 


36.5 


56.5 


77.4 


99.6 


123.0 


173.1 


160 


38.7 


59.8 


82.0 


105.4 


129.8 


182.2 


170 


41.8 


63.2 


86.5 


110.7 


136.7 


191.3 


180 


43.3 


66.6 


91.5 


116.6 


143.5 


200.4 


190 


45.6 


70.1 


97.7 


122.3 


150.3 


209.5 


200 


47.8 


73.5 


100.2 


128.1 


157.2 


218.7 


210 


50.1 


76.9 


104.8 


133.8 


164.0 


227.8 


220 


52.4 


80.2 


109.3 


139.5 


170.8 


236.9 


230 


54.7 


83.9 


113.8 


145.2 


177.7 


246.0 


240 


56.9 


86.8 


118.4 


150.9 


184.5 


255.1 


250 


59.2 


90.5 


123.0 


156.6 


191.3 


264.2 


260 


61.5 


94.0 


127.6 


162.3 


198.2 


273.3 


270 


63.8 


97.3 


132.1 


168.0 


205.0 


282.4 


280 


66.1 


100.8 


136.7 


173.7 


211.8 


291.5 


290 


68.3 


104.9 


141.2 


179.4 


218.7 


300.7 


300 


70.6 


107.6 


144.9 


185.1 


225.5 


309.7 


325 




116.2 
124.7 


157.2 

168.5 


199.3 
213.5 


242.6 
259.7 


332.5 


350 




355.3 


375 




133.2 
141.8 


179.9 
191.3 


227.8 
241.9 


276.7 
293.8 


378.1 


400 




400.9 


450 




158.9 
175.9 


214.1 
236.9 


270.5 
298.9 


328.9 
362.1 


446.4 


500 




492.0 









The weight of spigot and faucet joint = 0.2 metre. 
The weight of two flanges = 0.3 metre. 



Weight of Materials. 



301 



Weight of Bridge Rivets per Hundred. 

This table also applies to Button-headed Bolts. 



Before 
driving 

After 
driving 

Before 
driving 



Length 
of rivet 






Diameter of rivet, in inches. 






















under 


















head. 


% 


y 2 


% 


% 


Vs 


1 


tVs 


i% 


Inch. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


m 


5.7 


12.8 


22.0 


29.3 


43.9 


66.6 


93.3 


127.1 


% 


6.1 


13.5 


23.1 


30.9 


46.1 


69.4 


96.9 


131.5 


y 


6.5 


14.2 


24.1 


32.4 


48.2 


72.1 


100.4 


135.8 


78 


6.9 


14.8 


25.2 


34.0 


50.3 


74.9 


103.9 


140.2 


1 


7.3 


15.5 


26.3 


35.5 


52.5 


77.7 


107.4 


144.5 


7.7 


16.2 


27.4 


37.1 


54.6 


80.5 


110.9 


148.9 


2 


8.0 


16.9 


28.5 


38.7 


56.7 


83.3 


114.5 


153.2 


% 


8.4 


17.6 


29.6 


40.2 


58.8 


86.0 


118.0 


157.5 


% 


8.8 


18.3 


30.7 


41.8 


61.0 


88.8 


121.5 


161.9 


9.2 


19.0 


31.7 


43.3 


63.1 


91.6 


125.0 


166.2 


9.6 


19.7 


32.8 


44.9 


65.2 


94.4 


128.5 


170.6 


10.0 


20.4 


33.9 


46.5 


67.4 


97.2 


132.1 


174.9 


10.4 


21.1 


35.0 


48.0 


69.5 


99.9 


135.6 


179.3 


10.8 


21.8 


36.1 


49.6 


71.6 


102.7 


139.1 


183.6 


3 


11.2 


22.5 


37.2 


51.1 


73.7 


105.5 


142.6 


188.0 


% 


11.6 


23.2 


38.3 


52.7 


75.9 


108.3 


146.1 


192.3 


} 


11.9 


23.9 


39.3 


54.3 


78.0 


111.1 


149.7 


196.7 


12.3 


24.6 


40.4 


55.8 


80.1 


113.8 


153.1 


201.0 


12.7 


25.3 


41.5 


57.4 


82.3 


116.6 


156.7 


205.4 


Z8 


13.1 


26.0 


42.6 


58.9 


84.4 


119.4 


160.2 


209.7 


1 


13.5 


26.7 


43.7 


60.5 


86.5 


122.2 


163.7 


214.1 


13.9 


27.4 


44.8 


62.1 


88.6 


125.0 


167.3 


218.4 


4 


14.3 


28.1 


45.9 


63.6 


90.8 


127.8 


170.8 


222.8 


Vs 


14.7 


28.7 


46.9 


65.2 


92.9 


130.5 


174.3 


227.1 


M 


15.1 


29.4 


48.0 


66.7 


95.0 


133.3 


177.8 


231.4 


% 


15.5 


30.1 


49.1 


68.3 


97.2 


136.1 


181.3 


235.8 


I 


15.8 


30.8 


50.2 


69.9 


99.3 


138.9 


184.9 


240.1 


16.2 


31.5 


51.3 


71.4 


101.4 


141.7 


188.4 


244.5 


16.6 


32.2 


52.4 


73.0 


103.5 


144.4 


191.9 


248.8 


17.0 


32.9 


53.5 


74.5 


105.7 


147.2 


195.4 


253.2 


5 


17.4 


33.6 


54.5 


76.1 


107.8 


150.0 


198.9 


257.5 


i 


18.2 


35.0 


56.7 


79.2 


112.1 


155.6 


206.0 


266.2 


19.0 


36.4 


58.9 


82.3 


116.3 


161.1 


213.1 


274.9 


19.7 


37.8 


61.1 


85.5 


120.6 


166.7 


220.1 


283.6 


6 


20.5 


39.2 


63.2 


88.6 


124.8 


172.2 


227.1 


292.3 


7 


23.6 


44.7 


71.9 


101.1 


142.0 


194.5 


255.3 


327.1 


8 


26.8 


50.3 


80.6 


113.7 


158.9 


216.7 


283.4 


361.9 


9 


29.9 


55.9 


89.3 


126.2 


175.9 


239.0 


311.6 


396.6 


10 


33.0 


61.4 


98.0 


138.7 


193.0 


261.2 


339.7 


431.4 


12 


39.3 


72.5 


115.4 


163.7 


227.0 


305.7 


367.9 


501.0 



Weight of Two (2) Rivet Heads, 


in Pounds. 


.037 


.116 


.222 


.273 


.453 


.780 


1.16 


.032 


.082 


.147 


.246 


.369 


.545 


.746 



Weight of Body per Inch of Length, in Pounds. 

.031 i .056 | .087 I .125 I .170 I .223 I .282 



.348 



302 



Weight of Materials. 



Weight of Bolts per Hundred* 

Square Heads and Nuts. 
Dimensions in inches. 



Diameter. 



4 Vs I ty* V6 l% i 1 / 



Lb. 

3.9 
4.2 
4.6 
5.0 

5.4 
5.8 
6.2 
6.9 

7.6 
8.3 
9.0 
9.7 

10.4 
11.1 
11.8 
12.5 

13.2 



Lb. 

9.7 
10.5 
11.3 
12.1 

12.9 
13.7 
14.5 
16.1 

17.7 
19.2 
20.7 
22.2 

23.7 
25.2 

26.7 
28.2 

29.7 
33.1 
36.5 
40.0 

43.5 



Lb. 

20.4 
21.3 
22.4 
23.6 

25.0 
26.4 

27.8 
30.6 

33.4 
36.2 
39.0 

41.8 

44.6 
47.4 
50.2 
53.1 

56.0 
61.5 
67.0 
72.5 

78.0 
83.5 
89.0 
94.5 

100.0 
105.5 
111.0 



Lb. 

37.0 
37.9 
39.9 
42.0 

44.4 
46.2 
48.3 
52.5 

56.7 
60.9 
65.1 
69.2 

73.4 
77.6 
81.8 
86.0 

90.0 

98.0 

106.3 

114.6 

122.9 
131.2 
139.5 
148.0 

156.5 
165.0 
173.5 



Lb. 

58.0 
60.5 
63.2 
66.0 

69.0 
72.1 

75.2 
81.4 

87.6 

93.8 

100.0 

106.1 

112.2 
118.3 
124.4 
130.5 

136.6 
148.8 
161.0 
173.2 

184.4 
196.6 
208.8 
221.0 

233.2 
245.4 
257.6 



Lb. 



Lb. 



Lb 



97.7 
101.6 

105.6 
109.7 
113.8 
122.0 

130.2 
138.4 
146.6 
154.9 

163.2 
171.5 
179.8 
187.1 

195.4 

212.0 
229.0 
246.0 

263.0 
280.0 
297.0 
314.0 

331.0 
348.0 
365.0 



145 
149 

153 
158 
163 
174 

185 
196 
207 

218 

229 
240 
251 
262 

273 

295 
317 
339 

361 
383 
405 
427 

449 
471 
493 



Lb. 



200 
213 

226 
240 

255 
270 

285 
300 
315 
330 

345 
375 

405 
435 

465 
495 
525 
555 

585 
615 
645 



Lb 



289 
305 

322 
339 
356 
373 

390 
407 
434 
451 

468 
505 
540 
575 

610 
645 
680 
715 

750 

785 
820 



350 
370 

390 
410 
430 
450 

470 
490 
510 
530 

550 
590 
630 
670 

710 
751 
793 
835 

877 
919 
961 



Weight of Materials. 



303 



Weight of Bolts per Hundred. 

Hexagon Heads and Nuts. 
Dimensions in inches. 





Diameter. 


Length. 


























M 


% 


% 


% 


% 


Vs 


1 


tVs 


134 


iVs 


iy* 




Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


V/n 


3.4 


8.5 


17.7 


32.5 


49.0 














% 


3.7 


9.3 


18.6 


33.4 


51.5 














2 


4.1 


10.1 


19.7 


35.4 


54.2 


86.6 


128 










M 


4.5 


10.9 


20.9 


37.5 


57.0 


90.6 


132 










V* 


4.9 


11.7 


22.3 


39.9 


60.0 


94.6 


136 










% 


5.3 


12.5 


23.7 


41.7 


63.1 


98.7 


141 










3 


5.7 


13.3 


25.1 


43.8 


66.2 


102.8 


144 


174 


255 


310 


430 


X A 


6.1 


14.1 


26.6 


45.7 


69.4 


107.0 


151 


187 


271 


330 


450 


4 


6.8 


15.7 


29.4 


49.9 


75.6 


115.2 


162 


200 


288 


350 


470 


3^ 


7.5 


17.0 


32.2 


56.1 


81.8 


123.4 


173 


214 


305 


370 


495 


5 


8.2 


18.7 


35.0 


58.3 


88.0 


131.6 


184 


229 


322 


390 


520 


V* 


8.9 


20.2 


37.8 


62.4 


94.1 


139.9 


195 


244 


339 


410 


545 


6 


9.6 


21.7 


40.6 


66.6 


100.2 


148.2 


206 


259 


356 


430 


570 


X 


10.3 


23.2 


43.4 


70.8 


106.3 


156.5 


217 


274 


373 


450 


595 


7 


11.0 


24.7 


46.2 


75.0 


112.4 


164.8 


228 


289 


400 


470 


620 


% 


11.7 


26.2 


49.1 


79.2 


118.5 


172.1 


239 


304 


417 


490 


645 


8 


12.4 


27.7 


52.0 


83.2 


124.6 


180.3 


250 


319 


424 


510 


670 


9 




29.7 


54.8 


87.0 


130.8 


189.0 


262 


336 


465 


530 


700 


10 




33.1 


60.3 


95.0 


143.0 


206.0 


284 


366 


500 


570 


750 


11 




36.6 


65.8 


103.6 


155.2 


223.0 


306 


396 


535 


610 


800 


12 




40.1 


71.3 


111.9 


166.4 


240.0 


328 


426 


570 


650 


850 


13 






76.8 
82.3 


120.2 
128.5 


178.6 
190.8 


257.0 
274.0 


350 
372 


456 
486 


605 
640 


691 
733 


900 


14 






950 


15 






87.8 


137.0 


203.0 


291.0 


384 


516 


675 


775 


1000 


16 






93.3 

98.8 


145.5 
154.0 


215.2 
227.4 


308.0 
325.0 


416 

438 


546 
576 


710 
745 


817 
859 


1050 


17 






1100 


18 






104.3 


162.5 


239.6 


342.0 


460 


606 


780 


901 


1150 











304 



Screw Bolts. 



United States Standard Screw Threads. 



© 

a 
s 


© o 

3.2 


«4H 

o 

© ° o3 

S o 2 

5*"** 


08 

o 


'o 

o3 T3 
© o 


M © 

o3 "** 


■ d 

d bD 

.5 o 

ro *■■ 

O +j 

rd <0 


■ d 

d CQ 

,S © 

m 


. d 

d ^ 

5 s 
.g o 

M 


. -d 

d QC 

2 3 
2S ° 
-3 ^ 

O o 


© . 

© d 

*d o 


$ . 

dja 
H 


s 


y 


ill 


% 


§ 


% 


% 


© 


# 


no 


no 


In. 




Inch. 


Inch. 


Sq. in. 


Sq. in. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


In. 


% 


20 


.185 


.0062 


.049 


.027 


Y* 


A 


37 
"6¥ 


7 


^ 


A 


A 


18 


.240 


.0074 


.077 


.045 


19 
35 


H 


tt 


n 


A 


M 


% 


16 


.294 


.0078 


.110 


.068 


ft 


% 


M 


ti 


^ 


A 


A 


14 


.344 


.0089 


.150 


.093 


II 


II 


A 


w* 


A 


% 


K 


13 


.400 


.0096 


.196 


.126 


^ 


i§ 


l 


in 


K 


A 


1% 


12 


.454 


.0104 


.249 


.162 


31 
52 


29 
3 2 


V4 


i« 


A 


K 


« 


11 


.507 


.0113 


.307 


.202 


1A 


1 


1 7 


ik 


% 


A 


% 


10 


.620 


.0125 


.442 


.302 


1M 


1& 


1A 


in 


% 


H 


% 


9 


.731 


.0138 


.601 


.420 


ly 7 * 


1% 


i-M 


2A 


% 


ii 


1 


8 


.837 


.0156 


.785 


.550 


1% 


1A 


1% 


2M 


l 


if 


Vs 


7 


.940 


.0178 


.994 


.694 


Iff 


i% 


2A 


2A 


V/s 


1A 


u 


7 


1.065 


.0178 


1.227 


.893 


2 


lit 


2 T % 


2f| 


m 


1A 


Vs 


6 


1.160 


.0208 


1.485 


1.057 


2^ 


V/s 


017 
Z 32 


3^ 


Ws 


1A 


% 


6 


1.284 


.0208 


1.767 


1.295 


2% 


2A 


2% 


3§f 


VA 


1A 


% 


by 2 


1.389 


.0227 


2.074 


1.515 


2 T % 


VA 


931 
Z 32" 


3 5 ^ 


Ws 


1A 


% 


5 


1.491 


.0250 


2.405 


1.746 


2% 


m 


3A 


% 


1% 


m 


% 


5 


1.616 


.0250 


2.761 


2.051 


m 


2% 


Q13 
°32 


4A 


1% 


m 


2 


4K 


1.712 


.0277 


3.142 


2.302 


zy* 


3A 


3% 


W 


2 


m 


H 


4^ 


1.962 


.0277 


3.976 


3.023 


zy 2 


9 7 
°15 


*A 


m 


2^ 


2A 


y 


4 


2.176 


.0312 


4.909 


3.719 


oVs 


3H- 


±% 


m 


2^ 


2A 


% 


4 


2.426 


0312 


5.940 


4.620 


Vi 


4A 


m 


6 


2% 


214 


3 


3^ 


2.629 


.0357 


7.069 


5.428 


®A 


4A 


5% 


6Ji 


3 


2H 


M 


z% 


2.879 


.0357 


8.296 


6.510 


5 


411 


5B 


?A 


3K 


3A 


% 


&A 


3.100 


.03S4 


9.621 


7.548 


5% 


PL 5 
°T5 


6/ ? 


7-1! 


3^ 


3A 


% 


3 


3.317 


.0413 


11.045 


8.641 


5% 


5tt 


6§J 


8^ 


3% 


m 


4 


3 


3.567 


.0413 


12.566 


9.963 


6^ 


6A 


7& 


8ti 


4 


m 


& 


2% 


3.798 


.0435 


14.186 


11.329 


6K 


6A 


7A 


9A 


4^ 


4A 


% 


2% 


4.028 


.0454 


15.904 


12.753 


6% 


6H 


7fi 


9% 


4K 


4A 


% 


2% 


4.256 


.0476 


17.721 


14.226 


7^ 


?A 


8Ji 


1034 


4% 


4H 


5 


2^ 


4.480 


.0500 


19.635 


15.763 


% 


?A 


8?5 


lot* 


5 


4« 


Y± 


2^ 


4.730 


.0500 


21.648 


17.572 


8 


7« 


9A 


Hi! 


b% 


5A 


K 


2% 


4.953 


.0526 


23.758 


19.267 


8% 


8A 


9§f 


11% 


5% 


5A 


% 


2% 


5.203 


.0526 


25.967 


21.262 


8% 


8H 


10A 


u% 


5% 


5H 


6 


2M 


5.423 


.0555 


28.274 


23.098 


9^ 


9A 


10H 


121« 


6 


511 



Screw Bolts. 



305 



Whitworth Screw Bolts and Nuts. 



Size of 
bolt and 
thickness 

of nut. 


Number 
of threads 
per inch. 


Diameter 
at bottom 
of thread. 


Area at 

bottom of 

thread. 


Thickness 
of bolt head. 


Nut across 
plate. 


Nut across 
corners. 


Inch. 




Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Vs 


40 


.093 


.0067 


.109 


.338 


.390 


T% 


24 


.134 


.0141 


.164 


.448 


.517 


v± 


20 


.186 


.0271 


.219 


.525 


.606 


5 
13 


18 


.241 


.0458 


.273 


.601 


.694 


% 


16 


.295 


.0683 


.328 


.709 


.819 


7 
IB 


14 


.346 


.0940 


.383 


.820 


.947 


A 


12 


.393 


.1215 


.437 


.919 


1.06 


9 
16 


12 


.456 


.1633 


.492 


1.011 


1.16 


% 


11 


.508 


.2032 


.547 


1.101 


1.27 


H 


11 


.571 


.2560 


.601 


1.201 


1.38 


% 


10 


.622 


.3038 


.656 


1.301 


1.50 


H 


10 


.684 


.3674 


.711 


1.390 


1.60 


Vs 


9 


.733 


.422 


.766 


1.479 


1.70 


15 
16 


9 


.795 


.496 


.820 


1.574 


1.82 


1 


8 


.840 


.554 


.875 


1.670 


1.95 


Vs 


7 


.942 


.697 


.984 


1.860 


2.15 


% 


7 


1.067 


.894 


1.094 


2.048 


2.36 


% 


6 


1.161 


1.059 


1.203 


2.215 


2.55 


A 


6 


1.286 


1.30 


1.312 


2.413 


2.78 


Vs 


5 


1.369 


1.47 


1.422 


2.576 


2.97 


% 


5 


1.494 


1.75 


1.531 


2.758 


3.18 


Vs 


±A 


1590 


1.99 


1.641 


3.018 


3.48 


2 


^A 


1.715 


2.31 


1.750 


3.149 


3.63 


Vs 


±A 


1.840 


2.66 


1.859 


3.337 


3.85 


% 


4 


1.930 


2.92 


1.969 


3.546 


4.09 


y 8 


4 


2.055 


3.31 


2.078 


3.750 


4.33 


A 


4 


2.180 


3.73 


2.187 


3.894 


4.49 


Vs 


4 


2.305 


4.17 


2.297 


4.049 


4.67 


% 


%A 


2.384 


4.46 


2.406 


4.181 


4.82 


Vs 


3^ 


2.509 


4.92 


2.516 


4.346 


5.02 


3 


33^ 


2.634 


5.45 


2.625 


4.530 


5.23 



Whitworth threads are inclined at an angle of 55 degrees, and have 
one-sixth of the total depth of thread rounded off at the top and also at 
the bottom. 

The Whitworth system is the standard for Great Britain, and is also 
used extensively on the Continent pending the adoption of a satisfactory 
international metric screw thread system. In many of the leading ma- 
chine shops of France, Switzerland, Belgium, and Germany the bolts are 
made in English units with Whitworth threads, all other parts of the 
machines being in metric units. 

20 



306 



Pipe Standards. 



Wrought=iron Steam Pipe. 

United States Standard. 



Inner 
diameter. 


£ 

d 
M 

H 


o 

h 

03 

"S3 


o 

a 

2.1 

is 

H 


Inner 
diameter. 


DO 
03 

d 
M 


o 

a 

u 

03 


o 


"3 

a 

1 

o 
ft 


"3 

d 
o 


a 
1 


3 
O 
< 


as 

03 


Inch. 


Inch. 


Inch. 


Lb. 




Inch. 


Inch. 


Inch. 


Lb. 




% 


.270 


.068 


.24 


27 


VA 


4.508 


.246 


12.49 


8 


M 


.364 


.088 


.42 


18 


5 


5.045 


.259 


14.50 


8 


% 


.494 


.091 


.56 


18 


6 


6.065 


.280 


18.76 


8 


% 


.623 


.109 


.84 


14 


7 


7.023 


.301 


23.27 


8 


% 


.824 


.113 


1.12 


14 


8 


7.982 


.322 


28.18 


8 


1 


1.048 


.134 


1.67 


H^ 


9 


9.001 


.344 


33.70 


8 


% 


1.380 


.140 


2.24 


UK 


10 


10.019 


.366 


40.00 


8 


% 


1.611 


.145 


2.68 


UK 


11 


11.00 


.375 


45.00 


8 


2 


2.067 


.154 


3.61 


UK 


12 


12.00 


.375 


49.00 


8 


% 


2.468 


.204 


5.74 


8 


13 


13.25 


.375 


54.00 


8 


3 


3.067 


.217 


7.54 


8 


14 


14.25 


.375 


58.00 


8 


% 


3.548 


.226 


9.00 


8 


15 


15.25 


.375 


62.00 


8 


4 


4.026 


.237 


10.66 


8 













Whitworth or British Standard. 

Pipes having an internal diameter of % inch or % inch have 19 threads 
to the inch. Those of % inch, % inch, % inch, and % inch have 14 threads 
to the inch, and all otner sizes of pipes have 11 threads to the inch. 



03° 
S3 


Diameter. 


oJ 

N 

•a 
'3 

d 

1 

o 


Diameter. 


03* 

N 

•S3 

"3 

d 

1 

o 


Diameter. 


00 

"3 
a 

a 

o 


"3 

s 

03 

d 


"3 

e 

03 

"8 


"3 

d 

03 

d 


d 
u 

03 

H 


"3" 

d 
u 

03 

d 


'3 

d 
u 

03 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Vs 


.27 


.40 


2^ 


2.47 


2.87 


9 


8.94 


9.62 


H 


.36 


.54 


3 


3.07 


3.50 


10 


10.02 


10.75 


Vs 


.49 


.67 


K 


3.55 


4.00 


11 


11.00 


11.75 


V* 


.62 


.84 


4 


4.03 


4.50 


12 


12.00 


12.75 


% 


.82 


1.05 


% 


4.51 


5.00 


13 


13.25 


14.00 


1 


1.05 


1.31 


5 


5.04 


5.56 


14 


14.25 


15.00 


y* 


1.38 


1.66 


6 


6.06 


6.62 


15 


15.43 


16.00 


% 


1.61 


1.90 


7 


7.02 


7.62 


16 


16.40 


17.00 


2 


2.07 


2.37 


8 


7.98 


8.62 


17 


17.32 


18.00 



Cast-iron Pipe. 



307 



Standard Cast=iron Pipe. 

Metric System. 



Inside 
diam- 
eter. 


Thick- 
ness. 


Outside 
diameter. 


Weight 

per 
metre. 


Inside 
diam- 
eter. 


Thick- 
ness. 


Outside 
diameter. 


Weight 

per 
metre. 


Mm. 


Mm. 


Mm. 


Kilo. 


Mm. 


Mm. 


Mm. 


Kilo. 


40 


8.0 


56 


8.75 


375 


14.0 


403 


124.04 


50 


8.0 


66 


10.57 


400 


14.5 


429 


136.89 


60 


8.5 


77 


13.26 


425 


14.5 


454 


145.15 


70 


8.5 


87 


15.20 


450 


15.0 


480 


158.87 


80 


9.0 


98 


18.24 


475 


15.5 


506 


173.17 


90 


9.0 


108 


20.29 


500 


16.0 


532 


188.04 


100 


9.0 


118 


22.34 


550 


16.5 


583 


212.90 


125 


9.5 


144 


29.10 


600 


17.0 


634 


238.90 


150 


10.0 


170 


36.44 


650 


18.0 


686 


273.86 


175 


10.5 


196 


44.36 


700 


19.0 


738 


311.15 


200 


11.0 


222 


52.86 


750 


20.0 


790 


350.76 


225 


11.5 


248 


61.95 


800 


21.0 


842 


392.69 


250 


12.0 


274 


71.61 


900 


22.5 


945 


472.76 


275 


12.5 


300 


81.85 


1000 


24.0 


1048 


559.76 


. 300 


13.0 


326 


92.68 


1100 


26.0 


1152 


666.81 


325 


13.5 


352 


104.08 


1200 


28.0 


1256 


783.15 


350 


14.0 


378 


116.07 











Cast-iron Pipe. 

The following tables of dimensions and weights of standard cast-iron 
pipe for water mains are those adopted by the New England Water Works 
Association at its meeting in September, 1902. Ten classes of pipe are 
given, designated by the letters of the alphabet, the difference between 
the various classes being in the matter of thickness,— the inside diameter 
remaining constant for any size pipe, and the variation in thickness affect- 
ing the outside diameter. Table No. 1, herewith, gives the dimensions of 
the various sizes, and Table No. 2 gives the weights for standard 12-foot 
lengths of the various diameters and thicknesses. 

The various classes are required to stand hydrostatic tests, as follows : 

Pounds per square inch for diameters of 

20 inches Less than 

and larger. 20 inches. 

Class A 150 300 

Class B 200 300 

Class C 250 300 

Class D 300 300 

Class E 350 350 

Class F 350 350 



308 



Cast-iron Pipe. 



Table No. 1. 
General Dimensions of Pipes and Special Castings. 

.-.a — - 



n /? U 

3 I 5 !/f' 




Pipe-, +2-0— - 



JJ-y* 









Diam. of sockets. 


Depth of sockets. 






Nominal 
diameter. 


Classes. 


Actual 
outside 
diameter. 










u a" 




Pipe. 


Special 
castings. 


Pipe. 


Special 
castings. 


"b" 


Inch. 




Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch 


4 


A, C, E 


4.80 


5.60 


5.70 


3.0 


4.0 


1.50 


1.3 


4 


G, I, K 


5.00 


5.80 


5.70 


3.0 


4.0 


1.50 


1.3 


6 


A, C, E 


6.90 


7.77 


7.80 


3.0 


4.0 


1.50 


1.4 


6 


G, I 


7.10 


7.90 


7.80 


3.0 


4.0 


1.50 


1.4 


8 


A, C, E 


9.05 


9.85 


10.00 


3.5 


4.0 


1.50 


1.5 


8 


G, I 


9.30 


10.10 


10.00 


3.5 


4.0 


1.50 


1.5 


10 


A, B, C, D 


11.10 


11.90 


12.10 


3.5 


4.5 


1.50 


1.5 


10 


E, F, G, H 


11.40 


12.20 


12.10 


3.5 


4.5 


1.50 


1.5 


12 


A, B, C, D 


13.20 


14.00 


14.20 


3.5 


4.5 


1.50 


1.6 


12 


E, F, G, H 


13.50 


14.30 


14.20 


3.5 


4.5 


1.50 


1.6 


14 


A, B, C, D 


15.30 


16.10 


16.35 


3.5 


4.5 


1.50 


1.7 


14 


E, F, G, H 


15.65 


16.45 


16.35 


3.5 


4.5 


1.50 


1.7 


16 


A, B, C, D 


17.40 


18.40 


18.60 


4.0 


5.0 


1.75 


1.8 


16 


E, F, G, H 


17.80 


18.80 


18.60 


4.0 


5.0 


1.75 


1.8 


18 


A, B 


19.25 


20.25 


20.40 


4.0 


5.0 


1.75 


1.9 


18 


C,D 


19.50 


20.50 


20.40 


4.0 


5.0 


1.75 


1.9 


18 


E, F 


19.70 


20.70 


20.70 


4.0 


5.0 


1.75 


1.9 


20 


A, B 


21.30 


22.30 


22.50 


4.0 


5.0 


1.75 


2.0 


20 


C, D 


21.60 


22.60 


22.50 


4.0 


5.0 


1.75 


2.0 


20 


E, F 


21.90 


22.90 


23.00 


4.0 


5.0 


1.75 


2.0 


24 


A, B 


25.40 


26.40 


26.60 


4.0 


5.0 


2.00 


2.1 


24 


C,D 


25.80 


26.80 


26.60 


4.0 


5.0 


2.00 


2.1 


24 


E, F 


26.10 


27.10 


27.10 


4.0 


5.0 


2.00 


2.1 


30 


A, B 


31.60 


32.60 


32.60 


4.5 


5.0 


2.00 


2.3 


30 


C, D 


32.00 


33.00 


33.00 


4.5 


5.0 


2.00 


2.3 


30 


E, F 


32.40 


33.40 


33.40 


4.5 


5.0 


2.00 


2.3 


36 


A, B 


37.80 


38.80 


38.80 


4.5 


5.0 


2.00 


2.5 


36 


C, D 


38.30 


39.30 


39.30 


4.5 


5.0 


2.00 


2.5 


36 


E, F 


38.70 


39.70 


39.70 


4.5 


5.0 


2.00 


2.5 


42 


A, B 


44.00 


45.00 


45.00 


5.0 


5.0 


2.00 


2.8 


42 


C, D 


44.50 


45.50 


45.50 


5.0 


5.0 


2.00 


2.8 


42 


E, F 


45.10 


46.10 


46.10 


5.0 


5.0 


2.00 


2.8 


48 


A, B 


50.20 


51.20 


51.20 


5.0 


5.0 


2.00 


3.0 


48 


CD 


50.80 


51.80 


51.80 


5.0 


5.0 


2.00 


3.0 


48 


E, F 


51.40 


52.40 


52.40 


5.0 


5.0 


2.00 


3.0 


54 


A, B 


56.40 


57.40 


57.40 


5.5 


5.5 


2.25 


3.2 


54 


CD 


57.10 


58.10 


58.10 


5.5 


5.5 


2.25 


3.2 


54 


E, F 


57.80 


58.80 


58.80 


5.5 


5.5 


2.25 


3.8 


* 60 


A, B 


62.60 


63.60 


63.60 


5.5 


5.5 


2.25 


3.4 


60 


CD 


63.40 


64.40 


64.40 


5.5 


5.5 


2.25 


3.4 


60 


E, F 


64.20 


65.20 


65.20 


5.5 


5.5 


2.25 


4.0 



Cast-iron Pipe. 



309 



as en ^ ^ co co to to mhmm m a 
o^ooto osorf^o ccas^to ooooi*> g 

p 


Nominal diam- 
eter of pipe. 


t -1 !"* . p 1 
f- a ococo Vj^rosos bibbilft". Jp>. Jtx co co 2- 
O CO Cn ^7 O M M O -J en co <o -J tO GO M p 


Thickness 
of shell. 


o 

go" 

> 


CD <l OS rfU CO tO tO M HHH l_l 

CO en >-» co 00CDO05 »M\jO00 Os ►£». CO tO ~ 
OH- 'COO OClO'M h- ' h- 'Ml— i CJiMCOO C^ 

oooo oooo ocnoo o en o o • 


Weight per 
length. 


M J- 1 M M # m ■ .'.'.'.£? 

co to m o bco<tCTi bibbicn en" • • 2, 
OOOO OMtOO co o ^7 co o • • • p 


Thickness 
of shell. 


a 
w 


M • • 
OCOOen rf>wWbOH-« 1 — ' t— ' 1 — ' ... ,_, 

co ci o en to to to —7 en co o oo as ■ • • ~ 

O O -J OS ~~7 CO CO O M M CO OS oc • • • cr* 

oooo oooo ooenen en- • • 


Weight per 
length. 


h-» J-* M M M^ # s* 

en CO to M O eO CO <I OS as OS On en *■. M CO 2- 

o^ienco to i— ' o to tooiHvi co oo to as p 


Thickness 
of shell. 


Q 

Q 


H0<105 t^COtOM Ml— ' M i_j 

o oo co to oc as en co as m i— ' o <ioio:to tj 
ooo-a coocoto c h oi to to to en i— » c^ 
OOOO OOOO OOCJiO enenenen • 


Weight per 
length. 


l"" 1 t"" 1 1" 1 1" 1 t -1 1" 1 . .11! o 
<i en m to M O 00 <T ^7 ^7 as OS en • • • 2- 

o m o -<r WHOoto en o o i— L as • • • p 


Thickness 
of shell. 


p 


M M ... 

COOOOOS CnCOtOtO M M I- 1 ... ,. 
CO CO -7 CO CO CO --7 O <J On tO O <l ■ • • E 

oooovi ooi<too co o co oo o • • • cr 

OOOO OOOO OOOO en- • • 


Weight per 
length. 


M 
M M H» h-* M £-* # . __ B 

co ^ en lu. "to m co bo co^i^to as en m bo 2^ 

OtOCnO enoenen OenOen C05 0KO p 


Thickness 
of shell. 


Q 


en to co --7 on m co to m h ' h- 1 m i_i 




H-'co^i^r cocooto coascoo oo en co to ^ 
oen^o otooen m o o co o ^t oo co a - 
oooo oooo oooen eneneno • 


length. 


tO M M M M M M # . I .* I P 

m co ^7 en co "to o co bo bo ^-7 o as • ' • 2- 

oooco <i o co to oooko co • • • p 


Thickness 
of shell. 


Q 


a. co o oo as rf^ co to to m m m '.'.'. H 

en en os co w r. tc ^ Cvjo;o co • • • C7 
oooen oooifch l m o co co . **s>. . . . a 1 
oooo oooo oooo en- • . 


Weight per 
length. 


h- 1 
• • • • • . . . .... M 

co --7 ^j as en en ^ q 

III! *. '.'.'. ^ On cO CO MCOOtO p- 


Thickness 
of shell. 


Q 

P 


• • . . . . . . . (-1 H- 4 1— l l_j 

• ••• corfiM oo as m to £j 

• • . • .... . o en en 00 CO i— ' M C^ 
ooo enoen cl- 


Weight per 
length. 


inch. 

'.70 

.77 
.83 
.90 


Thickness 
of shell. 


9 

w 


CO en tO CO • • • £j 

cwm to • • • cr 

• ; • OOO en • • • 


Weight per 
length. 


• as en jsx § 

: \ : : : : : : : : : : : co^en £ 


Thickness 
of shell. 


9 
p" 

i— i 


' • • • '.'.'.'. '.'.'.'. I as ^ to t"J 

co^asa' 

.... .... .... • en o o • 


Weight per 
length. 


...... M 

:::.: : : : : \ : I: ::-k| 


Thickness 
of shell. 




; : i i ; i ! ; Mi; ! ; ! « g 

; f • • c/< * 


Weight per 
length. 



s 





n 

PC 


n 


s 


<-k 


fD 




r/) 


3 


V) 


r 

n 


o 

CA 


3 


p 


rt- 


3 


F 


a 




3 



H 

► 



o 



orq 



CO 



3 



CO 



310 



Cast-iron Pipe. 



Sizes and Weights of Cast=iron Pipe Connections. 



Crosses. 



Inch. 


Lb. 


2 


40 


3 


104 


3X2 


90 


4 


150 


4X 3 


114 


4X 2 


110 


6 


200 


6X 4 


150 


6X 3 


150 


8 


325 


8X 6 


265 


8X 4 


265 


8X 3 


225 


10 


510 


10 X 8 


415 


10 X 6 


388 


10 X 4 


338 


10 X 3 


350 


12 


700 


12 X 10 


650 


12 X 8 


615 


12 X 6 


540 


12 X 4 


525 


12 X 3 


495 


14 X 10 


750 


14 X 8 


635 


14 X 6 


570 


16 


1025 


16 X 14 


1070 


16X12 


1025 


16 X 10 


1010 


16 X 8 


825 


16 X 6 


700 


16 X 4 


650 


20 


1790 


20 X 12 


1370 


20X10 


1225 


20 X 8 


1000 


20 X 6 


1000 


20 X 4 


1000 


24 


2190 


24 X 20 


2020 


24 X 6 


1340 


30X20 


2635 


30 X 12 


2250 


30 X 8 


1995 



Tees. 



Inch. 

2 

3 

3X 2 

4 

4X 

4X 

6 

6X 

6X 

6X 



8X 



10 

10 X 8 
10 X 6 
10 X 4 
10 X 3 
12 

12 X 10 
12 X 8 
12 X 6 
12 X 4 
14X12 
14 X 10 
14 X 8 
14 X 6 
14 X 4 
14 X 3 
16 

16X14 
16 X 12 
16 X 10 
16 X 8 
16 X 6 
16 X 4 
20 

20 X 16 
20 X 12 
20 X 10 
20 X 8 
20 X 6 

20 X 4 

21 X 10 
24 

24 X 12 
24 X 8 
24 X 6 
30 

30 X 24 
30 X 20 
30 X 12 
30 X 10 
30 X 6 
36 

36 X 30 
36 X 12 



Lb. 

28 

76 

76 

100 

90 

87 

150 

130 

125 

120 

266 

252 

222 

220 

390 

330 

312 

292 

290 

565 

510 

492 

484 

460 

650 

650 

575 

545 

525 

490 

790 

850 

825 

890 

755 

630 

655 

1375 

1115 

1025 

1090 

900 

875 

845 

1465 

1875 

1425 

1375 

1375 

3025 

2640 

2200 

2035 

2050 

1825 

5140 

4200 

4050 



45° Branch Pipes. 



Inch. 

3 

6X 6X 4 

8 

8X 6 
24 

24 X 24 X 20 
30 
36 



Lb. 

90 

145 

300 

290 

2765 

2145 

4170 

10300 



Sleeves. 



Inch. 

2 

3 

4 

6 

8 

10 
12 
14 
16 
20 
24 
30 
36 



Lb. 

10 

20 

44 

65 

86 

140 

176 

208 

340 

500 

710 

965 

1500 



90° Elbows. 



Inch. 

2 

3 

4 

6 

8 
10 
12 
14 
16 
20 
21 



Lb. 

14 

34 
48 
110 
145 
225 
370 
150 
525 
900 
1100 



Plugs. 


Inch. 


Lb. 


2 


2 


3 


5 


4 


8 


6 


12 


8 


26 


10 


46 


12 


66 


14 


70 


16 


100 


20 


150 


24 


185 


30 


370 


Vs or 


45° 


Bends. 


Inch. 


Lb. 


3 


30 


4 


65 


6 


85 


8 


160 
190 


10 


12 


290 


16 


510 


20 


740 


24 


1425 


30 


2000 


i\ or 


22%° 


Ben 


ds. 


Inch. 


Lb. 
150 


6 


8 


155 


10 


165 


12 


260 


16 


500 


24 


1280 


30 


1735 



Reducers. 



Inch. 

3X 
4X 
4X 
6X 
6X 
8X 
8X 
8X 
10 X 
10 X 
10 X 
12 X 
12 X 
12 X 
12 X 
14 X 
14 X 
14 X 
14 X 
16 X 
16 X 
20 X 
20 X 
20 X 
20 X 
24 X 
30 X 
30 X 
36 X 



Lb. 

35 
42 
40 



126 
116 
116 
212 
150 
128 
278 
254 
250 
250 
475 
430 
340 
285 
475 
435 
690 
575 
540 
300 
745 
1305 
1385 
1730 



Caps. 



Inch. Lb. 



10 
12 



15 
25 
60 
75 
100 
120 



Drip- 
boxes. 



Inch. 

4 

8 

10 
20 



Lb. 

235 

355 

760 

1420 



Boiler Tubes. 



311 



Lap=welded American Charcoal Iron Boiler Tubes. 

Tables of Standard Sizes. 
IK? Morris, Tasker & Co. 



a 


a 








© 

ft 


© 




si 








C3 3 


o g 


S 4 *! 


Uo2 


S-i 
03 


© 

c3 


© 


"eS 






rs 2 


r3 ^ 


ssac 


O 1 g « 


_i 


«"g 


p< 


K"5 


c3 


M 

o 

3 


& a 






+J QQ -.03 


© 

"a 


1 


© o 


S 


H 


H 


H 




h! 


h-3 


H 


H 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Feet. 


Feet. 


Inch. 


Inch. 


Lb. 


1 


.856 


.072 


3.142 


2.689 


4.460 


3.819 


.575 


.785 


.708 


1/ 


1.106 


.072 


3.927 


3.474 


3.455 


3.056 


.960 


1.227 


.900 


^ 


1.334 


.083 


4.712 


4.191 


2.863 


2.547 


1.396 


1.767 


1.250 


1.560 


.095 


5.498 


4.901 


2.448 


2.183 


1.911 


2.405 


1.665 


2 


1.804 


.098 


6.283 


5.667 


2.118 


1.909 


2.556 


3.142 


1.981 


M 


2.054 


.098 


7.069 


6.484 


1.850 


1.698 


3.314 


3.976 


2.238 


^ 


2.283 


.109 


7.854 


7.172 


1.673 


1.528 


4.094 


4.909 


2.755 


y 4 


2.533 


.109 


8.639 


7.957 


1.508 


1.390 


5.039 


5.940 


3.045 


3 


2.783 


.109 


9.425 


8.743 


1.373 


1.273 


6.083 


7.069 


3.333 


¥ 


3.012 


.119 


10.210 


9.462 


1.268 


1.175 


7.125 


8.296 


3.958 


X 


3.262 


.119 


10.995 


10.248 


1.171 


1.091 


8.357 


9.621 


4.272 


3 4 


3.512 


.119 


11.781 


11.033 


1.088 


1.018 


9.687 


11.045 


4.590 


4 


3.741 


.130 


12.566 


11.753 


1.023 


.955 


10.992 


12.566 


5.320 


V, 


4.241 


.130 


14.137 


13.323 


.901 


.849 


14.126 


15.904 


6.010 


5 


4.720 


.140 


15.708 


14.818 


.809 


.764 


17.497 


19.635 


7.226 


6 


5.699 


.151 


18.849 


17.904 


.670 


.637 


25.509 


28.274 


9.346 


7 


6.657 


.172 


21.991 


20.914 


.574 


.545 


34.805 


38.484 


12.435 


8 


7.636 


.182 


25.132 


23.989 


.500 


.478 


45.795 


50.265 


15.109 


9 


8.615 


.193 


28.274 


27.055 


- .444 


.424 


58.291 


63.617 


18.002 


10 


9.573 


.214 


31.416 


30.074 


.399 


.382 


71.975 


78.540 


22.190 



Wrought=iron Welded Tubes. 

Extra strong. 



a 




C 
o 


02 

u 


60 

3 . P 


1 .* 


"3 ^ 


° ~© 

s a 




QQ © 


32 sh >i 

C ® w 

~ 2 S 

c« a +3 


02 S-, <U 

5-2 © • 

" ©^ bfl 

*aso 


B 2 


•5 .5 
o ,r 3 


2 x 

2h 


5A£ 


5 5 M 


3 - o s 


fc 


< 


H 


H 


<! 


<! 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


H 


.405 
.540 
.675 
.840 


.100 
.123 
.127 
.149 




.205 
.294 
.421 
.542 








V 






if 


.298 


.244 


1.050 


.157 


.314 


.736 


.422 


1 


1.315 


.182 


.364 


.951 


.587 


M 


1.660 


.194 


.388 


1.272 


.884 


V2 


1.900 


.203 


.406 


1.494 


1.088 


2 


2.375 


.221 


.442 


1.933 


1.491 


y 2 


2.875 


.280 


.560 


2.315 


1.755 


3 


3.5 


.304 


.608 


2.892 


2.284 


V2 


4.0 


.321 


.642 


3.358 


2.716 


4 


4.5 


.341 


.682 


3.818 


3.136 



312 



Wire Gauges. 



Different Standards for Wire Gauge in Use in the 
United States. 





Dimensions in decimal parts of an 


inch. 




Number 
of wire 
gauge. 


American, 

or Brown & 

Sharpe. 


Birming- 
ham, 
or Stubs'. 


Washburn & 
Moen Mfg. 

Co., Worces- 
ter, Mass. 


Trenton 

Iron Co., 

Trenton, 

N. J. 


United °\ 

States ns 

Standard, -yr. 


d Eng- 
li, from 
Brass 
rs. List. 


000000 






.46 
.43 




.46875 
.43750 




00000 






.45 




0000 


.460 000 


.454 


.393 


.40 


.40625 




000 


.409 640 


.425 


.362 


.36 


.37500 




00 


.364 800 


.380 


.331 


33 


.34375 







.324 950 


.340 


.307 


.305 


.31250 




1 


.289 300 


.300 


.283 


.285 


.28125 




2 


.257 630 


.284 


.263 


.265 


.26563 




3 


.229 420 


.259 


.244 


.245 


.25000 




4 


.204 310 


.238 


.225 


.225 


.23438 




5 


.181 940 


.220 


.207 


.205 


.21875 




6 


.162 020 


.203 


.192 


.190 


.20313 




7 


.144 280 


.180 


.177 


.175 


.18750 




8 


.128 490 


.165 


.162 


.160 


.17188 




9 


.114 430 


.148 


.148 


.145 


.15625 




10 


.101 890 


.134 


.135 


.130 


.14063 




11 


.090 742 


.120 


.120 


.1175 


.12500 




12 


.080 808 


.109 


.105 


.1050 


.10938 




13 


.071 961 


.095 


.092 


.0925 


.09375 




14 


.064 084 


.083 


.080 


.0800 


.07813 


083 


15 


.057 068 


.072 


.072 


.0700 


.07031 


072 


16 


.050 820 


.065 


.063 


.0610 


.06250 


065 


17 


.045 257 


.058 


.054 


.0525 


.05625 


058 


18 


.040 303 


.049 


.047 


.0450 


.05000 


049 


19 


.035 390 


.042 


.041 


.0390 


.04375 


040 


20 


.031 961 


.035 


.035 


.0340 


.03750 


035 


21 


.028 462 


.032 


.032 


.0300 


.03438 


0315 


22 


.025 347 


.028 


.028 


.0270 


.03125 


0295 


23 


.022 571 


.025 


.025 


.0240 


.02813 


0270 


24 


.020 100 


.022 


.023 


.0215 


.02500 


0250 


25 


.017 900 


.020 


.020 


.0190 


.02188 


0230 


20 


.015 940 


.018 


.018 


.0180 


.01875 


0205 


27 


.014 195 


.016 


.017 


.0170 


.01719 


01875 


28 


.012 641 


.014 


.016 


.0160 


.01563 


01650 


29 


.011 257 


.013 


.015 


.0150 


.01406 


01550 


30 


.010 025 


.012 


.014 


.0140 


.01250 


01375 


31 


.008 928 


.010 


.0135 


.0130 


.01094 


01225 


82 


.007 950 


.009 


.0130 


.0120 


.01016 


01125 


33 


.007 080 


.008 


.0110 


.0110 


.00938 


01025 


34 


.006 304 


.007 


.0100 


.0100 


.00859 


00950 


86 


.005 614 


.005 


.0095 


.0090 


.00781 


00900 



Weight of Wire. 



313 



Wire. — Iron, Steel, Copper, Brass. 

Weight, in Pounds, of 100 Feet. 
Birmingham Wire Gauge. 







Per 100 lineal feet. 




Number of 








gauge. 


Iron. 


Steel. 


Copper. 


Brass. 




Lb. 


Lb. 


•Lb. 


Lb. 


0000 


54.62 


55.13 


62.39 


58.93 


000 


47.86 


48.32 


54.67 


51.64 


00 


38.27 


38.63 


43.71 


41.28 





30.63 


30.92 


34.99 


33,05 


1 


23.85 


24.07 


27.24 


25.73 


2 


21.37 


21.57 


24.41 


23.06 


3 


17.78 


17.94 


20.30 


19.18 


4 


15.01 


15.15 


17.15 


16.19 


5 


12.82 


12.95 


14.65 


13.84 


6 


10.92 


11.02 


12.47 


11.78 


7 


8.586 


8.667 


9.807 


9.263 


8 


7.214 


7.283 


8.241 


7.783 


9 


5.805 


5.859 


6.630 


6.262 


10 


4.758 


4.803 


5.435 


5.133 


11 


3.816 


3.852 


4.359 


4.117 


12 


3.148 


3.178 


3.596 


3.397 


13 


2.392 


2.414 


2.732 


2.580 


14 


1.826 


1.843 


2.085 


1.969 


15 


1.374 


1.387 


1.569 


1.482 


16 


1.119 


1.130 


1.279 


1.208 


17 


.8915 


.900 


1.018 


.9618 


18 


.6363 


.6423 


.7268 


.6864 


19 


.4675 


.4720 


.5340 


.5043 


20 


.3246 


.3277 


.3709 


.3502 


21 


.2714 


.2740 


.3100 


.2929 


22 


.2079 


.2098 


.2373 


.2241 


23 


.1656 


.1672 


.1892 


.1788 


24 


.1283 


.1295 


.1465 


.1384 


25 


.1060 


.1070 


.1211 


.1144 


26 


.0859 


.0867 


.0981 


.0926 


27 


.0678 


.0685 


.0775 


.0732 


28 


.0519 


.0524 


.0593 


.0560 


29 


.0448 


.0452 


.0511 


.0483 


30 


.0382 


.0385 


.0436 


.0412 


31 


.0265 


.0267 


.0303 


.0286 


32 


.0215 


.0217 


.0245 


.0231 


33 


.0170 


.0171 


.0194 


.0183 


34 


.0130 


.0131 


.0148 


.0140 


35 


.0066 


.0067 


.0076 


.0071 


36 


.0042 


.0043 


.0048 


.0046 



314 



Sheet-metal Gauges. 



United States Standard Gauge for SYieet= 
iron and Steel, 1893. 



and Plate- 





Cm 

• © 


.* 


M 


© 


© 


CD 


© 


© ID 




.£ 


.2-* ,3 




u 


U 




Sh 


(-> »rt 


bo 

fee 

o 


~.2 

03 © 

11 


H O O 


2 

© 0Q 

« s 

S d "§ 


c3 m 

3 © 

3h © 
3d 3 oJ 
u 2 o 

a 3 3 


c$ so 

c* a . 

cc 3 02 

s-. o"3 

© P- o. 

A *2 


3 

a 

s a 
£\5 1 


e8 

Hi 


So 

is 3. 


© 


* ,= 2 


£ — ' ° 


+3-3 -p 


■+f .5 -d 






■^ oT'd 


^3 


O „rfl 


© »r+S 


o _r;3 


A -.£j 


•^ -i 


,3 "1 Sb 


A u bJO 


-c 2 « 


6 


a a; 3 




k x H 

P< <D '3 

©<3 a 

< 


© o > 






£S3 


03 fl ^ 


0000000 


K 


.500 000 


12.700 000 


320 


20.000 


9.072 


97.65 


215.28 


000000 


§§ 


.468 750 


11.906 250 


300 


18.750 


8.505 


91.55 


201.82 


00000 


A 


.437 500 


11.112 500 


280 


17.500 


7.938 


85.44 


188.37 


0000 


M 


.406 250 


10.318 750 


260 


16.250 


7.371 


79.33 


174.91 


000 


3 / 8 


.375000 


9.525000 


240 


15.000 


6.804 


73.24 


161.46 


00 


M 


.343 750 


8.731 250 


220 


13.750 


6.237 


67.13 


148.00 





A 


.312 500 


7.937 500 


200 


12.500 


5.670 


61.03 


134.55 


1 


A 


.281 250 


7.143 750 


180 


11.250 


5.103 


54.93 


121.09 


2 


H 


.265 625 


6.746 875 


170 


10.625 


4.819 


51.88 


114.37 


3 


fc 


.250 000 


6.350 000 


160 


10.000 


4.536 


48.82 


107.64 


4 


M 


.234 375 


5.953125 


150 


9.375 


4.252 


45.77 


100.91 


5 


A 


.218 750 


5.556 250 


140 


8.750 


3.969 


42.72 


94.18 


6 


if 


.203 125 


5.159 375 


130 


8.125 


3.6S5 


39.67 


87.45 


7 


A 


.187 500 


4.762 500 


120 


7.500 


3.402 


36.62 


80.72 


8 


tt 


.171875 


4.365 625 


110 


6.875 


3.118 


33.57 


74.00 


9 


5 
32 


.156 250 


3.968 750 


100 


6.250 


2.835 


30.52 


67.27 


10 


A 


.140 625 


3.571 875 


90 


5.625 


2.552 


27.46 


60.55 


11 


>8 


.125 000 


3.175 000 


80 


5.000 


2.268 


24.41 


53.82 


12 


7 


.109 375 


2.778 125 


70 


4.375 


1.984 


21.36 


47.09 


13 


A 


.093 750 


2.381250 


60 


3.750 


1.701 


18.31 


40.36 


14 


A 


.078125 


1.984 375 


50 


3.125 


1.417 


15.26 


33.64 


15 


T5S 


.070 312 500 


1.785 937 500 


45 


2.812 500 


1.276 


13.73 


30.27 


16 


A 


.062500 000 


1.587 500 000 


40 


2.500 000 


1.134 


12.21 


26.91 


17 


T6tf 


.056 250 000 


1.428 750 000 


36 


2.250 000 


1.021 


10.99 


24.22 


18 


A 


.050 000 000 


1.270 000 000 


32 


2.000 000 


.9072 


9.765 


21.53 


19 


7 


.043 750 000 


1.111250000 


28 


1.750 000 


.7938 


8.544 


18.84 


20 


5% 


.037 500 000 


.952 500 000 


24 


1.500 000 


.6804 


7.324 


16.15 


21 


oV* 


.034375 000 


.873 125 000 


22 


1.375 000 


.6237 


6.713 


14.80 


22 


A 


.031 250 000 


.793 750 000 


20 


1.250 000 


.5670 


6.103 


13.46 


23 


9 
35ff 


.028 125 000 


.714 375 000 


18 


1.125 000 


.5103 


5.493 


12.11 


24 


A 


.025000 000 


.635 000 000 


16 


1.000 000 


.4536 


4.882 


10.76 


25 


7 


.021 875 000 


.555 625 000 


14 


.875 000 


.3969 


4.272 


9.42 


26 


t3?t 


.018 750 000 


.476 250 000 


12 


.750 000 


.3402 


3.662 


8.07 


27 


Afe 


.017 187 500 


.436 562 500 


11 


.687 500 


.3119 


3.357 


7.40 


28 


*v 


.015 625 000 


.396875 000 


10 


.625 000 


.2835 


3.052 


6.73 


29 


9 


.014 062 500 


.357 187 500 


9 


.562 500 


.2551 


2.746 


6.05 


30 


A 


.012500 000 


.317 500 000 


8 


.500 000 


.2268 


2.441 


5.38 


31 


CI(T 


.010 937 500 


.277 812 500 


7 


.437 500 


.1984 


2.136 


4.71 


32 


T2S?7 


.010 156 250 


.257 968 750 


Gy 2 


.406 250 


.1843 


1.983 


4.37 


33 


3§ry 


.009 375000 


.238 125 000 


6 


.375 000 


.1701 


1.831 


4.04 


34 


T25JT 


.008 593 750 


.218 281 250 


&A 


.343 750 


.1559 


1.678 


3.70 


35 


5 


.007 812 500 


.198 437 500 


5 


.312 500 


.1417 


1.526 


3.36 


" 36 


1 'J HO 


.007 031 250 


.178593750 


4^ 


.281 250 


.1276 


1.373 


3.03 


37 


55 (J5 


.006 640 625 


.168 671875 


41 


.265 625 


.1205 


1.297 


2.87 


38 


T&5 


.006 250000 


.158 750 000 


4 


.250 000 


.1134 


1.221 


2.69 



Chains. 



315 



Crane Chains. 



CO '• 



[MK*;A-*-A** 



"D. B. Gr." Special Crane. 


Crane. 


a 
'3 

o 
%-( 

o 
© 


<i © 

£ ft 

ft 


o 

- ft 

© Qj . 


© 

o 


© 

o 
o 

ft 


© • 
© g 


"3 

<2 © 


© 

O 
O 
u 

ft 


M 

© • 

© i 

S-t © 

£ bC 

5* 


©1 

cS . 

111 

O 


Inch. 


Inch. 


Lb. 


Inch. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


% 


25 
32 


% 


Vs 


1932 


3 864 


1288 


1680 


3 360 


1120 


ft 


if 


1 


1ft 


2 898 


5 796 


1932 


2 520 


5 040 


1680 


Vs 


tt 


1* 


1M 


4 186 


8 372 


2 790 


3 640 


7 280 


2 427 


ft 


1 5 


2 


1% 


5 796 


11592 


3 864 


5 040 


10 080 


3 360 


K 


1H 


2^ 


1« 


7 728 


15 456 


5182 


6 720 


13 440 


4 480 


ft 


1» 


3ft 


1% 


9 660 


19 320 


6 440 


8 400 


16 800 


5 600 


% 


Iff 


4^ 


2ft 


11914 


22 828 


7 942 


10 360 


20 720 


6 907 


H 


1M 


5 


2M 


14 490 


28 980 


9 660 


12 600 


25 200 


8 400 


% 


l» 


5% 


2^ 


17 388 


34 776 


11592 


15120 


30 240 


10 080 


if 


2& 


6ft 


2H 


20 286 


40 572 


13 524 


17 640 


35 280 


11760 


% 


9 7 
Z 32 


8 


2% 


22 484 


44 968 


14 989 


20 440 


40 880 


13 627 


ft 


2M 


9 


3ft 


25 872 


51744 


17 248 


23 520 


47 040 


15 680 


1 


2» 


10ft 


3^ 


29 568 


59136 


19 712 


26 880 


53 760 


17 920 


ft 


z 32 


lift 


Q 5 
°T5 


33 264 


66 538 


22176 


30 240 


60 480 


20160 


% 


^32 


12% 


3% 


37 576 


75152 


25 050 


34160 


68 320 


22 773 


A 


Q 5 
°32 


1% 


3% 


41888 


83 776 


27 925 


38 080 


76160 


25 387 


% 


Q 7 
°35 


16 


4^ 


46 200 


92 400 


30 800 


42 000 


84 000 


28 000 


ft 


Q15 

d 35 


163^ 


4% 


50 512 


101 024 


33 674 


45 920 


91840 


30 613 


% 


3^ 


ISA 


4ft 


55 748 


111 496 


37165 


50 680 


101 360 


33 787 


ft 


3M 


19ft 


4% 


60 368 


120 736 


40 245 


54 880 


109 760 


36 587 


K 


3§i 


a* 


5 


66 528 


133 056 


44 352 


60 480 


120 960 


40 320 



The distance from centre of one link to centre of next is equal to the 
inside length of link, but in practice ^ inch is allowed for weld. This is 
approximate, and, where exactness is required, chain should be made so. 

For Chain Sheaves.— The diameter, if possible, should be not less than 
twenty times the diameter of chain used. 

Example. For 1-inch chain use 20-inch sheaves. 



316 



Window Glass. 



Window Glass. 

Number of Lights per Box of 50 Feet. 



Inch. 


No. 


Inch. 


No. 


Inch. 


No. 


Inch. 


No. 


6X 8 


150 


12 X 18 


33 


16X44 


10 


26X32 


9 


7X9 


115 


12 X 20 


30 


18 X 20 


20 


26 X 34 


8 


8X10 


90 


12 X 22 


27 


18 X 22 


18 


26 X 36 


8 


8X11 


82 


12 X 24 


25 


18 X 24 


17 


26 X 40 


7 


8X12 


75 


12X26 


23 


18X26 


15 


26 X42 


7 


8X13 


70 


12 X 28 


21 


18 X 28 


14 


26X44 


6 


8X14 


64 


12 X 30 


20 


18 X 30 


13 


26X48 


6 


8X15 


60 


12 X 32 


18 


18 X 32 


13 


26 X 50 


6 


8X16 


55 


12 X 34 


17 


18 X 34 


12 


26 X 54 


5 


9XH 


72 


13X14 


40 


18 X 36 


11 


26 X58 


5 


9X12 


67 


13X16 


35 


18X38 


11 


28X30 


9 


9X13 


62 


13 X 18 


31 


18X40 


10 


28 X 32 


8 


9X14 


57 


13 X 20 


28 


18 X44 


9 


28X34 


8 


9X15 


53 


13X22 


25 


20 X 22 


16 


28 X 36 


7 


9X16 


50 


13X24 


23 


20 X 24 


15 


28 X 38 


7 


9X17 


47 


13 X 26 


21 


20 X 26 


14 


28X40 


6 


9X 18 


44 


13X28 


19 


20 X 28 


13 


28 X 44 


6 


9X20 


40 


13X30 


18 


20 X 30 


12 


28X46 


6 


10 X 12 


60 


14X16 


32 


20 X 32 


11 


28 X 50 


5 


10 X 13 


55 


14 X 18 


29 


20X34 


11 


28 X 52 


5 


10 X 14 


52 


14X20 


26 


20X36 


10 


28X56 


4 


10 X 15 


48 


14 X 22 


23 


20 X 38 


9 


30 X 36 


7 


10 X 16 


45 


14 X 24 


22 


20X40 


9 


30 X 40 


6 


10 X 17 


42 


14 X 26 


20 


20 X 44 


8 


30 X 42 


6 


10X18 


40 


14 X28 


18 


20X46 


8 


30 X 44 


5 


10X20 


36 


14 X 30 


17 


20X48 


8 


30 X 46 


5 


10 X 22 


33 


14X32 


16 


20X50 


7 


30 X48 


5 


10X24 


30 


14 X 34 


15 


20 X 60 


6 


30 X 50 


5 


10X26 


28 


14 X 36 


14 


22 X 24 


14 


30X54 


4 


10X28 


26 


14 X 40 


13 


22 X 26 


13 


30 X 56 


4 


10X30 


24 


14 X 44 


11 


22 X 28 


12 


30X60 


4 


10 X 32 


22 


15 X 18 


27 


22 X 30 


11 


32 X 42 


5 


10 X 34 


21 


15 X 20 


24 


22 X 32 


10 


32X44 


5 


11X13 


50 


15 X 22 


22 


22 X 34 


10 


32 X 46 


5 


11X14 


47 


15 X 24 


20 


22 X 36 


9 


32 X 48 


5 


11X15 


44 


15 X 26 


18 


22 X 38 


9 


32X50 


4 


11X16 


41 


15 X 28 


17 


22 X 40 


8 


32 X 54 


4 


11X17 


39 


15 X 30 . 


16 


22 X 44 


8 


32 X 56 


4 


11X18 


36 


15 X 32 


15 


22 X 46 


7 


32 X 60 


4 


11X20 


33 


16 X 18 


25 


22X50 


7 


34 X 40 


5 


11 X 22 


30 


16 X 20 


23 


24 X 28 


11 


34X44 


5 


11 X 24 


27 


16 X 22 


20 


24 X 30 


10 


34 X 46 


5 


11 X 26 


25 


16 X 24 


19 


24 X 32 


9 


34 X 50 


4 


11 X 28 


23 


16 X 26 


17 


24 X 36 


8 


34 X 52 


4 


11 X 30 


21 


16 X 28 


16 


24 X 40 


8 


34 X 56 


4 


11X32 


20 


16 X 30 


15 


21 X 44 


7 


36 X 44 


5 


11X34 


19 


16 X 32 


14 


24 X 46 


7 


36 X 50 


4 


12 X 14 


43 


16 X 34 


13 


24 X 48 


6 


36 X 56 


4 


12 X 15 


40 


16 X 30 


12 


24 X 50 


6 


36 X 60 


3 


12 X 16 


38 


16 X 38 


12 


24 X 54 


5 


36 X 64 


3 


12X17 


35 


16 X 40 


11 


24 X 56 


5 


40 X 60 


3 



Slate. 



317 



Roofing Slate. 

A square of slating is 100 square feet of finished roofing. Slating is 
usually so laid that the third slate laps the first slate by three inches. To 
compute the number of slates of a given size required to cover a square of 
roof, subtract 3 inches from the length of the slate, multiply the remainder 
by the width of the slate, and divide by 2 ; the result is the number of 
square inches of roof covered per slate'. Divide 14,400 (the number of 
square inches in a square) by the number thus found, and the result will 
be the number of slates required for a square. 



Slate. 

Dimensions and Number per Square. 



Dimensions, 
in inches. 


Nu nib er 

per square. 


Dimensions, 
in inches. 


Number 
per square. 


Dimensions, 
in inches. 


Number 
per square. 


6X12 


533 


9X16 


246 


16X20 


137 


7X12 


457 


10X16 


221 


12 X 22 


126 


8X12 


400 


9X18 


213 


14 X 22 


108 


9X12 


355 


10X18 


192 


12 X 24 


114 


7X14 


374 


12X18 


160 


14 X 24 


98 


8X14 


327 


10 X20 


169 


16 X 24 


86 


9X14 


291 


11X20 


154 


14 X 26 


89 


10X14 


261 


12 X 20 


141 


16 X 26 


78 


8X16 


277 


14 X 20 


121 







Thickness, % inch, T 3 p inch, % inch, increasing by eighths to 1 inch. 
The weight of slate is about 174 pounds per cubic foot, or, per square 
foot of various thicknesses, as follows : 

Thickness, in inches.. % T \ % % % % % % 1 
Weight, in pounds 1.81 2.71 3.62 5.43 7.25 9.06 10.88 12.69 14.50 



318 



Tin Plate. 



Tin Plates (Tinned Sheet=steel). 



Brand 




IC 

29 

225 

Lb. 

108 
110 
132 
155 
178 
200 
230 
160 
190 


IX 

27 

225 

Lb. 

135 

138 
162 
193 

218 
248 
289 
195 
235 


IXX 

26 

225 

Lb. 
160 
165 
192 
230 
260 
290 
340 
222 
275 










Thickness, B. W. 
gauge 












Number of sheets pei 
box 














Inch. 
10 X 14 

12 X 12 

13 X 13 
14X14 

15 X 15 

16 X 16 

17 X 17 
10X20 
11X22 








































Net weight per box - 


























































Brand 




IC 

29 

112 

Lb. 
108 
216 
138 
160 
190 
220 
110 
132 
120 
130 
155 


IX 

27 

112 

Lb. 
135 
270 
158 
195 
235 
276 
138 
162 
148 
161 
193 


IXX 

26 

112 

Lb. 
160 
320 
178 
222 
275 
330 
165 
192 
174 
190 
230 


IXXX 

25 

112 

Lb. 

180 


IXXXX 

24^ 

112 

Lb. 

200 


IX 

27 

56 
Lb. 


IXX 


Thickness, B. W. 
gauge 




26 


Number of sheets pei 
box 




56 




Inch. 

14 X 20 
20 X 28 
18 X 18 
20 X 20 
22 X 22 
24X24 
12 X 24 
13X26 
14 X 22 
14X24 
14 X 28 
14X56 
14X31 

14 X 60 

15 X 21 

16 X 19 
16 X 20 
16 X 22 


Lb. 




180 








































































Net weight per box -j 


























185 


220 




178 


210 


240 














200 


240 




120 
120 
127 
138 


152 
147 
154 

170 


176 
170 

180 
200 

















































Brand 




DC 

28 

Lb. 
94 

130 

91 


DX 

25 

Lb. 
122 
180 

122 


DXX 

24 

Lb. 
143 
213 

143 


DXXX 

23 

Lb. 
164 
244 

164 


DXXXX 


Thickness, B. W. 
gauge 




22 


Net weight per box ( 
of 100 sheets \ 

Net weight per box of 
50 sheets 


Inch. 
12* X 17 
15 X21 

17 X25 


Lb. 

185 

275 

185 







Lead Pipe. 



319 





Weight and Thickness of Lead 


Pipe. 


© 

6 


oS 


© 


o 

3 


© 
a 

M 

o 


© 

3 © 

la 


bo 

c3 ft 


u 

© 

.-2 

6 




© 

to 

'© 


o 


© 

a 

M 

H 


CD 
i - 

s © 
-a fa 

© s 


bO 

© =3 


In. 




Lb. 


Oz. 


111. 


Lb. 


Lb. 


In. 




Lb. 


Oz. 


In. 


Lb. 


Lb. 


% 


AAA 


1 


12 


.180 


1968 


492 


1 


A 


4 





.210 


857 


214 


% 


AA 


1 


5 


.150 


1627 


406 


1 


B 


3 


4 


.170 


745 


186 


Vs 


A 


1 


2 


.130 


1381 


347 


1 


C 


2 


8 


.140 


562 


140 


Vs 


B 


1 





.125 


1342 


335 


1 


D 


2 


4 


.125 


518 


129 


% 


C 





14 


.110 


1187 


296 


1 


E 


2 





.100 


475 


118 


% 







10 


.087 


1085 


271 


1 




1 


8 


.090 


325 


81 


& 







9^ 


.080 


775 


193 


1H 


AAA 


6 


12 


.275 


962 


240 


% 


AAA 


3 





.250 


1787 


446 


iy 


AA 


5 


12 


.250 


823 


205 


% 




2 


8 


.225 


1655 


413 


w 


A 


4 


11 


.210 


685 


171 


% 


AA 


2 





.180 


1393 


343 


m 


B 


3 


11 


.170 


546 


136 


y 


A 


1 


10 


.160 


1285 


321 


iy 


C 


3 





.135 


420 


105 


y 


B 


1 


3 


.125 


980 


245 


m 


D 


2 


8 


.125 


350 


87 


y 


C 


1 





.100 


782 


195 


V4 




2 





.095 


322 


80 


y 


D 





9 


.065 


468 


117 


1>2 


AAA 


8 





.290 


742 


185 


y 







10 


.070 


556 


139 


iy 


AA 


7 





.250 


700 


175 


y 







12 


.090 


625 


156 


iy 


A 


6 


4 


.220 


628 


157 


% 


AAA 


3 


8 


.230 


1548 


387 


iy 


B 


5 





.180 


506 


126 


% 


AA 


2 


12 


.210 


1380 


345 


iy 


C 


4 


4 


.150 


430 


107 


% 


A 


2 


8 


.180 


1152 


288 


iy 


D 


3 


8 


.140 


315 


78 


% 


B 


2 





.160 


987 


246 


iy 




3 





.120 


245 


61 


% 


C 


1 


7 


.117 


795 


198 


m 


B 


5 









116 


% 


D 


1 


4 


.100 


708 


177 


m 


C 


4 









93 


% 


AAA 


4 


14 


.290 


1462 


365 


m 


D 


3 


10 


.125 


318 


79 


% 


AA 


3 


8 


.225 


1225 


306 


2 


AAA 


10 


11 


.300 


611 


152 


% 


A 


3 





.190 


1072 


268 


o 


AA 


8 


14 


.250 


511 


127 


3 A 


B 


2 


3 


.150 


865 


216 


2 


A 


7 





.210 


405 


101 


% 


C 


1 


12 


.125 


782 


195 


2 


B 


6 





.190 


360 


90 


% 


D 


1 


3 


.090 


505 


126 


2 


C 


5 





.160 


260 


65 


i 


AAA 


6 





.300 


1230 


307 


2 


D 


4 





.090 


200 


50 


i 


AA 


4 


8 


.230 


910 


227 

















320 



Roofing Materials. 



Corrugated Iron. 

The following table is calculated for sheets 30% inches wide "before 
corrugating. 



©" 

4 

.a s 


o 
H 


© o 

^ is 

bfi^ 


Weight per 
square foot, cor- 
rugated. 


Weight per square of 100 square feet, when laid, 
allowing 6 inches lap in length and 2% inches, 
or one corrugation, in width of sheet, for 
sheet lengths of 


©oS 
+* © C3 


,o be 

S.S 


5' 


6' 


7' 


8' 


9' 


10' 


Weigi 
squai 
galvs 




Inch. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


16 


.065 


2.61 


3.28 


365 


358 


353 


350 


348 


346 


2.95 


18 


.049 


1.97 


2.48 


275 


270 


267 


264 


262 


261 


2.31 


20 


.035 


1.40 


1.76 


196 


192 


190 


188 


186 


185 


1.74 


22 


.028 


1.12 


1.41 


156 


154 


152 


150 


149 


148 


1.46 


24 


.022 


.88 


1.11 


123 


121 


119 


118 


117 


117 


1.22 


26 


.018 


.72 


.91 


101 


99 


97 


97 


96 


95 


1.06 



Skylight and Floor Glass. 

Weight per Cubic Foot, 156 Pounds. 

Weight per Square Foot. 
Thickness, in inches. % T 3 s % % % 

Weight, in pounds 1.62 2.43 3.25 4.88 6.50 



8.13 9.75 



1 
13 



Flagging. 

Weight per Cubic Foot, 168 Pounds. 

Weight per Square Foot. 
Thickness, in inches ... 1 2 3 4 5 6 

Weight, in pounds 14 28 42 56 70 84 



8 
112 



Number and Weight of Shingles (Pine) per Square of 
100 Feet. 



Number of inches ex- 
posed to weather. 


Number of shingles 
per square. 


Weight per square, in 
pounds. 


4 


900 


216 


5 


800 
720 


192 
173 


6 


655 
600 


157 
144 



Building Material. 



321 



Shipping Weights of Corrugated Iron. 

United States Standard Gauge. 



No. 
16 . 
18 . 
20 . 
22. 
24 . 
26 . 



Black. 

Lb. per square foot. 

2.75 

2.20 

1.65 

1.38 

1.11 

0.84 



No. 
16 . 
18 . 
20 - 
22 . 
24 . 
26 . 



Galvanized. 

Lb. per square foot. 

2.91 

2.36 

1.82 

1.54 

1.27 

0.99 



Add to net surface 23 to 26 per cent, for roofing with 6-inch end laps. 
Add to net surface 20 to 22 per cent, for siding with 4-inch end laps. 
All side laps = 1 corrugation = 2% inches. 

Example. 1600 square feet roof = 2000 square feet sheeting = 2000 X 1.65 
pounds = 3300 pounds black corrugated iron. 



Weight of Roofing. 

Table for Computing Loads upon Roofs. 

Pounds per Square of 100 Square Feet. 

Yellow pine, Northern, sheathing, 1 inch thick 300 

Yellow pine, Southern, sheathing, 1 inch thick 400 

Spruce, sheathing, 1 inch thick 200 

Chestnut or maple, sheathing, 1 inch thick 400 

Ash or oak, sheathing, 1 inch thick 500 

Shingles, pine 200 

Slates, % incn thick 900 

Sheet-iron, ^ inch thick 300 

Sheet-iron, ^ inch thick, and laths 500 

Iron, corrugated 100 to 375 

Iron, galvanized, flat 100 to 350 

Tin 70 to 125 

Felt and asphalt 100 

Felt and gravel 800 to 1000 

Skylights, glass, ^ inch to % inch thick 250 to 700 

Sheet-lead 500 to 800 

Copper 80 to 125 

Zinc , 100 to 200 

Tiles, flat 1500 to 2000 

Tiles, flat, with mortar 2000 to 3000 

Tiles, pan 1000 



Timber Measurement. 

Two methods are in use for the measurement of timber : the method of 
board measure, and the use of the cubic foot. 

Board Measure, abbreviated B.M., employs as a unit one square foot 
of surface by one inch in thickness. For boards one inch thick the board 
measure, therefore, is equal to the number of square feet. For any thick- 
ness the board measure is obtained by multiplying the width in inches by 
the thickness in inches and by the length in feet, and dividing by 12. 
The table on page 322 gives the board measure for various widths and 
thicknesses for a length of one foot, and hence the tabular figures must be 
multiplied by the length of the board in feet. 

21 



322 



Board Measure. 



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Spikes and Nails. 



323 



Wrought Spikes. 

Size and Number in Keg of 150 Pounds. 



•• 00 

43 <D 


Diameter, in inches. 


43" <w 
^.3 




Diameter, in 


inches. 




^.2 


H 


t 5 5 


% 


& 


X 


M 


5 

T5 


y 8 


T 7 S 


H 


3 


2250 
1890 
1650 
1464 
1380 
1292 










7 
8 
9 
10 
11 
12 


1161 


662 
635 
573 


482 
455 
424 
391 


445 
384 
300 
270 
249 
236 


306 


3K 


1208 

1135 

1064 

930 

868 








256 








240 


ft 








222 


742 
570 










203 


6 












180 

















Wire Spikes. 

Size and Number to the Pound. 



Title. 


Number 


Length, 


Number 


Title. 


Number 


Length, 


Number 


of wire. 


in inches. 


per pound. 


of wire. 


in inches. 


per pound. 


lOd. 


7 


3 


50 


60d. 


1 


6 


10 


16d. 


6 


3^ 


35 


6^in. 


1 


6K 


9 


20d. 


5 


4 


26 


7 in. 





7 


7 


30d. 


4 


±% 


20 


8 in. 


00 


8 


5 


40d. 


3 


5 


15 


9 in. 


00 


9 


4K 


50d. 


2 


5K 


12 











Wire Nails. 

Length and Number to the Pound. 



r2 


43*" S 




d 

o 

■era 

tn O 


o 

a 
6 


© 

o 

C3 
© 


O oj 


6 


Casing and 
smooth and 
barbed fin- 
ishing. 


CO 
T3 

eS 
U 
43 
bJD 

a 

o 

o 


bib 

53 


o 

o 

o> • 

42 bC 

^•3 


© 
Us 
a 

43 
02 




2K 
2% 
3 

3>| 
4 

6 




















714 
469 
411 


























2d. 
3d. 


1200 


876 


710 




1558 


1550 
1140 

'760 


1350 




411 




3d. 
4d. 
5d. 
6d. 

7d. 


720 

432 

300 

252 

186 

132 

105 

87 

66 

51 

35 

27 

21 

15 

12 


568 

357 

235 

204 

139 

99 

90 

69 

53 

43 

31 

24 

18 


429 

274 

235 

157 

139 

99 

90 

83 

64 

59 

43 


142 
124 
92 
82 
62 
50 
38 
30 
23 


980 

760 

575 

350 

275 

190 

173 

137 

98 

81 

71 


913 

584 

410 

310 

238 

170 

150 

121 

97 

72 

54 

46 

36 


"i57* 
139 

99 
90 
67 
53 
43 


251 
209 
142 

*i82 
125 
114 

83 


251 
165 
142 

103 


'270 
204 


8d. 






9d. 

lOd. 






12d. 






16d. 








20d. 








30d. 










40d. 


















50d. 


















60d. 













































324 



Screws. 





Size and 


Weight 


of Lag Screws. 




Length, in 


Diameter, in inches. 


inches. 














% 


i 7 * 


K 


% 


% 




Lb. per 100. 


Lb. per 100. 


Lb. per 100. 


Lb. per 100. 


Lb. per 100. 


1% 

1% 


6.88 










7.50 


11.75 


16.88 






2 


8.25 
9.25 


12.62 

12.88 


17.18 
18.07 






2% 






23^ 


9.62 


13.28 


19.18 






3 


10.82 
11.50 


16.62 

18.18 


22.00 
24.00 


34.07 

35.88 




3K 




4 


13.31 


18.88 


26.82 


39.25 


64.00 


4^ 


14.82 


19.50 


28.25 


42.62 


67.88 


5 


16.50 


21.25 


30.37 


47.75 


71.37 


5^ 


17.37 


23.56 


33.88 


51.62 


79.37 


6 


18.82 


25.31 


35.37 


55.12 


86.62 


7 






38.94 






8 






44.37 


68.75 


97.50 


9 








77.00 


108.75 


10 








90.00 


124.75 













Dimensions of Wood Screws. 



a 


Threads 
per inch. 


Diameter 
of body. 


Diameter 
of flat 
head. 


Diameter 

of round 

head. 


Diameter 

of filister 

head. 


Lengths. 






Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


2 


56 


.0842 


.1631 


.1544 


.1332 


& to y 2 


3 


48 


.0973 


.1894 


.1786 


.1545 


& to % 


4 


32, 36, 40 


.1105 


.2158 


.2028 


.1747 


f\to % 


5 


32, 36, 40 


.1236 


.2421 


.2270 


.1985 


&to Vs 


6 


30,32 


.1368 


.2684 


.2512 


.2175 


T% tol 


7 


30,32 


.1500 


.2947 


.2754 


.2392 


% to iy a 


8 


30,32 


.1631 


.3210 


.2936 


.2610 


H to i^ 


9 


24, 30, 32 


.1763 


.3474 


.3238 


.2805 


X to 1% 


10 


24, 30, 32 


.1894 


.3737 


.3480 


.3035 


%toiy 2 


12 


20,24 


.2158 


.4263 


.3922 


.3445 


y 8 toi% 


14 


20,24 


.2421 


.4790 


.4364 


.3885 


%to2 


16 


16, 18, 20 


.2684 


.5316 


.4866 


.4300 


% to 234 


18 


16,18 


.2947 


.5842 


.5248 


.4710 


3^to2^ 


20 


16,18 


.3210 


.6368 


.5690 


.5200 


V 2 to2% 


22 


16, 18 


.3474 


.6894 


.6106 


.5557 


y 2 to-s 


24 


14,16 


.3737 


.7420 


.6522 


.6005 


y 2 to 3 


26 


14, 16 


.4000 


.7420 


.6938 


.6525 


%to3 


28 


14,16 


.4263 


.7946 


.7354 


.6920 


Vs to 3 


30 


14, 16 


.4520 


.8473 


.7770 


.7240 


1 to 3 



Lengths vary by 16ths from ft to 
from \y 2 to 3. 



by 8ths, from y 2 XoV/ 2 \ by 4ths, 



Extra Strong Pipe. 



325 



Wrought=iron Welded Extra Strong Pipe. 





Diameter. 




Thickness. 


Nominal weight 
per foot. 


Nominal 
internal. 


Actual 
external. 


Actual 
internal. 


Inch. 


Inch. 


Inch. 


Inch. 


Lb. 


Vs 


.405 


.205 


.100 


.29 


X 


.540 


.294 


.123 


.54 


% 


.675 


.421 


.127 


.74 


% 


.840 


.542 


.149 


1.09 


% 


1.050 


.736 


.157 


1.39 


1 


1.315 


.951 


.182 


2.17 


Wa 


1.660 


1.272 


.194 


3.00 


IK 


1.900 


1.494 


.203 


3.63 


2 


2.375 


1.933 


.221 


5.02 


2K 


2.875 


2.315 


.280 


7.67 


3 


3.500 


2.892 


.304 


10.25 


V/ z 


4.000 


3.358 


.321 


12.47 


4 


4.500 


3.818 


.341 


14.97 


5 


5.563 


4.813 


.375 


20.54 


6 


6.625 


5.750 


.437 


28.58 



Wrought=iron Welded Double Extra Strong Pipe. 






Diameter. 




Nominal weight 
per foot. 


Nominal 
internal. 


Actual 
external. 


Actual 
internal. 


Thickness. 


Inch. 


Inch. 


Inch. 


Inch. 


Lb. 


% 


.840 


.244 


.298 


1.70 


% 


1.0.50 


.422 


.314 


2.44 


1 


1.315 


.587 


.364 


3.65 


VA 


1.660 


.885 


.388 


5.20 


IK 


1.900 


1.088 


.406 


6.40 


2 


2.375 


1.491 


.442 


9.02 


2K 


2.875 


1.755 


.560 


13.68 


3 


3.500 


2.284 


.608 


18.56 


3K 


4.000 


2.716 


.642 


22.75 


4 


4.500 


3.136 


.682 


27.48 


5 


5.563 


4.063 


.750 


38.12 


C 


6.625 


4.875 


.875 


53.11 



326 



Boiler Tubes and Pressure Pipe. 






Lap=welded Charcoal Iron Boiler Tubes. 







Length of tube per 








square foot of 


Nominal 




Thickness. 




weight 








External 


Internal 


per foot. 


External. 


Internal. 




surface. 


surface. 




Inch. 


Inch. 


Inch. 


Feet. 


Feet. 


Lb. 


3 


2.782 


.109 


1.273 


1.373 


3.33 


3% 


3.010 


.120 


1.175 


1.260 


3.96 


3% 


3.260 


.120 


1.091 


1.172 


4.28 


3.510 


.120 


1.018 


1.088 


4.60 


4 


3.732 


.134 


.955 


1.024 


5.47 


41 4 

4% 


3.982 


.134 


.899 


.959 


5.82 


4.232 


.134 


.849 


.902 


6.17 


4.482 


.134 


.804 


.852 


6.53 


5 


4.704 


.148 


.764 


.812 


7.58 


5^ 


4.954 


.148 


.728 


.771 


7.97 


5>^ 


5.204 


.148 


.694 


.734 


8.36 


6 


5.670 


.165 


.637 


.673 


10.16 



Double Galvanized Spiral Riveted Pressure Pipe.. 

For Compressed Air. 



Inside 
diameter, 


Thickness. 


Approxi- 
mate weight 
per foot, in 

pounds. 


Approximate 

bursting 
pressure, in 
pounds, per 
square inch. 


Safe working 
pressure, in 
pounds, per 
square inch. 


in inches. 


B. W. G. 


Inches. 


3 


20 


.035 


2^ 


900 


300 


4 


20 


.035 


3 


700 


220 


5 


20 


.035 


4 


550 


175 


6 


18 


.049 


5 


700 


220 


7 


18 


.049 


6 


600 


185 


8 


18 


.049 


7 


500 


150 


9 


18 


.049 


8 


450 


135 


10 


16 


.065 


11 


500 


150 


11 


16 


.065 


12 


450 


135 


12 


16 


.065 


14 


400 


120 


13 


16 


.065 


15 


380 


115 


14 


14 


.083 


20 


470 


140 


15 


14 


.083 


22 


450 


135 


16 


14 


.083 


24 


400 


120 


18 


14 


.083 


29 


370 


110 


20 


14 


.083 


34 


325 


100 


22 


12 


.109 


40 


365 


110 


24 


12 


.109 


50 


335 


100 



A variety of joints can be used to connect lengths, but the surest are 
bolted joints where the pipe is to carry an excessive pressure. Flanged, 
leaded, and cement joints may also be conveniently used according to 
pressure and permanency of pipe line. 



Riveted Hydkaulic Pipe. 



327 



Riveted Hydraulic Pipe. 

Pel ton Water-wheel Company. 



.2 


o^'S 




+^ rj-J 


« . 


.2 


1^1 




-**" 'd 


U *? QD 


.2 £* 




d B g 

■a 5" 


B'% £ 


& 2 'd 


o3 ©"„! 


is! • 

d E S ® 


d x m 

-2 2.3 

cs 3 f 




e<£ d 
■M.-, d 


Diame 
of pi 
inch* 


•2 * °°. i 

rd 3gQ bfi 
H 


'lea 




A d O 
hD <D Ph 

'3.2 a 


•2©. 2 
p 


_3 _£ 3 '<& 
o o8 " % 
A B GO &c 

H 






-d e8 O 

bC <u Ph 

[3 .2 d 


3 


18 


.05 


810 


2.25 


18 


12 


.109 


295 


25.25 


4 


18 


.05 


607 


3.00 


18 


11 


.125 


337 


29.00 


4 


16 


.062 


760 


3.75 


18 


10 


.140 


378 


32.50 


5 


18 


.050 


485 


3.75 


18 


8 


.171 


460 


40.00 


5 


16 


.062 


605 


4.50 


20 


16 


.062 


151 


16.00 


5 


14 


.078 


757 


5.75 


20 


14 


.078 


189 


19.75 


6 


18 


.050 


405 


4.25 


20 


12 


.109 


265 


27.50 


6 


16 


.062 


505 


5.25 


20 


11 


.125 


304 


31.50 


6 


14 


.078 


630 


6.50 


20 


10 


.140 


340 


35.00 


7 


18 


.050 


346 


4.75 


20 


8 


.171 


415 


45.50 


7 


16 


.062 


433 


6.00 


22 


16 


.062 


138 


17.75 


7 


14 


.078 


540 


7.50 


22 


14 


.078 


172 


22.00 


8 


16 


.062 


378 


7.00 


22 


12 


.109 


240 


30.50 


8 


14 


.078 


472 


8.75 


22 


11 


.125 


276 


34.50 


8 


12 


.109 


660 


12.00 


22 


10 


.140 


309 


39.00 


9 


16 


.062 


336 


7.50 


22 


8 


.171 


376 


50.00 


9 


14 


.078 


420 


9.25 


24 


14 


.078 


158 


23.75 


9 


12 


.109 


587 


12.75 


24 


12 


.109 


220 


32.00 


10 


16 


.062 


307 


8.25 


24 


11 


.125 


253 


37.50 


10 


14 


.078 


378 


10.25 


24 


10 


.140 


283 


42.00 


10 


12 
11 


.109 
.125 


530 
607 


14.25 


24 


8 


.171 
.200 


346 

405 


50.00 


10 


16.25 


24 


6 


59.00 


10 


10 


.140 


680 


18.25 


26 


14 


.078 


145 


25.50 


11 


16 


.062 


275 


9.00 


26 


12 


.109 


203 


35.50 


11 


14 


.078 


344 


11.00 


26 


11 


.125 


233 


39.50 


11 


12 


.109 


480 


15.25 


26 


10 


.140 


261 


44.25 


11 


11 


.125 


553 


17.50 


26 


8 


.171 


319 


54.00 


11 


10 


.140 


617 


19.50 


26 


6 


.200 


373 


64.00 


12 


16 


.062 


252 


10.00 


28 


14 


.078 


135 


27.25 


12 


14 


.078 


316 


12.25 


28 


12 


.109 


188 


38.00 


12 


12 


.109 


442 


17.00 


28 


11 


.125 


216 


42.25 


12 


11 


.125 


506 


19.50 


28 


10 


.140 


242 


47.50 


12 


10 


.140 


567 


21.75 


28 


8 


.171 


295 


58.00 


13 


16 


.062 


233 


10.50 


28 


6 


.200 


346 


69.00 


13 


14 


.078 


291 


13.00 


30 


12 


.109 


176 


39.50 


13 


12 


.109 


407 


18.00 


30 


11 


.125 


202 


45.00 


13 


11 


.125 


467 


20.50 


30 


10 


.140 


226 


50.50 


13 


10 


.140 


522 


23.00 


30 


8 


.171 


276 


61.75 


14 


16 


.062 


216 


11.25 


30 


6 


.200 


323 


73.00 


14 


14 


.078 


271 


14.00 


30 


% 


.250 


404 


90.00 


14 


12 


.109 


378 


19.50 


36 


11 


.125 


168 


54.00 


14 


11 


.125 


433 


22.25 


36 


10 


.140 


189 


60.50 


14 


10 


.140 


485 


25.00 


36 


& 


.187 


252 


81.00 


15 


16 


.062 


202 


11.75 


36 


.250 


337 


109.00 


15 


14 


.078 


252 


14.75 


36 


T 5 6 


.312 


420 


135.00 


15 


12 


.109 


352 


20.50 


40 


10 


.140 


170 


67.50 


15 


11 


.125 


405 


23.25 


40 


1% 


.187 


226 


90.00 


15 


10 


.140 


453 


26.00 


40 


% 


.250 


303 


120.00 


16 


16 


.062 


190 


13.00 


40 


s 


.312 


378 


150.00 


16 


14 


.078 


237 


16.00 


40 


% 


.375 


455 


180.00 


16 


12 


.109 


332 


22.25 


42 


10 


.140 


162 


71.00 


16 


11 


.125 


379 


24.50 


42 


& 


.187 


216 


94.50 


16 


10 


.140 


425 


28.50 


42 


M 


.250 


289 


126.00 


18 


16 


.062 


168 


14.75 


42 


* 


.312 


360 


158.00 


18 


14 


.078 


210 


18.50 


42 


.375 


435 


190.00 



328 



Weight of Wrought-iron and Copper Pipe. 



Weight of Wrought=iron Pipe. 

Metric System. 
Weight, in Kilogrammes, per Metre. 









Thickness, in millimetres. 






r-l * 


















e| 




















2 


3 


4 


5 


6 


7 


8 


10 


Mm. 


Kg. 


Kg. 


Kg. 


Kg. 


Kg. 


Kg. 


Kg. 


Kg, 


10 


.58 


.95 


1.36 


1.83 


2.34 


2.90 


2.51 


4.87 


13 


.73 


1.17 


1.65 


2.20 


2.78 


3.41 


4.09 


5.60 


15 


.83 


1.31 


1.85 


2.43 


3.07 


3.75 


4.51 


6.09 


20 


1.07 


1.68 


2.34 


3.04 


3.80 


4.60 


5.46 


7.30 


25 


1.31 


2.05 


2.83 


3.65 


4.54 


5.46 


6.43 


8.50 


30 


1.56 


2.41 


3.42 


4.26 


5.26 


6.31 


7.40 


9.74 


35 


1.80 


2.78 


3.80 


4.87 


5.99 


7.16 


8.38 


10.96 


40 


2.05 


3.14 


4.26 


5.48 


6.72 


8.00 


9.31 


12.18 


45 


2.29 


3.51 


4.77 


6.09 


7.45 


8.86 


10.32 


13.39 


50 


2.53 


3.87 


5.26 


6.69 


8.18 


9.72 


11.30 


14.61 


55 


2.78 


4.24 


5.75 


7.30 


8.91 


10.57 


12.27 


15.83 


60 


3.02 


4.60 


6.23 


7.92 


9.64 


11.42 


13.25 


17.04 


70. 


3.51 


5.33 


7.21 


9.13 


11.10 


13.01 


15.20 


19.48 


80 


3.99 


6.06 


8.18 


10.35 


12.56 


14.83 


17.14 


21.91 



Weight of Copper Pipe. 

Metric System. 
Weight, in Kilogrammes, per Metre. 



u 






Thickness, in millimetres. 






•— 1 ** 


















e| 


















M 


2 


3 


4 


5 


6 


7 


8 


10 


Mm. 


Kg. 


Kg. 


Kg. 


Kg. 


Kg. 


Kg. 


Kg. 


Kg. 


10 


.68 


1.10 


1.58 


2.12 


2.71 


3.37 


4.07 


5.66 


13 


.85 


1.36 


1.92 


2.55 


3.22 


3.96 


4.75 


6.50 


15 


.96 


1.53 


2.15 


2.83 


3.56 


4.35 


5.24 


7.07 


20 


1.24 


1.95 


2.71 


3.53 


4.41 


5.34 


6.33 


8.48 


25 


1.52 


2.38 


3.28 


4.24 


5.26 


6.33 


7.46 


9.90 


30 


1.81 


2.80 


3.85 


4.93 


6.11 


7.32 


8.60 


11.31 


35 


2.09 


3.22 


4.41 


5.66 


6.96 


8.31 


9.73 


12.72 


40 


2.38 


3.65 


4.98 


6.36 


7.80 


9.30 


10.86 


14.14 


45 


2.66 


4.07 


5.54 


7.07 


8.65 


10.29 


11.99 


15.55 


50 


2.94 


4.50 


6.11 


7.78 


9.50 


11.28 


13.12 


16.96 


55 


3.22 


4.92 


6.67 


8.48 


10.35 


12.27 


14.25 


18.38 


60 


3.51 


5.34 


7.24 


9.19 


11.20 


13.26 


15.38 


19.79 


70 


4.07 


6.19 


8.37 


10.60 


12.89 


15.24 


17.64 


22.62 



Standard Flanges. 



329 



Standard Flanges. 

The following standard dimensions for pipe flanges were prepared by a 
committee of the American Society of Mechanical Engineers in 1892. 





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© . 
ft g 

.a^a 
ft o 

S3 ft 
°8 
$^ 

© -+J 


a 

o 

3 

■9 

re 
« 


u - 
© 
© 

a 

'•B 

© 

G 

OS 

5 


CO 

© 

© 

bJD 
G 
c3 

N 


© 

fcO 

a 
e6 

qa 

© <D 

■5 © 

'c a 


© 
1 

© 




4h 



A3 

5 

a 


of 

Sh 
© 

© 

a 
© 

"E 


pq 



a 

a 

a 

© 

+a 

pq 


u 4 

a a 

C/2 


2 


.409 


T 7 6 


460 


6 


% 


• 2 


4% 


4 


V2 % 


2 


825 


2A 


.429 


IB 


550 


ij 


6% 


ii 

8 


2% 


5% 


4 


y 2 % 


234 


1050 


3 


.448 


72 


690 


^A 


2% 


6 


4 


23^ 
2% 
2% 


1330 


H 


.466 


700 


L s 


8 


13 

l¥ 


234 


6% 


4 


2530 


4 


.486 


^ 


800 


>? 


8% 


15 
T6 


1 


734 


4 




2100 


U 


.498 


2 


900 


9% 


15 
16 


7% 


8 


3 


1430 


5" 


.525 


8 


1000 


1 


10 


if 


2% 


83^ 


8 


3 


1630 


6 


.563 


A 


1060 


n% 


1 


2% 
2% 
2% 
2% 


9% 


8 


3 


2360 


' 7 


.600 


% 


1120 


1 8 


123^ 


*A 


10% 

11% 


8 


3% 


3200 


8 


.639 


% 


1280 


L - 


13K 

14% 


1% 


8 




4190 


9 


.678 


J* 
k 


1310 


A 


13^ 


13 


12 


Y* A 


3% 
3% 


3610 


10 


.713 


1330 


A 


16 


1A 


3 


14% 


12 


H % 


2970 


12 


.790 


H 


1470 


A 


183^ 


1M 


3% 


m 


12 


% % 


4280 


14 


.864 


Vs 


1600 


A 


21 


1% 
1% 


3% 


12 


%i 


^ 


4280 


15 


.904 


15 
16 


1600 


T 3 6 


22% 


3% 
3% 


20 


16 


Y* 1 


434 


3660 


16 


.946 


1 


1600 


T 3 6 


233| 


1A 


21% 
22% 


16 


ki 


434 

4% 


4210 


18 


1.02 


*A 


1690 


A 


25 


lx 9 6 


3% 

3% 
3% 


16 


1 1% 

1 V/s 


4540 


; 20 


1.09 


i% 


1780 


3 
T6 


27% 
293^ 


m 


25 


20 


5 


4490 


22 


1.18 




1850 


% 


in 

134 1% 

Ws 2 


27% 


20 


1 1% 


5% 


4320 


^ 24 


1.25 


1920 


M 


31% 32 
33% 3434 


3% 4 


29% 29% 
31% 31% 
33% 34 


20 


1 1% 


5% 


5130 


26 


1.30 


1A 


1980 


M 


3% 4% 


24 


1 1% 


5% 


5030 


28 


1.38 


1% 


2040 


% 


36 363^ 


115 Ol 
X T6 Z 16 


4 4% 
4 Ws 

4% 4% 

4% 5% 


28 


1 1% 


6 


5000 


- 30 


1.48 


1^2 


2000 


38 38% 


1% 2% 


35% 36 


28 


i%i% 


6% 


4590 


; 36 


1.71 


1% 


1920 


% 


44% 45% 


1% 23^ 
1% 2^ 


42 42% 


32 


13^1% 


6% 


5790 


42 


1.87 


2 


2100 


% 


51 52% 


48% 49% 


36 


1341% 

WsVA 


7H 


5700 


1 48 


2.17 


234 


2130 


X 4 


57% 59% 


2 2% 


54% 56 


44 


7% 


6090 



Sizes up to 24 inches are designed for 200 pounds or less. 

Sizes from 24 to 48 inches are divided into two scales, one for 200 pounds, 
the other for less. 

The two sizes of bolts given are for medium and high pressures. 

The sudden increase in diameters at 16 inches is due to the possible 
insertion of wrought-iron pipe, making, with a nearly constant width of 
gasket, a greater diameter desirable. 

When wrought-iron pipe is used, if thinner flanges than those given are 
sufficient, it is proposed that bosses be used to bring the nuts up to the 
standard lengths. This avoids the use of a reenforcement around the pipe. 

Figures in the third, fourth, fifth, and last columns refer only to pipe 
for 200 pounds pressure. 

The above standards, while not officially adopted, are used by many 
manufacturers. 



330 



Extra Heavy Flanges. 



Extra Heavy Flanges. 

Standard dimensions for extra heavy flanges for pipe fittings and 
valves, adopted by leading manufacturers in the United States, January 1, 
1902. 



ft 


o 


o 


O^' 




+S 


6 
ft 


o 


o 


■si 


«m 


£ 


ft 




BD 

4> 0J 




o 


o 


ft 


u 


CD 




O 

u 


o 

.Q 


o 

0> 




1 = 




II 


o 


o 


0) b£ 


5 SP 

.2 3 




So 


O 
CD 


N 


.2<e 


^CP 


.S.o 


Sjo 




N 


.2<p 


,<=« 


.2 -a 


P .0 


N 


CO 


ft 


H 


ft 


fc 


CO 


CO 


ft 


H 


ft 


ft 


CO 


In. 


Inch. 


Inch. 


Inch. 




Inch. 


In. 


Inch. 


Inch. 


Inch. 




Inch. 


2 


6X 


% 


5 


4 


1 


9 


16 


i 3 4 


14 


12 


P 


*% 


7^ 
8% 


1 


5% 


4 


10 


17K 


i% 


15% 


16 


3 


iy fl 


6y 8 


8 


12 


20 


2 


17 y 4 


16 


«K 


9 


ly a « 


7 X 4 


8 


?/« 


14 


22% 

23% 


2 1 /* 


20 


20 


4 


10 


14 


7 7 4 


8 


1 

/8 


15 


2 T 3 * 


21 


20 


1 


4X 


10% 


1A 


8% 


8 


16 


25 


32 

2% 

2^ 
2^ 
2% 


22% 

24% 
26f| 

31)| 


20 


1 


5 

6 


11 
12% 




11 7 4 


8 
12 


18 
20 


27 
29% 


24 
24 


1 


7 


14 


1% 


12 


22 


31% 


28 


i% 


8 


15 


l 5 /8 


13 


12 


24 


34 


28 


iy 8 



The foregoing table includes the following features : bolt holes are in 
multiples of four, in order to enable the positions of connections to be 
varied by right angles ; bolt holes to be drilled to straddle vertical axis ; 
the distance between bolt centres not to exceed 3% inches, which is ac- 
complished on all but the 2%-inch size ; distance from centre of bolt to 
edge of the flange should always equal or exceed the diameter of bolt 
plus % inch for 9-inch valves and under, and diameter of bolt plus not 
less tnan % inch for sizes larger. 

The bolt circle diameters, as above stated, will allow the use of calking 
recess on pipe flanges, provided such device is specified. 

The above standard sizes have been adopted by the following firms, 
and will be furnished by other firms to order : 

The Eaton, Cole & Burnham Company Bridgeport, Conn. 

Chapman Valve Manufacturing Company Indian Orchard, " 

Walworth Manufacturing Company Boston, Mass. 

Crane Company Chicago, 111. 

The Pratt & Cady Company Hartford, Conn. 

Jenkins Bros New York City. 

General Fire Extinguisher Company Providence, R. I. 

Builders' Iron Foundry Providence, R. I. 

Jarecki Manufacturing Company Erie, Penna. 

Crosby Steam Gauge and Valve Company Boston, Mass. 

The Kennedy Valve Manufacturing Company New York City. 

The Ludlow Valve Manufacturing Company Troy, N. Y. 

The Lunkheimer Company Cincinnati, Ohio. 

The Michigan Brass and Iron Works Detroit, Mich. 

The Kelly & Jones Company New York City. 

Eastwood Wire Manufacturing Company Belleville, N. J. 

National Tube Company Pittsburg, Penna. 

Collin Valve Company Boston, Mass. 

Rensselaer Manufacturing Company Troy, N. Y. 

The Mason Regulator Company Boston, Mass. 

MeNab & Ilarlin Manufacturing Company New York City. 

The John Davis Com pan v Chicago, 111. 

Watson & McDaniel Company Philadelphia, Penna. 

Ross Valve Company Troy, N. Y. 

Edward P. Bates Syracuse, N. Y. 



Cast-iron Pipe Fittings. 



331 



Dimensions of Cast=iron Pipe Fittings. 

As made by Best & Co., Pittsburg, Pa. 




4— 



*f 



For Pressures from 50 to 1000 Pounds. 
= thickness of body. F = thickness of flanges. 



Size. 


50 1b. 


100 lb. 


150 lb. 


200 lb. 


300 lb. 


500 lb. 


1000 lb. 


B 


F 


^ 


.p 


B 


,P 


B 


F 


B 


F 


B 


.P 


B 


F 


Inch. 


In, 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


1 


% 


% 


Vs 


13 
16 


K 


H 


A 


13 
16 


A 


13 
16 


K 


7-< 
/8 


K 


K 


VA 


1% 


% 


7 
16 


15 
16 


A 


it 


K 


1 


K 


1 


17 
32 


1 


17 
32 


l 


1% 


A 


% 


A 


H 


7 
16 


15 
16 


k 


1 


K 


1 


% 


IK 


% 


iK 


2 


A 


K 


7 
T6 


15 

18 


7 
16 


if 


A 


1 


A 


1 


Vs 


IK 


% 


iK 


2K 


K 


1 


k 


1 


K 


1 


9 
16 


1 


9 
16 


1 


11 

16 


IK 


25 
32 


iK 


3 


K 


1 


K 


1 


K 


1 


K 


IK 


% 


IK 


% 


IK 


II 


iK 


3K 


9 
16 


IK 


9 
16" 


1^ 


A 


1H 


% 


IK 


% 


IK 


K 


IK 


1 


iK 


4 


A 


1% 


9 
TS 


1^ 


A 


1^ 


11 

16 


IK 


1 1 
16 


IK 


K 


i% 


IK 


2 


*K 


A 


IK 


A 


1% 


A 


1>8 


11 

16 


IK 


ii 


IK 


K 


i% 


IK 


2 


5 


A 


IK 


A 


IVs 


% 


1¥ 


% 


IK 


K 


IK 


l 


iK 


IK 


2K 


6 


A 


IK 


9 

T6 


IK 


H 


1% 


H 


m 


if 


IK 


iK 


2 


IK 


2K 


7 


9 
16 


IK 


% 


134 


11 

16 


1A 


H 


IK 


ii 


IK 


iK 


2K 


IK 


2K 


8 


A 


IK 


% 


1M 


K 


IK 


% 


IK 


l 


IK 


IK 


2K 


iK 


2K 


9 


A 


Ws 


K 


IK 


% 


IK 


if 


IK 


1A 


IK 


IK 


2K 


iK 


2K 


10 


9 

15 


IVs 


ti 


m 


H 


1A 


i 


i% 


IK 


i% 


iK 


2K 


2 


2% 


11 


K 


1A 


tt 


ik 


Vs 


lie 


1A 


IK 


1A 


iK 


iK 


2K 


2K 


2K 


12 


K 


1A 


% 


1A 


K 


1% 


IK 


2 


1A 


2 


iK 


2% 


2K 


3 


*0. D. 






























14 


u 


1A 


i§ 


1% 


15 
16 


1% 


1A 


2K 


1A 


2K 


2 


3 


2% 


3K 


15 




l- 7 


if 

K 


Hi 


if 
1 


115 


l- 3 - 


2K 
2K 
2% 
2K 
2K 
2% 

2K 
3 


IK 
l- 9 - 


2K 
2% 
2K 
2% 
3 










16 


H 9 u 


1% 

1% 

1-- 


?, 


IK 
1% 
IK 
IK 
l 11 










18 


K 


1 5 K 

?, 


K 


lit 


2A 

2K 
2K 
2% 

2n 


iii 










20 


13 


15 


1% 
1_3_ 


IK 

9 










22 


13 


1 


9 To 










24 


K 


1A 


2K 

2% 
2^ 
2K 

?5^ 


Hu 


2K 

2K 
2K 
2K 
2K 
2% 
2K 


3K 

3K 
3K 
3% 
4 










26 


15 


l- 5 - 


113 










28 


15 


?, 


1_3_ 


13^ 


2% 

2ii 


He 










30 


1 


?, 


1M 


l-7_ 


? 


3K 
3% 
3K 
3K 










32 


1 


?, 


ik 

1 T 5 B 


l 1 ^ 


^Tu- 


9 K 










34 


1A 


v% 


*A 




2A 

2A 


4K 
4K 










36 


1A 


2V 8 


IK 


2% 


1% 


Q 3 

°T6 



















* 0. D. 14 inches and larger is for lap-weld steel pipe whose outside diameter is 
of sizes given. 



332 



Steel Fittings and Flanges. 



Dimensions of Steel Fittings and Flanges. 

■*-F As made by Best & Co., Pittsburg, Pa. 

For Pressures from 150 to 1000 Pounds. 

B = thickness of body. 
F = thickness of flange. 

— diameter of male. 

1 = diameter of female 
m = height of male. 
/ = depth of female. 






Steel Fittings. 


Cast and Rolled Steel Flanges. 


a 


'n • 


Size. 


150 to 
300 lb. 


500 1b. 


1000 lb. 


150 and 
200 lb. 


300 lb. 


500 lb. 


1000 lb. 


u u *-> 
o ftS 

PR 




B 


i? 1 


B 


F 


5 


i^ 



In. 


J 
In. 



In. 


I 
In. 



In. 


I 
In. 



In. 


I 
In. 


m 
In. 


1/ 


Inch. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


1 


% 


1 


A 


1 


A 


1 


2- 5 - 


2% 


9 5 
^TB 


2% 


2% 


2 t 5 b 


2% 


2A 


% 


A 


1M 


7 
16 


1 


% 


1 


K 


1 


2i 9 b 


2% 


2t 9 b 


2% 


2% 


2t 9 b 


2% 


2t 9 b 


K 


A 


IK 


7 
16 


1 


% 


1 


X 


1 


2il 


2% 


211 


2% 


3 


3f X B 


3 


3A 


>4 


A 


2 


7 
T6 


1A 


A 


lf 3 6 


A 


13. 
X 16 


3 t 5 b 


3% 


Q 5 
<*TB 


3% 


3% 


3i 9 b 


3% 


3A 


^ 


A 


2% 


K 


l^ 


9 
16 


lr 3 6 


A 


1 3 

L TB 


qi5 
^TB 


4 


Q15 
6 Te 


4 


4 


4A 


4 


4A 


M 


A 


3 


K 


1% 


^ 


1 5 

J-fe 


% 


1 5 

-Lib 


411 


5 


414 
^lB 


5 


4% 


41i 


4% 


411 


% 


A 


3% 


9 

16 


1t 3 b 


% 


1 5 


% 


1t 5 b 


5 t 7 b 


5% 


P* 7 

°TB 


5% 


G 


6A 


6 


«A 


^ 


A 


4 


A 


ll 3 6 


11 
16 


1 7 
ifB 


11 

16 


1t 7 b 


6A 


6% 


6A 


6% 


6 


6A 


6 


6A 


% 


A 


4% 


9 
16 


1A 


11 
16 


-•-16 


11 
16 


1 7 


6t 9 b 


6% 


6t 9 b 


6% 


7% 


7 r 5 B 


7% 


7i 5 b 


M 


A 


5 


9 

16 


ll 3 B 


^ 


1 7 


3 A 


1 7 


7 t 9 b 


7% 


7i 9 b 


7% 


7% 


7i 5 b 


7% 


7A 


M 


A 


6 


A 


1 3 


13 
TB 


ll 9 6 


13 
16 


ll 9 6 


8t 9 b 


8% 


8t 9 b 


8% 


8% 


8 t 9 b 


8% 


8 t 9 b 


M 


A 


7 


^ 


1 5 
X IB 


if 


1t 9 6 


H 


ll 9 B 


9i 9 b 


9% 


9A 


9% 


9% 


91b 


9% 


Oil 
y TB 


% 


A 


8 


5^ 


1A 


^ 


1ft 


l 


Itt 


Ht 9 b 


U% 


nil 


12 


10% 


101b 


10% 


101 6 


% 


A 


9 


^ 


1/6 


15 
TB 


lit 


1A 


iH 


12i 9 b 


12% 


1211 


13 


11% 


101b 


11% 


1111 


}i 


A 


10 


1 1 
16 


1A 


1 


2 


1% 


2 


13H 


13% 


14B 


14% 


12% 


12fi 


13 


13A 


K 


A 


11 


1 1 
TS 


1% 


1A 


2A 


1A 


9 i 

Z TB 


1411 


14% 


15H 


15% 


14 


14A 


15 


15A 


% 


A 


12 


% 


1% 


i% 


2A 


1A 


2A 


16t 3 b 


16% 


16H 


16% 


15 


i&A 


15 


ISA 


% 


A 


*0.D. 


































14 


13 

16 


1% 


1A 


2% 


1/6 


2% 


17 t 5 b 


17% 


171b 


18 


17 


17A 


17 


17A 


5 

IS 


% 


15 


13 


1% 

1% 
9 










18j 8 e 


18% 
19% 
21% 
23% 
26% 
28% 

30^ 


1q To 


19% 

20% 
22% 
24% 
27% 
29% 










A 
A 

_r> 


% 
% 
% 
% 
% 


16 


15 










lq TC 


9 °A 










18 










91 11 


99 To 










20 


9 A 










23H 
26A 
2*A 

30A 


9 4i 3 c 










S 


22 


1 


2% 
2A 

2% 








27A 

29A 










A 

A 


24 


1A 

i% 




















1 








I 






26 






28 
30 


1A 

i% 


2% 
2% 


Elbows, Tees, 
and Crosses, 200 
pounds and 
above. Female 


32A 

34 T 7 6 


32% 

34% 


Flanges and Fittings 150 

pounds, and below plain face. 

For pressures 200 pounds and 


32 


1J4 


2K, 


36^ 


36% 


above. Fittings, Female; 


34 


ll 5 8 


2% 


on all ends. 


38A 


38% 


Valves, Male ; according to di- 
mensions on this table. 


36 


1% 


2«^ 4 




»<>!;! 


4oy 8 





*0. D. 14 inches and larger is for lap-weld steel pipe whose outside diameter is 
of sizes given. 



Pipe Fittings. 



333 



Dimensions of Pipe Fittings. 

As made by Best & Co., Pittsburg, Pa. 

Pipe Bends. 

Made of Standard and Extra Heavy Pipe. 




R = radius. 



H = centre to face. 



U= centre to centre. 



Size 


Standard radius. 


Minimum radius. 




R 


H 


U 


R 


H 


?7 


Inch. F 


t. Id. F 


t. In. 


Ft. 


In. F 


t. In. F 


t. In. I 


't. In. 


Vs 


. 3 


. 3% 




6 


. 1 


• 1% 


2 


H 


• 3% ■ 


• 4% 




7 


. m . 


• 2}£ 


■ 2% 


Vs 


. 4 


. 5% 




8 


. i% • 


. 2% 


3 


% 


. 5 


• 6% 




10 


• m • 


. 3% 


• 3% 


% 


. 6 


. 7% 




12 


. 2 


. 3% 


. 4 


1 


. 7 


. 9^ 




14 


• 2% . 


• 4% 


• 4% 


v-A 


. 8 


. 10% 




16 


. 2% . 


. 5^ 


• 5% 


1% 


. 10 


. 12% 




20 


. 3% . 


. 6 


• 6% 


2 


. 12 


. 15 


2 


4 


• 4% . 


• 7% 


9 


2% 


. 14 


• 17% 


2 


4 


. 6 


• 9% 


. 12 


3 


. 18 


. 22 


3 




. 7 


. 11 


. 14 


3% 


. 20 


72 


3 


4 


. 9 


. 13% 


. 18 


4 


. 24 


2 5 


4 




. 12 


. 17 


. 24 


4% 


2 2 


2 .7% 


4 


4 


. 15 


. 20% 


2 6 


5 


2 6 


3 .. 


5 




. 18 


. 24 


3 .. 


6 


3 .. 


3 6% 


6 




. 24 


2 6% 


4 .. 


7 


4 .. 


4 7 


8 




2 6 


3 1 


5 .. 


8 


4 6 


5 2 


9 




3 2 


3 10 


6 4 


10 


5 6 

7 6 


6 6 

8 8 


11 




4 4 
6 6 


5 4 


8 8 


12 


15 




7 8 ] 


L3 .. 


*0. D. 
















14 


8 .. 


9 6 


16 




7 6 


9 .. ] 


L5 .. 


16 1 


.. 1 


1 10 


20 
24 




8 6 1 

9 6 1 


4 ] 

1 6 ] 


L7 .. 


18 1 


2 .. 1 


4 .. 


L9 .. 


20 1 


4 .. 1 


6 .. 


28 


1 


6 1 


2 6 \ 


>1 .. 


22 1 


6 .. 1 


8 .. 


32 


1 


16 1 


3 6 S 


IS .. 


24 1 


8 .. 2 


.. 


36 


1 


2 6 1 


4 6 5 


>5 .. 



* 0. D. 14 inches and larger is for lap-weld steel pipe whose outside diameter is 
of sizes given. 

Bends below heavy lines made in two pieces. 



334 



Pipe Fittings. 



Dimensions of Pipe Fittings. 

As made by Best & Co., Pittsburg, Pa. 
Pipe Bends. 

Made of Standard and Extra Heavy Pipe. 



Long Radius, for Pressures 
from 50 to 300 Pounds. 



4 



K--H-* 1 




H 






<Mj^ 



■n 



-H-*--H'^\ 

R = radius. 
H= centre to face. 
h = centre to face, on short 
end. 



Size. 


R 


H 


Inch. 


Ft. In. 


Ft. In. 


2 


.. 5i| 


. . 7 


2% 


.. 6% 


-. 7% 


3 


• • 7% 


.. 8% 


3% 


.. 7% 


.. 9% 


4 


•• 8 r % 


.. 10 


4% 


.. 9^ 


• • 10% 


5 


• • 9t 7 5 


.. 11 


6 


. . 1034 


.. 12 


7 


.. li a 


.. 13 


8 


. . 12% 


.. 14 


10 


■ • 14& 


. . 16% 


12 


.. 16% 


.. 19 


*O.D. 






14 


• • ISri 


.. 21 


16 


. . 21^ 


. . 23% 


18 


. . 23% 


2 2 


20 


2 1A 


2 4 


22 


2 3% 


2 6 


24 


2 4% 


2 8 



Inch. 
4% 
5 

5% 
6 

6% 
7% 

7% 

8% 

9 

9% 
11% 
12% 

14 
16 

17 
18 
19 
21 



Extra Long 
Radius, for 
Pressures 
from 50 to 300 
Pounds. 




S = radius. 
C = centre to 
face. 



Ft. In. 

Q 5 

. 10% 

. 11% 

• 12% 

. is* 

• ih% 

■ Hit 
. 16% 

• 17H 

. 19% 
. 22i§ 
2 23% 



5A 

3A 

6% 

8% 



C 



Ft. In. 

. 10% 

• H% 

. 12% 
. 14 
. 15 

• 15% 

. 16% 
. 18 
. 19% 
. 21 
2 % 
2 4% 

2 7% 

2 11% 

3 3 
3 6 

3 9 

4 .. 



45° Elbows and Y's, 
for Pressures from 50 
to 150 Pounds. 




/ = radius. 
6 = centre to face. 
I = centre to face. 
a = centre to face, on 
short end. 



/ 


b 


I 


Inch. 


Inch. 


Ft. In. 


4% 


2% 


.. 9 


5 


3 3 % 


.. 10 


5H 


3% 


.. 11% 


51i 


3% 


.. 12% 


6% 


4% 


.. 13% 


7% 


4K 


.. 14% 


m 


4% 


.. 16% 


8A 


5% 


. . 17% 


9% 


5% 


.. 18% 


9% 


6 


.. 21 


11 


6% 


.. 24 


12 


7% 


2 3% 


13% 


7% 


2 5 


14% 


8 3 % 


2 7% 


15% 


9% 


2 10% 


16 3 % 


9% 


3 1% 


18% 


10% 


3 5% 


20 


11% 


3 8% 



In. 

2% 

3 

3 

3% 

3% 

3% 

3% 
4% 
4% 
4% 
5 



6% 
7% 
7% 
8% 
8% 
9% 



* 0. I). 14 inches and larger is for lap-weld steel pipe whose outside diameter is 
of sizes given. 



Angle, Globe, and Check Valves. 



335 



Dimensions of Angle, Globe, and Check Valves. 

As made "by Best & Co., Pittsburg, Pa. 





Angle. 


Globe. 


Check, 




mil o 

i i i • 


jfej 


r^ 


— [— q 








m 


c 




TTJ 


s 


jp 






U H* 




k--f — »t 


P 7 ->1 


H = centre to face. 


Z = face to face. 


I = face to face. 


o = centre 


to top of wheel, 


o = centre to top of 


c = centre to top 


when open. 


wheel, when open. 


of cover. 


Size 


Hy. and Ex. Hy. 


Hy. and Ex. Hy. 


Hy. and Ex. Hy. 




H 





1 





1 


c 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


2% 


4% 


12^ 

17i| 
18% 
19% 


9% 
12% 


12M 


8 


5% 


3 


6% 
6% 


1634 


11 


6% 


3% 


13% nu 

14 18K 


12 


7% 


4 


7 


13% 
15% 


8% 


5 


Ws 


15% 


19% 


10 


6 


m 

10% 
12M 


2131 

23^| 


17% 


21% 
23^| 


17 


11 


7 


19% 


18 


11% 
12% 
14% 
16% 

2o|| 
22 


8 


25% 
35K 


21 


25% 


19% 


10 


24% 


29% 


22% 


12 


14 


28 


35% 


25 


14 


15 


39 


30 


39 


28 


16 

18 


15% 


42% 


30% 


42% 


32 
34 
36 
38 


20 










24 


22 










26% 
<2&2 


24 










42 



















Hy. for 150 pounds pressure. 
X Hy. for 200 pounds pressure. 

Diameter of Wheels for 
Valves. 




Dimensions of hydraulic 
valves to 5000 pounds on ap- 
plication. 

Angle Check Valves to 
order. 



Size of valves, in inches. 



Angle and Globe Valves 
Lt., Std., and Hy. Gate Valves 
X Hy., XX Hy. Gate Valves 

Hvd. Gate Valves 

X Hyd. Gate Valves 



1 


1% 


1% 


2 


2% 


3 


3% 


4 


4% 


5 


6 

16 


7 
16 








8 


9 


10 


12 


12 


14 


14 


4i% 


6 


6 


7 


7 


10 


10 


10 


10 


12 


1P> 


13 


^6 
4% 


6 


6 


7 


8M 


10 


10 


12 


12 


13 


14 


14 


6 


6 


7 


a* 


12 


12 


14 


14 


14 


16 


16 


6 


6 


7 


8% 


10 


12 


12 


14 


16 


16 


18 


20 



Size of valves, in inches. 


9 


10 


12 


14 


15 


16 


18 


20 

30 
30 


22 

32 
32 


24 

32 

36 


26 

32 


28 
36 


30 


Angle and Globe Valves 

Lt.,Std., and Hy. Gate Valves 
X Hy., XX Hy. Gate Valves. 
Hyd . Gate Valves 


18 

14 
16 

18 
24 


20 
15 
18 
20 
24 


24 
15 
20 
20 
30 


30 
18 
22 
24 
30 


'26' 
24 


24 
24 


24 
30 


36 


X Hyd. Gate Valves 









Butterfly Valves. 

Light, Standard, and X Hy.— Dimensions in inches. 

Size of valve 4 5 6 7 8 10 12 14 15 16 18 20 22 24 

Length, face to face.. 4* 5* 6 7 8 10 12 12£ 13 13f 15 16* 18* 19* 
Double Butterfly Valves to order. 



-§- 



Wrjt-A 



336 



Gate Valves. 



Inside Screw. 




Dimensions of Gate Valves. 

As made by Best & Co., Pittsburg, Pa. 



Pressures Indicated. 



Light, 


501b. 


Standard, 


100 lb. 


Hy., 


150 lb. 


XHy., 


200 lb. 


XX Hy., 


3001b. 


Hyde., 


500 lb. 


X Hyde., 


1000 lb. 



k — ; — * 



I = face to face. 

L = face to face by-pass valves. 

S = centre to top of wheel. 

O = centre to top of stem, when open. 



Outside Screw and Yoke. 
No By-pass. With By-pass. 






Light and Standard. 


Hy., 


XHy., XX Hy. 


Hyde, XHydc. 


Size. 














1 


S 





£ 


1 


S 





L 


1 


S 


Inch. 


In. 


Ft. 


In. 


Ft. 


In. 


In. 


In. 


Ft. 


In. 


Ft. In. 


In. 


Inch. 


Inch. 


1 
























6% 


8% 


i% 
























7% 


10% 


i% 
























11% 


12% 


2 


8 




11% 


.. 


14 














12 


14% 


2% 


8% 


.. 


12% 




15% 




9% 




12% 


.. 15 




14 


17 


3 


8% 




14 




19% 




11% 




16% 


.. 19% 




14% 


18 


3% 














11% 




18% 


.. 21% 




14% 


18 


4 


9% 




16% 




23 




12 




19 


.. 23% 




20% 


22 


4% 


10% 




19 




24 




13% 




21% 


.. 26% 




21% 


23 


5 


9% 




19% 




26 




15 




22% 


.. 27% 


18% 


22% 


24 


6 


10% 




22% 




31% 




15% 




25% 


•• 31% 


19 


24 


25% 


7 


11% 




24 




36% 




16% 




29% 


.. 36% 


20 


25 


28 


8 


11% 




25% 


3 


4 


18% 


16% 




32% 


3 5% 


20% 


26 


29% 


9 


12% 


.. 


26% 


3 


7% 


19% 


17 




34% 


3 8% 


21% 


26 


32% 


10 


13% 




30% 


3 


10% 


21 


18 


3 


1% 


4 % 


22% 


27 


35 


12 


W% 




33% 


4 


6% 


22 


19% 


3 


8% 


4 8% 


23% 


30% 


39% 


14 


15% 


3 


IK 


5 


2% 


24 


21% 


4 


2 


5 4% 


25% 


32% 


43% 


15 














21% 


4 


5% 


5 9% 


2b% 




16 


18% 


3 


7 


6 


% 


26% 




4 


8 


6 % 


32% 


Necks. 


18 


20 


3 


10% 


7 


% 


26% 




5 


2% 


6 9% 


33% 


1 


H — — H 4 


20 


21 


4 


2 


7 


4% 


27 




5 


8% 


7 5% 


35% 


lilt 


22 


22% 


4 


6% 


8 




32 




6 


2% 


8 1% 


35% 


24 


24 


4 


10 


8 


8 


32% 




6 


8 


8 8% 


35% 


<^ps> 


26 


2b 


5 


2% 


9 


4% 


32% 




Cast-iron, cast-stee 


1 ; forged. 


28 


28 


5 


5% 


10 


1 


34% 


Size 4 to 6, 8 in. II. Size 


7 to 10, 8% in. JET. 


30 


30 


5 


9 


10 


7 


36 




Size 12 to 16, 9 


in. H. 



Standard Pipe Unions. 



337 



00 o 3: 

B'o- B 

rt- 3 p 

o'er? go. 



« 3 3. 




srw w 


* 3^ 


3 $ 


bS 



U.S. Standard 
^ Thread 




^ 


co 


CO 


to 


to 


M 


M 


M 


K 




0D\ 


K 


*F 


- 


rfs> 
Its. 


CO 

o 
o 


CO 
£>> 
O 
M 


to 
<I 


to 

8 

OS 


bo 

•cc 


Cn 

CO 

to 


OS 


CO 
CO 

to 


^7 
CO 

co 


b 

CO 

o 


Its. 

CO 
OS 


S3 

Cn 


fco 


4.026 


CC 

cc 


CO 

b 

CS 


to 

OS 

Co 


to 

b 

OS 


h- 1 

b 
o 


H- 1 

CO 


b 

Its. 

CO 


bo 
to 


b 
to 

CO 


CO 


CO 


to 

^7 

O 


04 


.374 




CO 
(ta. 


CO 

3 


to 

CO 


to 

to 

— ' 


to 


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CO 


M 

OS 
CO 




M 

CO 

OS 


CO 

to 


1 


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4*. 


b 


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CO 

co 


to 

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to 


to 

*>> 
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to 


l-» 

>JSv 


£ 


f- 1 
OS 


so 

o 


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Cn 
co 


cn 


5.22 


00 


CO 


CO 
Cn 


to 

CO 

Cn 


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to 

CC 


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CO 




CO 
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o 


b 

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5.47 


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3 


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5.75 


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CO 


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£ 


£ 


M 


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u- 


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to 


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3 

a 









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s 

or; 






C 

3 # 
5" 

3 



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s 



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B 



o 

C 
5" 

3 

CA 



13* 



22 



338 



Machine and Set Screws. 



Machine Screws. 



Screw gauge 
size. 


Threads per 
inch. 


Outside diam- 
eter, in inches. 


Approximate 

diameter, in 

inches. 


Tap drill, 

B. & S. drill 

gauge. 


2 


56 


.08420 


5 


No. 49 


3 


48 


.09730 


& 


No. 45 


4 


36 


.11050 


£ 


No. 42 


5 


36 


.12360 


Vs 


No. 38 


6 


32 


.13680 


& 


No. 35 


7 


32 


.15000 


5 
35 


No. 30 


8 


32 


.16310 


£ 


No. 29 


9 


30 


.17630 


M 


No. 27 


10 


24 


.18940 


& 


No. 25 


11 


24 


.20206 


H 


No. 21 


12 


24 


.2158 


& 


No. 17 


13 


22 


.2289 


15 
6¥ 


No. 15 


14 


20 


.2421 


if 


No. 13 


15 


20 


.2552 


H 


No. 8 


16 


18 


.2684 


17 


No. 6 


17 


18 


.2816 


£ 


No. 2 


18 


18 


.2947 


u 


No. 1 


19 


18 


.3079 


6 


H" 


20 


16 


.3210 


u 


M" 


22 


16 


.3474 


M 


9 // 

5? 


24 


16 


.3737 


% 


w 


26 


16 


.4000 


H 


3i" 


28 


14 


.4263 


tf 


11// 

55 


30 


14 


.4526 


T ? B 


§r 



Set Screws. 



Outside 

diameter, in 

inches. 


Short diame- 
ter of square 


Threads per 


Size of tap 


Lengths under 


head, 
in inches. 


inch. 


drill. 


head, in inches. 


% 


% 


20 


No. 5 


%to3 


6 
TB 


& 


18 


17// 

B¥ 


% to 3}4 


% 


Ys 


16 


W 


%to3^ 


T ? B 


A 


14 


%" 


%to3% 


V* 


V* 


12 


31" 


%to4 


h 


9 
IB 


12 


ti" 


% to 4^ 


% 


% 


11 


w 


%to4^ 


% 


3 A 


10 


w 


1 to A% 


% 


Vs 


9 


if" 


l^to5 


1 


1 


8 


%" 


IK to 5 



Sleeve Nuts. 



339 



Standard Sleeve Nuts and Upsets. 

Passaic Rolling Mill Company. 















,~-; 






B~ 


Jtc 


^\ 




tV 








^T\\ 


-*-*l ( 




"if* 




^ lil^^j 


/ J 


U 









^43 


f 


Diame- 
ter of 

• 

rods. 


Side 
of 

■ 
rods. 


© 
w 

a, 

o 

© 

I 

5 


1 

o 

h3 


o 
u 

© 


O 

h 

© 

© . 

a a 


a © 


© 
> 

© 


© w 


© 
> 

© 
© 

O 
?*»' 

!© fl 


Additional 

length of rod 

required for 

one upset. 


• 


■ 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 




Inch. 


Lb. 


Inch. 


Inch. 


% 


% 


1 


4 


2M 


2% 


8 


S% 


4 


3% 


4% 


! % 


% 


1% 


4 


2>i 


2% 


7 


sy 2 


5 


3M 


3% 


l 


Vs 


m 


4^ 


2% 


2% 


6 


9% 


7 


4% 


5 


V/s 


1 


V4 


4K 


2% 


3 5 


6 


934 


8 


4M 


4M 


W 


V/s 


Ws 


4^ 


2% 


^ 5 


53^ 


9K 


9 


3% 


3^ 


h Ws 


VA 


Ws 


5 


3M 


3% 


5 


1034 


13 


5M 


4^ 


Vi 


i% 


2 


5 


3M 


3% 


4^ 


iom 


13 


4% 


4 


IVs 


V4 


2^ 


5 


3% 


4A 


■ 4K 


10^ 


16 


434 


33^ 


1% 




2M 


&A 


3% 


4 T % 


4K 


11 


18 


4^ 




<! i% 


Ws 


2% 


53^ 


4 


4% 


4 


1134 


21 


4 


43^ 


3 2 


i% 


2^ 


5^ 


4 


4% 


4 


UU 


22 


3% 


4 


2^ 


i% 


2«% 


6 


4% 


5% 


4 


12 


29 


3% 


4 


2^ 


2 


2% 


6 


4% 


53^ 


33^ 


12M 


33 


4^ 


43^ 


2^ 


2^ 


3}^ 


6 


5K 


5if 


3^ 


12^ 


40 


5 


4% 


1 23^ 


23^ 


3^ 


6 


5^ 


6% 


334 


12% 


47 


4^ 


4 


3 




3% 


6 


5% 


6% 


3 


13 


58 


4 





340 



Eye Bars. 



Standard Steel Eye Bars. 

Passaic Rolling Mill Company. 



-h-M 




Width 
of bar. 



Minimum 

thickness 

of bar. 



Diameter 
of head. 



Diameter 
of largest 
pin-hole. 



Sectional area of 

head on lines S — S 

in excess of that 

in body of bar. 



Additional length 

of bar beyond 
centre of pin-hole 
to form one head. 



Inch. 
3 
3 
4 
4 
5 
5 

6 
6 

7 

8 

10 



Inch. 

% 



1 

i% 



9% 
io% 
n% 

12% 

13% 

14% 

16 

18 

23 



Inch. 

2A- 1 - 

Qll 
d T6 
Q15 
6 X6 

4% 
4% 
5% 

4% 

5% 

5% 

7 

9 



Per cent. 
42 
42 
37% 
39 
41 
41 

42 

42 

43 

37% 

40 



Inch. 
14% 
18% 
18% 
23% 
21 
25% 

22 

26% 

28 

32% 

40 



Notes on Passaic Steel Eye Bars. 

Passaic standard steel eye bars are forged without the addition of ex- 
traneous metal and without welds of any kind, and are guaranteed under 
the conditions given in the above table to develop the full strength of 
the bar when tested to destruction. 

The maximum sizes of pin-holes, given in the above table, allow an 
excess in the net section of the head over that of the body of the bar of 
40 per cent, when the thickness of the head is the same as the thickness of 
the body of the bar. The thickness of the head is usually ^ of an inch 
thicker than the body of the bar; and where a number of eve bars are to 
be placed closely together, as at a joint, the thicknesses of the heads should 
be considered % of an inch greater than the bodies of the bars, in order to 
allow for the increased thickness of the heads and for the usual roughness 
of forged work. 

Unless otherwise specified, the steel manufactured by the Passaic Roll- 
ing Mill Company for the use of eye bars is open-hearth medium steel, 
conforming with the standard specifications of the Association of Ameri- 
can Steel Manufacturers. 

All eye bars are finished to length, and the eyes bored at the specified 
distances, centre to centre, according to United States standard measure- 
ments. 

Eye bars having larger or smaller heads than the above standards can 
be furnished by special arrangement. 



Pins and Nuts. 



341 



Standard Pins and Nuts. 

Passaic Rolling Mill Company. 



ir- 
i 



-$ — i 




*-s-** 



= grip. L = G -{-% inch. 



8-J 



d 


T 


8 


Short 
diameter 
of nut. 


Long 

diameter of 

nut. 




Diameter 
of pin. 


Diameter 
of thread. 


Length of 
thread. 


Weight of 
one nut. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Lb. 


1t% 


1 


IK 


1% 
1% 
3M 


2 




i ll 


1 


IK 


2 




m 


*K 


iK 


3% 


1.5 


in 


IK 


iK 


3M 


3% 


1.5 


2 T 3 6 


IK 


iK 


334 


3% 


1.5 


2/b 


1% 


iK 


3K 


3% 


1.5 


m 


2 


iK 


3% 


4% 


2.5 


m 


2M 


IK 


4K 


534 


3.0 


V Q 3 


2K 


iK 


4K 


5M 


2.8 


1 ^ 7 


2K 


iK 


4K 


5M 


2.8 


»H 


2% 


iK 


4% 


5K 


3.0 


qis 


3 


iK 


4% 


5K 


3.0 


4% 


3K 


iK 


5K 


6K 


3.8 


! 45 /« 


3K 


iK 


5K 


634 


3.8 


4% 


4 


iK 


6 


7 


6.7 


5% 

: 

5% 


4 


2 


6 


7 


6.7 


4 


2 


7 


8 


9.1 


7 


5 


2K 


8 


9^ 


12.0 


8 


6 


2^ 


ioK 


12 


18.8 


; 9 


7 


2^ 


iok 


12 


22.8 















342 



Wire Rope. 



Standard Wire Hoisting Rope. 

John A. Roebling's Sons Company. 

Composed of 6 Strands and a Hemp Centre, 19 Wires to the Strand. 

Swedish Iron. 



Trade 
number. 



Diameter, 
in inches. 



Approximate 

circumference, 

in inches. 



Weight per 

foot, 
in pounds. 



Approximate 
breaking strain, in 
tons of 2000 pounds. 



1 
2 
3 
4 
5 

5K 



9 
10 

10^ 

10% 

10a 

10b 

10c 

lOd 

1 
2 
3 
4 
5 



8 

9 

10 

10K 
10^ 
10% 
10a 
10b 
10c 
lOd 



2% 
2^ 
2M 
2 

1% 

V4 

m 
i 



2K 

2 



% 

5 

4% 

4^ 

4 

3^ 

3 

2% 

2% 
2 

1% 
IX 

1% 

IVb 

1 



Cast-steel. 

7^ 

5^ 
5 

4% 

434 

4 

z x A 

3 

2% 

2^ 
2 

1% 
IK 

1^ 

1 



11.95 
9.85 
8.00 
6.30 
4.85 
4.15 
3.55 

3.00 

2.45 

2.00 

1.58 

1.20 

.89 

.62 

.50 

.39 

.30 

.22 

.15 

.10 

8.00 
6.30 
4.85 
4.15 
3.55 
3.00 
2.45 
2.00 
1.58 
1.20 

.89 
.62 
.50 
.39 
.30 
.22 
.15 
.10 



Wire Rope. 



343 



Transmission or Haulage Rope. 

John A. Roebling's Sons Company. 
Composed of 6 Strands and a Hemp Centre, 7 Wires to the Strand. 
Swedish Iron. 



Trade 


Diameter 


Approximate 


Weight per 


Approximate 


number. 


in inches. 


circumference, 
in inches. 


foot, 
in pounds. 


breaking strain, in 
tons of 2000 pounds. 


11 


IK 


i% 


3.55 


34.0 


12 


W 


*% 


3.00 


29.0 


13 


VA 


4 


2.45 


24.0 


14 


IVs 


3K 


2.00 


20.0 


15 


1 


3 


1.58 


16.0 


16 


Vs 


2% 


1.20 


12.0 


17 


% 


v-a 


.89 


9.3 


18 


tt 


2K 


.75 


7.9 


19 


% 


2 


.62 


6.6 


20 


9 


1% 


.50 


5.3 


21 


K 


IK 


.39 


4.2 


22 


T 7 6 


IK 


.30 


3.3 


23 


Vs 


IK 


.22 


2.4 


24 


5 
IS 


l 


.15 


1.7 


25 


A 


% 
Cast-st< 


.125 
iel. 


1.4 


11 


V4 


4% 


3.55 


68.0 


12 


M 


4K 


3.00 


58.0 


13 


IK 


4 


2.45 


48.0 


14 


IK 


3K 


2.00 


40.0 


15 


l 


3 


1.58 


32.0 


16 


K 


2% 


1.20 


24.0 


17 


% 


2K 


.89 


18.6 


18 


H 


2K 


.75 


15.8 


19 


% 


2 


.62 


13.2 


20 


9 
16 


1% 


.50 


10.6 


21 


K 


IK 


.39 


8.4 


22 


T5 


IK 


.30 


6.6 


23 


3 /8 


IK 


.22 


4.8 


24 


5 

is 


l 


.15 


3.4 


25 


9 
32 


K 


.125 


2.8 



344 



Wire Rope. 



Extra Strong Crucible Cast=steel Rope. 

John A. Roebling's Sons Company. 
Composed of 6 Strands and a Hemp Centre, 19 Wires to the Strand. 



Trade 
number. 


Diameter, 
in inches. 


Approximate 

circumference, 

in inches. 


Weight per 

foot, 
in pounds. 


Approximate 
breaking strain, in 
tons of 2000 pounds. 




2% 


s% 


11.95 


266.0 




2K 


?K 


9.85 


222.0 


1 


2M 


?K 


8.00 


182.0 


2 


2 


6K 


6.30 


144.0 


3 


1% 


*K 


4.85 


112.0 


4 


1 5 K 


5 


4.15 


97.0 


5 


VA 


4% 


3.55 


84.0 


&K 


1% 


4K 


3.00 


72.0 


6 


1M 


4 


2.45 


58.0 


7 


V/s 


3K 


2.00 


49.0 


8 


l 


3 


1.58 


39.0 


9 


% 


2% 


1.20 


30.0 


10 


% 


2K 


.89 


22.0 


10% 


% 


2 


.62 


15.8 


10K 


I 9 B 


m 


.50 


12.7 


10% 


K 


v/* 


.39 


10.1 


10a 


7 
16 


ix 


.30 


7.8 


106 


% 


ik 


.22 


5.78 


10c 


& 


i 


.15 


4.05 


lOd 


34 


7 Wires to th 


.10 

3 Strand. 


2.70 


11 


IK 


4% 


3.55 


79.0 


12 


i% 


4M 


3.00 


68.0 


13 


1M 


4 


2.45 


56.0 


14 


i^ 


3K 


2.00 


46.0 


15 


l 


3 


1.58 


37.0 


16 


Vs 


2% 


1.20 


28.0 


17 


% 


2K 


.89 


21.0 


18 


\h 


2K 


.75 


18.4 


19 


% 


2 


.62 


15.1 


20 


i% 


1% 


.50 


12.3 


21 


K 


IK 


.39 


9.70 


22 


/ e 


IK 


.30 


7.50 


23 


Vs 


IK 


.22 


5.58 


24 


5 
13 


l 


.15 


3.88 


25 


9 
32 


K 


.125 


3.22 



Steel Wire. 



345 



Weight, Length, and Strength of Steel Wire. 

John A. Roebling's Sons Company. 



Number, 


Diam- 


Area, in 


Breaking load 


Weight, in 


pounds. 


Number 


Roebling 
gauge. 


eter, in 
inches. 


square 
inches. 


at rate of 
100,000 pounds 






of feet 






in 2000 






per square inch. 


Per 1000 feet. 


Per mile. 


pounds. 


000000 


.460 


.166 191 


16619.0 


558.4 


2948.0 


3 582 


00000 


.430 


.145 221 


14522.0 


487.9 


2576.0 


4 099 


0000 


.393 


.121 304 


12130.0 


407.6 


2152.0 


4 907 


000 


.362 


.102 922 


10292.0 


345.8 


1826.0 


5 783 


00 


.331 


.086 049 


8605.0 


289.1 


1527.0 


6 917 





.307 


.074 023 


7402.0 


248.7 


1313.0 


8 041 


1 


.283 


.062 902 


6290.0 


211.4 


1116.0 


9 463 


2 


.263 


.054 325 


5433.0 


182.5 


964.0 


10 957 


3 


.244 


.046 760 


4676.0 


157.1 


830.0 


12 730 


4 


.225 


.039 761 


3976.0 


133.6 


705.0 


14 970 


5 


.207 


.033 654 


3365.0 


113.1 


597.0 


17 687 


6 


.192 


.028 953 


2895.0 


97.3 


514.0 


20 559 


7 


.177 


.024 606 


2461.0 


82.7 


437.0 


24191 


8 


.162 


.020 612 


2061.0 


69.3 


366.0 


28 878 


9 


,148 


.017 203 


1720.0 


57.8 


305.0 


34 600 


10 


.135 


.014 314 


1431.0 


48.1 


254.0 


41584 


11 


.120 


.011 310 


1131.0 


38.0 


201.0 


52 631 


12 


.105 


.008 659 


866.0 


29.1 


154.0 


68 752 


13 


.092 


.006 648 


665.0 


22.3 


118.0 


89 525 


14 


.080 


.005 027 


503.0 


16.9 


89.2 


118 413 


15 


.072 


.004 071 


407.0 


13.7 


72.2 


146 198 


16 


.063 


.003 117 


312.0 


10.5 


55.3 


191 022 


17 


.054 


.002 290 


229.0 


7.70 


40.6 


259 909 


18 


.047 


.001 735 


174.0 


5.83 


30.8 


343 112 


19 


.041 


.001 320 


132.0 


4.44 


23.4 


450 856 


20 


.035 


.000 962 


96.0 


3.23 


17.1 


618 620 


21 


.032 


.000 804 


80.0 


2.70 


14.3 


740 193 


22 


.028 


.000 616 


62.0 


2.07 


10.9 


966 651 


23 


.025 


.000 491 


49.0 


1.65 


8.71 




24 


.023 


.000 415 


42.0 


1.40 


7.37 




25 


.020 


.000 314 


31.0 


1.06 


5.58 




26 


.018 


.000 254 


25.0 


.855 


4.51 




27 


.017 


.000 227 


23.0 


.763 


4.03 




28 


.016 


.000 201 


20.0 


.676 


3.57 




29 


.015 


.000 177 


18.0 


.594 


3.14 




30 


.014 


.000 154 


15.0 


.517 


2.73 




31 


.0135 


.000 143 


14.0 


.481 


2.54 




32 


.0130 


.000 133 


13.0 


.446 


2.36 




33 


.0110 


.000 095 


9.5 


.319 


1.69 




34 


.0100 


.000 079 


7.9 


.264 


1.39 




35 


.0095 


.000 071 


7.1 


.238 


1.26 




36 


.0090 


.000 064 


6.4 


.214 


1.13 





This table was calculated on a basis of 483.84 pounds per cubic foot for 
steel wire. 

The breaking loads were calculated for 100,000 pounds per square inch 
throughout, simply for convenience, so that the breaking loads for wires 
of any strength per square inch may be quickly determined by multi- 
plying the values given in the table by the ratio between the strength per 
square inch and 100,000. Thus, a No. 15 wire, with a strength per square 

i kq Ann 

inch of 150,000 pounds, will break with a load of 407 X ^~~ = 610.5 

100,000 
pounds. 

It must not be thought from this table that steel wire invariably has a 
strength of 100,000 pounds per square inch. As a matter of fact, it ranges 
from 45,000 pounds per square inch for soft annealed wire to over 400,000 
pounds per square inch for hard wire. 



346 Strength of Materials. 

STRENGTH OF MATERIALS. 

When a body is subjected to the action of external forces certain defor- 
mations are produced. These deformations are called strains, and the 
forces by which they are produced are called stresses. 

The application of the external stresses is opposed by the production of 
internal stresses. The extent to which these internal stresses are capable 
of resisting the external stresses constitutes the strength of the material. 

Thus, when a man lifts a weight so that it is suspended from his arm, 
stresses of sufficient magnitude to sustain the weight against the action of 
gravity are produced in the muscles. In like manner, a weight suspended 
from an iron rod produces internal stresses upon the fibres of the metal ; 
and since equilibrium exists, — the weight being sustained, — the internal 
forces must balance the external ones. 

Since the internal forces are brought into play by the production of 
deformation, or strain, it follows that every force, however slight, when 
acting upon a resistant body must produce some deformation, in order 
that the internal fibre stress by which the external force is to be opposed 
shall appear. No body, therefore, is absolutely rigid ; since, if a body 
could be entirely rigid, no internal stresses could be produced and no ex- 
ternal stresses could be resisted. 

In the use of materials of engineering it is necessary to know the extent 
to which they may safely be subjected to certain external forces. It is 
also important to know the extent to which they become deformed under 
determinate loads, as well as the manner in which the stresses and strains 
are distributed. These various properties constitute the resistance of the 
materials, and it is upon a knowledge of the resistance of materials that 
the ability to make a correct distribution of them in any given structure 
depends. 

The manner in which the fibres of a material act in resisting deforma- 
tion is not entirely understood. Apparently, the first and smaller defor- 
mations act only to separate the particles to distances within their range 
of attraction for each other, so that, when the external force is removed, 
the original relation of the particles is resumed. When, however, the 
deformation becomes sufficiently great for the range of attraction of the 
particles to be exceeded, the original relations are not resumed upon the 
removal of the external stresses, but a portion of the deformation remains, 
the structure of the material being more or less broken down. This is 
very clearly shown by the change in the appearance of the polished sur- 
face of a metal under stress, the bright surface suddenly becoming dulled 
when the stress exceeds a magnitude which affects the permanent struct- 
ure. This was first observed by Professor J. B. Johnson, as long ago as 
1892, and has recently been further investigated in France by M. Fremont. 

Further increase of external stress after the structure of the material 
has broken down is followed by rapidly-increasing deformation and rup- 
ture. 

It is obvious that no material should ever be subjected to such stresses 
in practice as will result in the breaking down of its molecular structure, 
since no further effective resistance can then be expected of it. It is, 
therefore, of little importance to know the force which produces rupture 
in a material. The important thing to know is the magnitude of the load 
at which the break-down begins, so that the structure under consideration 
may be so proportioned that this load is approached only within a cer- 
tain known limit. In other words, it is not the breaking load which is 
required, but the permissible fibre stress to which the material may be sub- 
jected. 

The external forces which act to cause strains in a material may produce 

Tension, 

Compression or Crushing, 

Bending, 

Shearing, 

Torsion ; 

and, in most instances, several of these actions are produced at the same 
time. 

Up to the point at which the molecular structure of the material breaks 
down under stress, the deformation produced exists only during the appli- 



Strength of Materials. 347 

cation of the stress, and the material returns to its original dimensions 
and form upon the removal of the stress. This property of returning to 
its original form and dimensions is called the elasticity of the material. 
Since a body does not return to its original dimensions and form when 
loaded to the point of structural break-down, its elasticity is then said to 
have been overcome, or its elastic limit reached. 

Up to the elastic limit the deformation of a body is directly proportional 
to the load, — that is, the strain is proportional to the stress. If a certain 
elongation is observed with a load of 100 pounds, double that elongation 
will be produced by 200 pounds, and so on until the elastic limit is reached. 
This is known as Hooke's Law. 

If the material is subjected to a continually-increasing load in a testing 
machine provided with an autographic recorder, the line of the record 
will be a straight one, making a constant angle to the axes ; while as soon 
as the elastic limit is closely approached it becomes a curve, the curvature 
rapidly changing until rupture occurs. 

The determination of the true elastic limit has been a matter of much 
discussion. Theoretically, it is the point at which " set" first occurs ; prac- 
tically, it is often assumed to correspond to the load at which the weigh- 
beam of the testing machine drops, showing the sudden yield of the 
material. An examination of autographic test diagrams show T s that the 
departure from Hooke's law begins gradually, not suddenly, the deviation 
from a straight line being at first slight, but rapidly increasing. 

The true elastic limit is determined as the point at which strain ceases 
to be proportional to stress; but this point is not readily determined in 
practice, except with precise and accurate testing machines, and hence 
the yield point, or point at which the drop of the beam of the ordinary 
testing machine occurs, is usually substituted for it. While admitting that 
this is not absolutely correct, it is quite within the working limits of accu- 
racy under existing shop conditions. 

In practice, the loads or stresses upon a body are expressed in units of 
weight per unit of area, as pounds per square inch or kilogrammes per 
square centimetre. Elongations are expressed either as the extension of a 
unit of length or in percentages of the length of test-piece. 

The Modulus of Elasticity of a material is the result obtained by 
dividing the stress per unit of area by the strain per unit of length. If we 
call the stress per unit of area = S, and the corresponding elongation per 
unit of length = e, we have 

Modulus of elasticity = E = — . 

Thus, if an elongation of 0.01 is produced in a bar of 10 inches in length 
and 1 square inch cross-section by a load of 30,000 pounds, the modulus of 
elasticity will be 

^^30,000,000. 

Since this is constant up to the yield point, it may be used for the determi- 
nation of the elongation produced by any other load. Thus, 

S S 



E 30,000,000' 

and any value of e can be obtained for any given value of S. 

In the use of any material in the construction of a framework, a ma- 
chine, or any kind of mechanism, it is most important to use judgment 
and common sense in a careful examination of the case under considera- 
tion before attempting to apply any of the rules or tables. The actual 
magnitude and direction of the forces acting should be determined as 
closely as possible, for we may be well assured that they will be in action 
whether taken into account or not. It has been well said that "theory 
takes into account all the conditions which can be ascertained, but prac- 
tice has to take into account all the conditions there are." 

In the design of structural work, such as bridges, roofs, buildings, etc., 
the size and direction of action of loads can generally be determined with 
a fair degree of accuracy, the principal uncertainty being as to the action 
of wind pressure. In machine design, however, the stresses are much 



348 Strength of Materials. 

more difficult of determination, the number and complex action of forces 
often rendering determinate analysis impossible. Under such circum- 
stances recourse must often be had to empirical rules, based upon the 
experience gained in practice. In such cases, also, careful exercise of 
judgment is demanded, in order that one may be assured that the case 
under consideration is similar to those from which the experience has 
been derived ; and too frequently errors have been made by blindly follow- 
ing the precedent set by some excellent authority, but wholly inapplicable 
to the case in hand. 

Before applying any rules, tables, or formulas, the end to be obtained 
should be intelligently considered. In some instances it is the actual 
strength of the material which must be taken into account, but in ma- 
chine design this is not often the case. More generally, it is the stiffness 
which must be considered. It is always necessary that a machine should 
retain the relative position of its parts to such an extent that the move- 
ments may continue within determinate limits of accuracy, and that no 
undue binding or friction be created in the running parts. 

Steam machinery must be so rigid that valve seats, etc., will remain 
tight and true, lathe beds must not spring under heavy cuts, planer up- 
rights must stand firmly to their work ; and all these and many other parts 
must be made far heavier than would be necessary for mere strength, in 
order that ample rigidity may be obtained. 

In many instances the principal value to be obtained from a study of 
the distribution of stresses in a machine is to ascertain the relative disposi- 
tion of the material, and not the absolute strength to be used. Experience 
has shown how heavy certain portions must be, in order that deflection or 
spring may be kept within working limits, while a graphical analysis of 
the distribution of the stresses will then show where metal may safely be 
spared and where it must be lavishly disposed. 

It must be remembered, also, that break-downs usually occur by reason 
of unusual or abnormal stresses. Machines rarely break down under reg- 
ular working loads. It is when some sudden shock occurs that the rupture 
takes place. While it is not to be expected that provision can be made for 
all accidents, yet the possible accidents should be considered in the origi- 
nal design ; and often a little forethought as to whence the unusual stress 
may be expected will materially modify the disposition of the material. 

Bearing the preceding considerations in mind, the following rules, 
formulas, and tables may be used, as representing both theory and prac- 
tice. 

Tension. 

By far the greater number of tests of materials are made by pulling a 
test- piece, and observing or recording successive stages in the strains pro- 
duced by the increasing stresses. The points usually observed are 

Elastic Limit, 
Ultimate Strength, 
Ductility, 
Stiffness, 
Resilience. 

As already stated, the elastic limit is the point at which the strain ceases 
to be proportional to the stress. In testing machines which are not pro- 
vided with a recording attachment the nearest approach which can usually 
be had to this value is the stress observed at the moment of the drop of 
the beam. When a diagram of the test is automatically produced the 
point at which the line distinctly deviates from a straight one, at a definite 
angle with the axes, shows the elastic limit. 

The ultimate strength is found when the material yields so rapidly that 
no further increase in load can be made. Both the elastic limit and the 
ultimate strength are always referred to the original area of the test speci- 
men. In general, the elastic limit is reached at a stress about equal to 
six-tenths of the ultimate, but this varies for different materials and con- 
ditions. 

The ductility of a material when subjected to tension is measured by 
the elongation in a given length or by the reduction of fractured area. 



Strength of Materials. 



349 



The stiffness is measured by the angle which the test-line makes with 
the coordinate axes, the portion within the elastic limit alone being con- 
sidered. 

Resilience is the amount of work performed in the production of strain 
by stress. It is, therefore, expressed in terms of force by length, usually in 
inch-pounds. When a piece is strained to the elastic limit, the work re- 
quired is called the elastic resilience. When the load is applied gradually, 
the work done is equal to one-half the product of the stress at the elastic 
limit by the extension. W T hen the load is applied instantaneously, the 
elastic deformation is double that produced by the same load applied 
slowly. When the force is applied by a drop, producing percussion, the 
product of the weight by the fall will give the work. 

An examination of the following table, from data of the Pencoyd Iron 
Works, will serve to show the relations which exist in open-hearth basic 
steel, such as is used in structural work. 



Open=hearth Basic Structural Steel. 

Pencoyd Iron Works. 



Percentage 


Tensile strength, in pounds, per 
square inch. 


Ductility. 


of carbon. 


Ultimate 
strength. 


Elastic 
limit. 


Stretch in 8 
inches. 


Reduction of 
fractured area. 








Per cent. 


Per cent. 


.08 


54000 


32500 


32 


60 


.09 


54800 


33000 


31 


58 


.10 


55700 


33500 


31 


57 


.11 


56500 


34000 


30 


56 


.12 


57400 


34500 


30 


55 


.13 


58200 


35000 


29 


54 


.14 


59100 


35500 


29 


53 


.15 


60000 


36000 


28 


52 


.16 


60800 


36500 


28 


51 


.17 


61600 


37000 


27 


50 


.18 


62500 


37500 


27 


49 


.19 


63300 


38000 


26 


48 


.20 


64200 


38500 


26 


47 


.21 


65000 


39000 


25 


46 


.22 


65800 


39500 


25 


45 


.23 


66600 


40000 


24 


44 


.24 


67400 


40500 


24 


43 


.25 


68200 


41000 


23 


42 



The predominant elements other than carbon in the above steels aver- 
age as follows : manganese, 0.40 per cent. ; phosphorus, 0.04 per cent. ; 
sulphur, 0.05 per cent. Any increase of these constituents is attended by 
an increase of tensile strength and a diminished ductility. The tensile 
strength of steel is also affected to some extent by the heat treatment to 
which it has been subjected. Bessemer or open-hearth acid steel will 
generally show a higher tensile strength than basic steel, owing to the 
higher proportion of phosphorus, sulphur, and manganese present. 



350 Strength of Materials. 

For convenient distinguishing terms it is customary to classify steel in 
three grades: "mild or soft," "medium," and "hard," and although the 
different grades blend into each other, so that no line of distinction exists, 
in a general sense the grades below 0.15 carbon may be considered as 
"soft" steel, from 0.15 to 0.30 carbon as "medium," and above that " hard" 
steel. Each grade has its own advantages for the particular purpose to 
which it is adapted. The soft steel is well adapted for boiler plate and 
similar uses, where its high ductility is advantageous. The medium grades 
are used for general structural purposes, while harder steel is especially 
adapted for axles and shafts and any service where good wearing surfaces 
are desired. Mild steel has superior welding property as compared to hard 
steel, and will endure higher heat without injury. Steel below 0.10 carbon 
should be capable of doubling flat without fracture, after being chilled 
from a red heat in cold water. Steel of 0.15 carbon will occasionally sub- 
mit to the same treatment, but will usually bend around a curve whose 
radius is equal to the thickness of the specimen ; about 90 per cent, of 
specimens stand the latter bending test without fracture. As the steel 
becomes harder, its ability to endure this bending test becomes more ex- 
ceptional, and when the carbon ratio becomes 0.20, little over 25 per cent, 
of specimens will stand the last-described bending test. Steel having 
about 0.40 per cent, carbon will usually harden sufficiently to cut soft iron 
and maintain an edge. 

Compression. 

When a material is subjected to a compressive load a crushing action is 
produced. This is frequently misunderstood, many assuming that the 
material is really compressed into a smaller volume than before. As a 
matter of fact, the only reduction in volume which can be produced is 
that permitted by the presence of voids in the material, the matter being 
pressed into the spaces existing in it. Liquids, in which no voids exist, 
are practically incompressible, while most metals may be materially in- 
creased in density under the hammer or the forging press ; but it must be 
understood in all such cases that the increased density is due to the 
reduction in voids, and not the crowding of the actual particles of the 
metal closer together. 

Crushing, however, is the usual effect of a heavy compressive stress, 
the material spreading in some other dimension as the yielding occurs 
along the line of compression. For ductile materials no definite point of 
rupture can be determined, since the change of shape becomes too great 
before any sign of rupture appears. Brittle materials, such as cast-iron, 
stone, bricks, cement, etc., have crushing points which may be more 
clearly determined. Many materials show a fairly distinct elastic limit 
under compression, the upsetting being proportional to stress within such 
limit. 

The manner of rupture under crushing is a matter of less importance 
than the determination of a safe working stress, and this is generally 
taken as the upsetting or yield point. For brittle materials, in which no 
such yield point can be determined, the actual crushing load must be 
used, the safe working load being made a certain proportion of the crush- 
ing load. 

Shearing. 

By shearing is understood the resistance which a material opposes to 
displacement in a plane. This action rarely, if ever, takes place alone. 
When a cutting edge begins to shear a bar, for example, true shearing 
takes place only for a very short distance, the material then bending and 
flowing down with further pressure, so that with a thick bar the fibres are 
torn apart before the shearing edge has passed entirely through, and the 
divided piece falls off, the fracture clearly indicating the combing actions 
to which it has been subjected. These actions are still more clearly 
shown by polishing the surface of the metal and etching it to bring out 
the distortion of the fibres. The relation of the shearing to the tensile 
strength cannot be expressed as any definite ratio, varying with the mate- 
rials and their disposition. 



Strength of Materials. 351 

Bending. 

When a body, such as a beam, is subjected to the action of a force 
producing deflection, there are reactions at the supports, and if no motion 
is produced these external forces must be equal to each other, or in equi- 
librium. In like manner, these external forces are opposed by internal 
forces acting upon the fibres of the material. In the case of a horizontal 
beam, the fibres in the upper portion are subjected to compression and 
those in the lower portion to tension, there being a portion between these 
where the reversal of stress takes place and where the fibre stress is zero. 

In such materials as steel and wrought-iron the resistance to compres- 
sion and tension may be taken as equal, and this neutral axis, as it is called, 
then coincides w r ith the centre of gravity of the section of the beam. 
When the beam is of symmetrical section the neutral axis naturally coin- 
cides with the centre of figure. If the beam is to resist the external forces, 
the internal stresses upon its fibres at any point must be equal to the bend- 
ing moment of the external forces at the same point. The sum of the 
moments of the internal forces about the neutral axis is called the moment 
of resistance. 

This moment of resistance is determined as follows : 

Let S be the stress per unit of area in the extreme outer fibre of the 
cross-section ; a, the cross-section of a fibre ; y, the distance of any other 
fibre from the neutral axis. Then the moment of any fibre stress at a 
distance, y, from the neutral axis will be 

v y 

and the sum of all the fibre-stress moments of the cross-section, taken with 
reference to the neutral axis, is 

! 

The quantity 2a?/ 2 , or the sum of all the elements of the area multiplied 
by the squares of their respective distances from the neutral axis, is called 
l the moment of inertia of the section, and is always symbolized as I, so 
that we have for the moment of resistance of any section 

v 

The value of the moment of inertia depends upon the form of the 
cross-section ; the value of v is also dependent upon the shape of the sec- 
tion, while the value of & the maximum permissible fibre stress, is gov- 
erned by the material. These formulas are true only when the material is 
subjected to strains within the elastic limit, and the value of S should 
always be chosen within that limit. As a general rule, the maximum 
fibre stress should not exceed one-half the elastic limit of the material. 

Since both J and v depend upon the shape of the section, we may con- 
sider them by themselves, and write the moment of resistance 

M= £— . 

v 

The factor — , or the moment of inertia divided by the distance of the 
v • 

extreme fibre from the neutral axis, is called by Reuleaux and by Unwin 
the Section Modulus. It may be called 



The radius of gyration of any section may be obtained by taking the 
square root of the quotient obtained by dividing the moment of inertia by 



352 Strength of Materials. 

the area of the section. Thus, if R be the radius of gyration, I the mo- 
ment of inertia, and A the area of the section, we have 

This will be seen to be of use in connection with struts and pillars. 

We thus see that an expression for the internal forces in a body sub- 
jected to bending stresses — such as a beam — has been obtained, and that it 
contains but two elements, the fibre stress on the material and the shape 
of the cross-section of the beam. It is only necessary, therefore, to place 

this expression for the moment of resistance, S — , equal to the moment of 

the external forces, to have their relation fully expressed. Thus, for a 
cantilever or projecting beam of a length, I, carrying a load, W, at its 
extremity, we have 

Wl = £— , or W= ~ • — • 

v I v 

For a cantilever carrying a load, W, uniformly distributed, the lever arm, 
I, is one-half as long, and we have 

I v 
For a beam carrying a load, W, in the middle, we have 

nr a S J 

W = 4— . — ; 

I c 
and for a beam carrying a load, W, uniformly distributed, we have 

I c 

In the preceding formulas IF is the load, in pounds, which will produce 
a fibre stress, S, in pounds ; I being the length of the beam, in inches ; 2, 
the moment of inertia of the section ; and v, the distance of the most 
remote fibre from the neutral axis. By taking S as about one-half the 
elastic limit of the material, the proper working load, W, can readily be 
determined. Good practice takes S at 14,000 pounds for wrought-iron and 
16,000 pounds for structural steel. Other values will be tabulated hereafter. 

The determination of the value of the moment of inertia for the section 
used is evidently the principal feature in the problem. Most of the impor- 
tant sections have been reduced to formulas, as in the following tables : 



Section Elements. 



353 



Elements of Usual Sections. 

Pencoyd Iron Works. 

Moments refer to horizontal axis, as shown. This table is intended for 
convenient application where extreme accuracy is not important. Some 
of the terms are only approximate ; those marked * are correct. Values 
for radius of gyration in flanged beams apply to standard minimum sec- 
tions only. A = area of section. 



Shape of 
section. 


Moment of inertia. 


Section 
modulus. 


Distance of 
base from 
centre of 
gravity. 


Least radius of 
gyration. 




K--W 




6ft? * 

12 


bh?* 

6 


h 

2 




- 







Least side * 
3.46 








■# 


ft** 

12 


0.1178/i3* 




h * 




3.46 




.g4_&4* 

12 


i/jB 4_54* 
A B 


B 

2 




1 - 




* 

4r 




4 
* 


/ £2 _|_ 52 * 

V 12 




.k 


— -B-*. 




\?~br-i, 


36 


bh?* 
24 


v*h 


The least of 
the two : 

h 6 * 

or — 

4.24 4.9 




MP* 

12 














1 /r~^ 




662 + 665! + ^2 
36(25 + 6i) 




^26 + 6i 












" 2 


N ** 




■0- 


^LZ>2* 
16 


-42>* 

8 


2) 

2 


2> * 


!<— D— *| 


,0.0491(2)* — d*)* 


0.0982^ ^ a 


2) 
2 




XV (2)2 + d2)* 










23 







354 




Section Elements. 






Elements of Usual Sections. 






Pencoyd Iron Works. 


4 


Moments refer to horizontal axis, as shown. This table is intended for 
convenient application where extreme accuracy is not important. Some 
of the terms are only approximate ; those marked * are correct. Values 
for radius of gyration in flanged beams apply to standard minimum sec- 
tions only. A = area of section. 


Shape of 
section. 


Moment of 
inertia. 


Section 
modulus. 


Distance of 
base from 
centre of 
gravity. 


Least radius of 
gyration. 


-i4^ 


0.1098r4 * 


Fi = 0.1908r3* 
W 2 = 0.2587r 3 


0.4244r 


0.0699/' 2 * 


|<-F>i 




HbJ 


0.78546a 3 * 


0.7854fra 2 * 






via 










D 


AJW 

10.4 


Ah 

7.4 


h 

3.5 


h 


p==Vl 


5 


k—fc--H 




n 


AW 
9.9 


4ft 

6.7 


h 
3.1 


hb 


mi 


2.6(ft + 6) 


K-6-ai 










JV 


AW 


Ah 


ft 


ft 


! if | 


19 


9.5 


2 


. 4.74 


k— 7l --*i 










M 

K--6— ^ 


AW 
10.9 


4ft 

7.6 


ft 

3.3 


b 
4.66 




AW 
6.1 


Ah 
"3.0 


ft 

2 






- — a 


6 
5.2 


f 





AW 
6.73 


4ft 

3.3 


ft 
2 


6 




3.56 


F 


<6-H 











Moment of Inertia. 



355 



Moment of Inertia of Rectangles, 



A X I S. 



Depth, 
in 






Width of rectangle, 


in inches. 






inches. 


v± 


% 


X 


% 


% 


Vs 


1 


6 


4.50 


6.75 


9.00 


11.25 


13.50 


15.75 


18.00 


7 


7.15 


10.72 


14.29 


17.86 


21.44 


25.01 


28.58 


8 


10.67 


16.00 


21.33 


26.67 


32.00 


37.33 


42.67 


9 


15.19 


22.78 


30.38 


37.97 


45.56 


53.16 


60.75 


10 


20.83 


31.25 


41.67 


52.08 


62.50 


72.92 


83.33 


11 


27.73 


41.59 


55.46 


69.32 


83.18 


97.06 


110.92 


12 


36.00 


54.00 


72.00 


90.00 


108.00 


126.00 


144.00 


13 


45.77 


68.66 


91.54 


114.43 


137.31 


160.20 


183.08 


1-1 


57.17 


85.75 


114.33 


142.92 


171.50 


200.08 


228.67 


15 


70.31 


105.47 


140.63 


175.78 


210.94 


246.09 


281.25 


16 


85.33 


123.00 


170.67 


213.33 


256.00 


298.67 


341.33 


17 


102.35 


153.53 


204.71 


255.89 


307.06 


358.24 


409.42 


18 


121.50 


182.25 


243.00 


303.75 


364.50 


425.25 


486.00 


19 


142.90 


214.34 


285.79 


357.24 


428.68 


500.14 


571.58 


20 


166.67 


250.00 


333.33 


416.67 


500.00 


583.33 


666.67 


21 


192.94 


289.41 


385.88 


482.34 


578.81 


675.28 


771.75 


22 


221.83 


332.75 


443.67 


554.58 


665.50 


776.42 


887.33 


23 


253.48 


380.22 


506.96 


633.70 


760.44 


887.18 


1013.92 


24 


288.00 


432.00 


576.00 


720.00 


864.00 


1008.00 


1152.00 


25 


325.52 


488.28 


651.04 


813.80 


976.56 


1139.32 


1302.08 


26 


366.17 


549.25 


732.33 


915.42 


1098.50 


1281.58 


1464.67 


27 


410.06 


615.09 


820.13 


1025.16 


1230.19 


1435.22 


1640.25 


28 


457.33 


686.00 


914.67 


1143.33 


1372.00 


1600.67 


1829.33 


29 


508.10 


762.16 


1016.21 


1270.26 


1524.31 


1778.36 


2032.42 


30 


562.50 


843.75 


1125.00 


1406.25 


1687.50 


1968.75 


2250.00 


31 


620.65 


930.97 


1241.30 


1551.62 


1861.94 


2172.26 


2482.60 


32 


682.67 


1024.00 


1365.33 


1706.67 


2048.00 


2389.33 


2730.67 


33 


748.69 


1123.03 


1497.38 


1871.72 


2246.06 


2620.40 


2994.76 


34 


818.83 ' 


1228.25 


1637.67 


2047.08 


2456.50 


2865.92 


3275.33 


35 


893.23 


1339.84 


1786.46 


2233.07 


2679.68 


3126.30 


3572.92 


36 


972.00 


1458.00 


1944.00 


2430.00 


2916.00 


3402.00 


3888.00 


37 


1055.27 


1582.90 


2110.54 


2638.17 


3165.80 


3693.44 


4221.08 


38 


1143.17 


1714.75 


2286.33 


2857.92 


3429.50 


4001.08 


4572.67 


39 


1235.81 


1853.72 


2471.62 


3089.53 


3707.44 


4325.34 


4943.24 


40 


1333.33 


2000.00 


2666.67 


3333.33 


4000.00 


4666.67 


5333.33 



356 



Steuctural Sections. 






Moments of Inertia of Standard Sections. 

Pencoyd Iron Works. 

i 

x When not otherwise specified, the inertia is the greatest 

around centre of gravity, or for horizontal axis in figures. 



*-rn 

I 



fFJt 
lit 



.4 = total area of section. 

I Beam Section. 

s = taper of flange. I = k — 



_ bft 3 — ck 3 cs 3 csl' 2 
- ^ + Is" + "T"' 



2s 
3 * 



mb 3 kt 3 , 



(b — t\ 3 



K^HM)- 



♦IN- 



Channel Section. 

s = taper of flange. 



bh 3 — — (k*—l*) 



1 = 



12 



b — I" 



2m63 + ^+ - (6 4 — * 4 ) 
V J, axis z?/ = Ad 2 . 



fc/2 c 



Deck Beam Section. 

£ = taper of flange. 

o = ??i — 



m 3 (6 — 




a = area of bulb. 



(b — t)& s{b — t)o 2 



ak* nt 3 
J, axis xy = — + — + 



12 



d = 



a(2ft - *) + t(h - kf- + (6 — Op 2 + *(6 — (p + y) 



2^4 



Structural Sections. 



357 



Tee Section. 

t<? + &#* — (b — t)a? 






" 



I, axis xy = 
d- 



U* + bd3—(b — t)(d- 







3 


fV> 


+ (h- 


-/)< 3 




12 




bp 


+ *(fc 2 


~/ 2 ) 




2A 





{ 

b 

£ a 
a i — t— ■ 



J, 



Angle Section. 

— — , for even or uneven 



J, axis tw = 



angles. 
t(b- 



V 



• rfi) 3 + Mi 3 —(h — t) (d x — t)* 



for 



uneven angles. 

a-?/ passes through centre of gravity parallel to ee. 

2# — 2(d — 4 + t lb — Ud — -Ml 
J, axis xy = — ^= 



1 









X 



L , for even 



A close approximation for the latter is the following : 
i, axis ??/ = -==-, for even angles. 



! /, 



axis xy -- 



cV 



AhW 



13(^2 + fc2) 

b$ + t(W — P) 



for uneven angles. 



24 

^ 2 + £(5 2 - 



, for even and uneven angles. 



3T! 



*I^ 



r£gh 



■^) 



24 



, for uneven angles. 



In uneven angles the distance from centre of gravity in direction of the 
long leg exceeds that in the direction of the short leg by half the differ- 
ence in the length of the two legs. 

In angles and tees of equal legs and thickness 



d = Y±(b+ -!*), nearly. 



Inertia of Compound Shapes. 

" The moment of inertia of any section about 
any axis is equal to the J about a parallel axis 
passing through its centre of gravity + the area 
of the section multiplied by the square of the 
distance between the axes." 

By use of this rule, the moments of inertia or 
radii of gyration of any single sections being 
known, corresponding values can readily be ob- 
tained for any combination of these sections. 

Example 1. A combination of two 9-inch 






M 



H 



D— 



*±Jr* 



channels of 3.89 square inches section and two 12" x %" plates, as shown. 



358 Compound Sections. 



Axis AB of Section. 
J for two channels, column VI., page 362, = 95.78 



19 Y 25 3 
I for two plates = — —^ — X 2 = .03125 

6 (area of plates) X 4-% 2 = 128.34375 



■ 



128.375 



I for combined section = 224.155 

which, divided by area (13.78), gives 16.27 = R 2 or 4.03 radius of combined 
section. 

Axis CD. 

Find distance, d = (.60), from column XII., page 363, then obtaining the 
distance (4.17) between axes CD and EF. 

I for two channels around axis EFirom column VI. = 3.54 

Area of channels X square of distance = 7.78 X 4.17 2 = 135.286 

5 v 12 3 
J for two plates = ^ = 72.000 

I for combined section — 210.826 



T 



"r •,, t> a- t « / 210.826 aM 

u. - 8-C" --», ^ Radius of gyration = A/-^™— = 3.91. 



ts By similar methods inertia or radius of gyration for 

{ any combination of shapes can readily be obtained. 

"f "~~g Example 2. A " built-up beam" composed of 

. ? 4 angles 3" X 3" X W- 

XJc-^ 2 plates 8" v x/J* 

K — \ _ 1 nloto 1^/1 



3 



1 plate 15" X %". 



D 



Axis AB. 



Jof two 8" X %" plates = 8 ^ X 2 == .167 

4- 8 (area) X 7% 2 (square of distance, d) = 480.500 



480.667 
Jof one 15" X %" plate = £ = 105.469 

I of four 3" X 3" X %" angles = 4 X 1.25 = 5.000 
+ 5.76 (area) x 6.66 2 (square of distance, d 1 ) = 255.488 

Inertia of combined section around AB = 

x> q . r « / 846.624 

Radius of gyration = •A/- T ^ oor =6.61. 
\ 19.385 

Radius of Gyration of Compound Shapes. 

In the case of a pair of any shape without a web the value of R can 
always be readily found without considering the moment of inertia. 

The radius of gyration for any section around an axis parallel to another 
axis passing through its centre of gravity is found as follows : 

Let r — radius of gyration around axis through centre of gravity ; R = 
radius of gyration around another axis parallel to above; d = distance 
between axis. 

R = j/cP + r 2. 



Compound Sections. 



359 



"When r is small, R may be taken as equal to d without 
material error. Thus, in the case of a pair of channels 
latticed together, or a similar construction. 

Example 1. Two 9" channels of 3.89 square inches section 
| placed 5.68" apart ; required the radius of gyration around 
axis CD for combined section. 

Find in column X., page 362, r = .67 and r 2 = 0.45. 

Find distance from base of channel to neutral axis, same 
page, = .60, this added to one-half the distance between 
the two bars, 2.84" = 3.44" d, and d 2 = 11.8336. 

Radius of gyration of the pair as placed = 



160 



Y 11.8336 + 0.45 = 3.505. 



sn inp 



f O F 



The value of R for the whole section in relation to the axis AB is the 
same as for the single channel, to be found in the tables. 

Example 2. Four 3" X 3" X %" angles, placed 
as shown, form a column of 10 inches square ; 
required the radius of gyration. 

Find in column VIII., page 381, r = .93 and 
r 2 = .8649. 

Find distance from side of angle to neutral 
axis, same page, = .84. Subtract this from half 
the width of column = 5 — .84 = 4.16 = d, or 
distance between two axes, d 2 = 17.3056. 

Radius of gyration of four angles as placed = 




A— ■ 



]/ 17.3056 + .8649 = 4.26. 

When the angles are large, as compared with 
the outer dimensions of the combined section, the radius of gyration can 
be taken without serious error from the table of radii of gyration for 
square columns, on page 353. 



Elements of Pencoyd Structural Shapes. 

In the following tables various fundamental properties of rolled sections 
are given, whereby the strength or stiffness of each can be readily deter- 
mined. 

The calculations are made for the least and greatest thickness of the 
various shapes ; intermediate thicknesses of these can be approximated by 
interpolation. 

Moments of Inertia for the sections are obtained as hereafter de- 
scribed. 



Radius of Gyration, equal to -\j 
resistance of struts or columns. 
Section Modulus, equal to 



Inertia 



is used for determining the 



Inertia 



, is used 



distance from axis to extreme fibres' 
for determining transverse strength in beams, etc. 

Coefficient for Safe Load is the calculated load, in net tons, on a beam 
one foot between supports, that produces fibre strains of 16,000 pounds per 
square inch. A corresponding load for any beam is found by dividing this 
coefficient by the length of span in feet. 

Coefficients for Deflection are based on a modulus of elasticity of 
28,000,000 pounds. They apply to beams one foot long, bearing one ton 
(2000 pounds). The deflection of any beam, in inches, is found by multi- 
plying its coefficient by the load in net tons and by the cube of the length 
in feet. 

Maximum Load, in Net Tons, indicates the greatest load that a beam, 
however short, should carry, unless its web is reinforced to prevent crip- 
pling. This load is obtained by the formula : 

2xdt x = 8 tons. 

W= — — ^ — d = depth of beam. 



1 + 



I 2 



3000J2 



t = thickness of web. 
4 = d X secant 45° (P = 



2d 2 ). 



360 



Structural Sections. 



Elements of Pencoyd Beams. 







c~ 


\ 


" n. 








/ 


l u 




I. 


II. 


III. 


IY. 


V. 


VI. 


VII. 


VIII. 


Size, 
in 


Section 
number. 


Area, in 
square 
inches. 


Weight 
per foot, 

in 
pounds. 


Moments of inertia. 


Square of radius 
of gyration. 


inches. 


Axis AB. 


Axis CD. 


Axis AB. 


Axis 
CD. 


24 
24 


240B 
244B 


23.53 
29.42 


80.0 
100.0 


2111.40 
2497.30 


42.84 
57.53 


89.73 

84.88 


1.82 
1.96 


20 
20 


200B 
207B 


19.10 
29.42 


65.0 
100.0 


1179.71 
1649.55 


27.72 
55.57 


61.76 
56.07 


1.45 

1.89 


18 
18 


180B 
187B 


16.13 
26.46 


55.0 
90.0 


809.05 
1187.99 


21.17 

46.03 


50.16 
44.90 


1.31 
1.74 


15 
15 


150B 
158B 


12.35 
23.54 


42.0 
80.0 


443.71 
773.84 


14.43 
40.69 


35.93 
32.87 


1.17 
1.73 


12 
12 


120B 
127B 


9.27 
19.12 


31.5 
65.0 


218.71 
403.48 


9.45 
28.93 


23.59 
21.10 


1.02 
1.51 


10 
10 


100B 
103B 


7.34 
11.75 


25.0 
40.0 


123.07 
175.48 


6.81 
12.36 


16.77 
14.93 


0.93 
1.05 


9 
9 


90B 
93B 


6.17 
10.30 


21.0 
35.0 


84.94 
112.76 


5.06 
7.25 


13.77 
10.95 


0.S2 
0.70 


8 
8 


80B 
83B 


5.29 
7.50 


18.0 
25.5 


57.36 
69.14 


3.72 
4.70 


10.84 
9.22 


0.70 
0.63 


7 
7 


70B 
72B 


4.42 

5.88 


15.0 
20.0 


36.61 
42.55 


2.64 
3.20 


8.28 
7.24 


0.60 
0.54 


6 
6 

6 


60B 
68B 

68B 


3.60 
7.03 
to 
8.15 


12.25 
23.90 

to 
27.70 


22.09 
41.98 

45.36 


1.83 
7.89 

8.99 


6.14 
5.97 

5.57 


0.51 
1.12 

1.10 


5 
5 


50B 
52B 


2.87 
4.34 


9.75 
14.75 


12.12 
15.18 


1.21 
1.67 


4.22 
3.50 


0.42 
0.39 


4 
4 


40B 
43B 


2.20 
3.08 


7.50 
10.50 


5.90 
7.07 


0.76 
1.00 


2.68 
2.30 


0.34 
0.32 


3 
3 


30B 
32B 


1.62 
2.20 


5.50 
7.50 


2.43 

2.87 


0.45 
0.59 


1.50 
1.30 


0.28 
0.27 



Structural Sections. 



361 



Elements of Pencoyd Beams. 



rv 



u 



f\ 



IX. 


X. 


XL 


XII. 


XIIL 


XIV. 


XY. 


IT. 


I. 


Kadius of 
gyration. 


Section 
modu- 
lus. 


Coefficient 
for great- 
est safe 
load, in 
net tons. 


Coefficient for 
deflection. 


Maximum 
load, in 
net tons. 


Weight 

per 
foot, in 
pounds. 


c ® 


Axis 
AB. 


Axis 
CD. 


Axis 
AB. 


Distribu- 
ted load. 


Centre 
load. 


S3 -FH 

00 


9.47 
9.21 


1.35 
1.40 


176.0 

208.1 


938.4 
1109.9 


.000 00076 
.000 00064 


.000 00122 
.000 00103 


75.8 
143.4 


80.0 
100.0 


24 
24 


7.86 
7.49 


1.20 
1.37 


118.0 
165.0 


629.2 

889.8 


.000 00137 
.000 00097 


.000 00217 
.000 00155 


74.2 

184.8 


65.0 

100.0 


20 
20 


7.08 
6.70 


1.14 
1.32 


89.9 
132.0 


479.4 
712.9 


.000 00198 
.000 00135 


:000 00317 
.000 00216 


65.6 

178.2 


55.0 
90.0 


18 
18 


5.99 
5.73 


1.08 
1.32 

1.01 
1.23 


59.2 
103.2 

36.5 
67.3 


315.5 

550.3 

194.4 

358.7 


.000 00357 
.000 00207 


.000 00578 
.000 00331 


47.6 
162.6 

35.6 
134.4 


42.0 
80.0 

31.5 
65.0 


15 
15 


4.86 
4.59 


.000 00727 
.000 00397 


.000 01172 
.000 00635 


12 
12 


4.10 
3.86 


0.96 

1.03 


24.6 
35.1 


131.3 

187.2 


.000 0129 
.000 0091 


.000 0208 
.000 0146 


27.0 

78.8 


25.0 
40.0 


10 
10 


3.71 
3.31 


0.91 
0.84 


18.9 
25.1 


100.7 
133.6 


.000 0185 
.000 0142 


.000 0302 
.000 0227 


21.2 
93.8 


21.0 
35.0 


9 
9 


3.29 
3.04 


0.84 
(K79 


14.3 
17.3 


76.5 
92.2 


.000 0275 
.000 0231 


.000 0447 
.000 0371 


19.4 
58.8 


18.0 
25.5 


8 
8 


2.88 
2.69 


0.78 
0.74 


10.5 
12.2 


55.8 
64.8 


.000 0433 
.000 0376 


.000 0700 
.000 0603 


17.2 
43.2 


15.0 
20.0 


7 
7 


2.48 
2.44 


0.71 
1.06 


7.4 
14.0 


39.3 

74.6 


.000 0717 
.000 0370 


.000 1161 
.000 0591 


13.8 
30.8 


12.25 
23.90 


6 
6 


2.36 


1.05 


15.1 


80.6 


.000 0342 


.000 0547 


50.2 


27.70 


6 


2.05 

1.87 


0.65 
0.62 


4.9 
6.1 


25^ 

32.4 


.000 1305 
.000 1054 


.000 2115 
.000 1689 


11.0 
36.8 


9.75 
14.75 


5 
5 


1.64 
1.52 


0.58 
0.57 


3.0 
3.5 


15.7 
18.9 


.000 2671 
.000 2263 


.000 4346 
.000 3627 


8.2 
23.4 


7.50 
10.50 


4 

4 


1.23 
1.14 


0.53 
0.52 


1.6 
1.9 


8.6 
10.2 


.000 6452 
.000 5575 


.001 0552 
.000 8934 


5.4 
15.6 


5.50 
7.50 


3 
3 



362 



Stkuctural Sections. 





Elements of 


Pencoyd Channels. 

A 








-O- 


(L- 


_L 

-H — 
4 


=J-»° 






I. 


ii. 


III. 


IV. 


V. 


VI. 


VII. 


VIII. 


IX. 


X. 


Size, 
in 


Section 
number. 


Area, 

in 
square 
inches. 


Weight 

per 
foot, in 
pounds. 


Moments of 
inertia. 


Square of 
radius of 
gyration. 


Radius of 
gyration. 


inches. 


Axis 
AB. 


Axis 
CD. 


Axis 
AB. 


Axis 
CD. 


Axis 
AB. 


Axis 
CD. 


15 
15 


150C 
155C 


9.69 
16.17 


33.0 
55.0 


311.21 
469.85 


3.10 
17.20 


32.12 
29.06 


0.84 
1.06 


5.67 
5.39 


0.91 
1.03 


13 
13 


130C 
130C 


9.39 

to 

14.27 


31.9 
to 

48.5 


238.26 
306.25 


11.48 
16.22 


25.38 
21.47 


1.22 
1.14 


5.04 
4.63 


1.11 
1.07 


12 
12 

12 


120C 

128C 

128C 


6.02 
6.01 
to 
9.40 


20.5 
20.5 
to 
32.0 


129.27 
123.98 

164.30 


3.90 
3.10 

4.42 


21.47 
20.63 

17.48 


0.65 
0.52 

0.47 


4.63 
4.54 

4.18 


0.81 
0.72 

0.69 


10 
10 


100C 
104C 


4.41 
10.29 


15.0 
35.0 


67.11 
124.61 


2.28 
5.99 


15.22 
12.11 


0.52 
0.58 


3.90 
3.48 


0.72 
0.76 


9 
9 

9 


90C 
95C 

95C 


3.89 
5.98 
to 
8.23 


13.25 
20.30 
to 
28.00 


47.89 
70.21 

85.40 


1.77 
3.99 

5.17 


12.31 
11.65 

10.32 


0.45 
0.66 

0.63 


3.51 
3.41 

3.21 


0.67 
0.81 

0.79 


8 

8 


80C 

84C 


3.31 
6.25 


11.25 

21.25 


32.51 
51.85 


1.32 
2.97 


9.82 
8.30 


0.40 
0.48 


3.13 

2.88 


0.63 
0.69 


7 
7 


70C 
74C 


2.86 

5.81 


9.75 
19.75 


21.37 
35.85 


0.98 
2.49 


7.47 
6.17 


0.34 
0.43 


2.73 
2.48 


0.59 
0.66 


G 
6 


60C 
65C 


2.35 
4.46 


8.00 
15.10 


13.07 
25.15 


0.69 
5.20 


5.56 
5.64 


0.29 
1.17 


2.36 
2.38 


0.54 
1.08 


5 
5 


50C 
52C 


1.91 
3.38 


6.50 
11.50 


7.37 
10.43 


0.47 
0.82 


3.86 
3.09 


0.25 
0.24 


1.96 
1.76 


0.50 
0.49 


4 
4 


40C 
42C 


1.54 
2.13 


5.25 
7.25 


3.74 
4.52 


0.32 2.4:') 
0.44 2.12 


0.21 
0.21 


1.56 

1.46 


0.45 
0.46 


3 
3 


30O 
32C 


1.18 
1.76 


4.00 
6.00 


1.61 
2.05 


0.20 
0.31 


1.36 
1.16 


0.17 
0.18 


1.17 
1.07 


0.41 
0.42 


2 


22C 
20C 

20C 


1.12 

0.87 
1.06 


3.80 
2.90 
3.60 


0.80 
0.48 
0.54 


0.19 
0.08 
0.11 


0.71 
0.55 
0.51 


0.17 
0.10 
0.10 


0.85 
0.74 
0.71 


0.42 
0.31 
0.32 


m 


17C 


0.33 


1.13 


0.15 


0.01 


0.46 


0.03 


0.67 


0.16 



Stktjctukal Sections. 



363 



Elements of Pencoyd Channels. 

& 4 U — j — 4 -£p 



XI. 


XII. 


XIII. 


XIY. 


XV. 


XVI. 


I. 


Distance, 
d, from 
base to 


Section 
modulus. 


Coefficient 

for greatest 

safe load, in 

net tons. 


Coefficient for 
deflection . 


Maximum 
load, in 
net tons. 


Size, in 
inches. 


neutral 
axis. 


Axis AB. 


Distributed 
load. 


Centre 
load. 


0.79 
0.95 


41.5 
62.7 


221.3 
334.1 


.000 00514 
.000 00340 


.000 00826 
.000 00546 


45.0 
135.4 


15 
15 


1.01 


36.7 


195.5 


.000 00651 


.000 01042 


43.4 


13 


0.97 


47.1 


251.3 


.000 00507 


.000 00811 


130.0 


13 


0.70 
0.62 


21.6 
20.7 


114.9 
110.2 


.000 01237 
.000 01290 


.000 01986 
.000 02072 


23.4 
24.2 


12 
12 


0.62 


27.4 


146.0 


.000 00974 


.000 01564 


82.4 


12 


0.64 
0.76 


13.4 
24.9 


71.6 
132.9 


.000 02384 
.000 01284 


.000 03838 
.000 02067 


16.4 
106.8 


10 
10 


0.60 
0.74 


10.6 
15.6 


56.8 
83.2 


.000 03341 
.000 02210 


.000 05379 
.000 03536 


15.6 
39.0 


9 

9 


0.75 


19.0 


101.2 


.000 01817 


.000 02907 


79.0 


9 


0.57 
0.66 


8.1 
13.0 


43.4 
69.1 


.000 04921 
.000 03086 


.000 07923 
.000 04968 


13.6 
57.0 


8 
8 


0.54 
0.65 


6.1 
10.2 


32.6 
54.6 


.000 07487 
.000 04463 


.000 12054 
.000 07185 


13.2 
57.8 


7 
7 


0.51 
1.07 


4.4 

8.4 


23.2 
44.7 


.000 12242 
.000 06170 


.000 19709 
.000 09872 


10.8 

28.2 


6 
6 


0.49 
0.50 


3.0 
4.2 


15.7 
22.3 


.000 21710 
.000 15340 


.000 34953 
.000 24697 


9.2 

21.4 


5 
5 


0.46 
0.46 


1.9 
2.3 


10.0 
12.1 


.000 42781 
.000 35398 


.000 68877 
.000 56991 


8.2 
18.4 


4 
4 


0.43 
0.45 


1.1 
1.4 


5.7 

7.3 


.000 99377 
.000 78050 


.001 59997 
.001 25660 


6.0 
16.0 


3 
3 


0.47 
0.36 
0.37 


0.7 
0.5 
0.5 


3.8 
2.6 
2.9 


.002 00000 
.003 33333 
.002 96980 


.003 22000 
.005 36666 
.004 78138 


8.6 
6.6 
9.6 


2 


0.18 


0.2 


0.9 


.010 66672 


.017 17342 


2.0 


m 



364 



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368 Struts. 



Thrust. 

Bodies subjected to thrust, such as columns, struts, etc., generally fail 
by bending sideways,— this showing the practical impossibility of main- 
taining the thrust in the exact axial line. As in the case of beams, the 
shape of the cross-section of the column is an important element in the 
supporting power, but with this must be considered the length and the 
manner in which the ends are held. 

The cross-section is best represented by the least radius of gyration, 
usually indicated by r. The length of the column or strut being taken as 

I, in inches, we have the ratio, — , as representing the proportions of the 

column. The manner of supporting the ends are classified according to 
the extent to which the column is secured and the degree to which it is 
maintained in the line of the thrust. 

Owing to the complex nature of the stresses in columns it is difficult to 
determine the maximum fibre stresses, and the various formulas which 
have been devised are the consequence of attempts to embody the results 
of experimental investigations. These have been conducted to determine 
the crippling loads required for the various conditions, the safe load then 
being taken as a certain portion of the crippling load, the latter being 
divided by a so-called factor of safety. 

The following discussion of the subject, prepared by Mr. James Christie 
to accompany the tabulated results of his experiments for the Pencoyd 
Iron Company, represent standard current practice. 

Struts are generally classified in four divisions, with respect to the 
manner in which the ends are secured,— viz., "fixed-ended," " flat-ended," 
"hinged-ended," and "round-ended." 

In the class of "fixed ends" the struts are supposed to be so rigidly 
attached at both ends to the contiguous parts of the structure that the 
attachment would not be severed if the member was subjected to the ulti- 
mate load. "Flat-ended" struts are supposed to have their ends flat and 
normal to the axis of length, but not rigidly attached to the adjoining 
parts. "Hinged ends" embrace the class which have both ends properly 
fitted with pins or ball and socket joints of substantial dimensions, as 
compared with the section of the strut, the centres of these end joints 
being practically coincident with an axis passing through the centre of 
gravity of the section of the strut. " Round-ended" struts are those which 
have only central points of contact, such as balls or pins resting on flat 
plates, but still the centres of the balls or pins coincident with the proper 
axis of the strut. 

If in hinged-ended struts the balls or pins are of comparatively insig- 
nificant diameter, it will be safest in such cases to consider the struts as 
round-ended. 

If there should be any serious deviation of the centres of round or 
hinged ends from the proper axis of the strut there will be a reduction of 
resistance that cannot be estimated without knowing the exact condi- 
tions. 

When the pins of hinged-end struts are of substantial diameter, well 
fitted and exactly centred, experiment shows that the hinged-ended will 
be equally as strong as flat-ended struts. But a very slight inaccuracy of 
the centring rapidly reduces the resistance to lateral bending, and, as it is 
almost impossible in practice to uniformly maintain the rigid accuracy 
required, it is considered best to allow for such inaccuracies to the extent 
given in the tables, which are the average of many experiments. 

It is considered good practice to increase the factors of safety as the 
length of the strut is increased, owing to the greater inability of the long 
struts to resist cross strains, etc. For similar reasons it is considered advisa- 
ble to increase the factor of safety for hinged and round ends in a greater 
ratio than for fixed or flat ends. 

Presuming that one-third of the ultimate load would constitute the 
greatest safe load for the shortest struts, the following progressive factors 
of safety are adopted for the increasing lengths : 

3 + .01 — for flat and fixed ends. 



Struts. 



369 



3 -f .015 — for hinged and round ends. 
r 

I = length of strut. r = least radius of gyration. 

From the above we derive the following factors of safety : 



I 

r 


Fixed 

and flat 

ends. 


Hinged 

and 
round 
ends. 


I 

r 


Fixed 

and flat 

ends. 


Hinged 

and 
round 

ends. 


l 
r 


Fixed 

and flat 

ends. 


Hinged 
and 

round 
ends. 


20 


3.2 


3.30 


110 


4.1 


4.65 


200 


5.0 


6.00 


30 


3.3 


3.45 


120 


4.2 


4.80 


210 


5.1 


6.15 


40 


3.4 


3.60 


130 


4.3 


4.95 


220 


5.2 


6.30 


50 


3.5 


3.75 


140 


4.4 


5.10 


230 


5.3 


6.45 


60 


3.6 


3.90 


150 


4.5 


5.25 


240 


5.4 


6.60 


70 


3.7 


4.05 


160 


4.6 


5.40 


250 


5.5 


6.75 


80 


3.8 


4.20 


170 


4.7 


5.55 


260 


5.6 


6.90 


90 


3.9 


4.35 


180 


4.8 


5.70 


270 


5.7 


7.05 


100 


4.0 


4.50 


190 


4.9 


5.85 


280 


5.8 


7.20. 


Oeuab 






Cast=ii 


'on Cc 


lumns 









but their use is not to be recommended for buildings where the iron frame- 
work must be rigid and afford sufficient lateral stability. The manner in 
which cast-iron columns are connected together and the mode of attaching 
beams and girders to them does not permit of obtaining sufficient rigidity 
for such buildings. Cast-iron columns have more or less internal strains, 
due to the unequal cooling of the metal in the moulds, which makes it 
necessary to employ a large factor of safety. No cast-iron column should 
be used in a building with a factor of safety less than 8. Particular atten- 
tion should be paid to the designing of the cast-iron brackets for support- 
ing the beams and girders, in order that they may not be subjected to 
large internal strains, making them liable to break off under a sudden 
shock. The tables on pages 391 and 392 furnish an easy method of deter- 
mining the safe loads on round and square cast-iron columns. Where the 
loads are eccentrically applied, producing bending strains in the columns, 
cast-iron columns are inadmissible, because of their inability to resist such 
strains. 



24 



370 



Struts. 



Table No. 1. 
Struts of Wrought=iron or Extreme Soft Steel. 

Destructive pressure, in pounds, per square inch. 



Length. 


Fixed ends. 


Flat ends. 


Hinged ends. 


Round ends. 


Least radius of 


gyration. 










20 


46000 


46000 


46000 


44000 


30 


43000 


43000 


43000 


40250 


40 


40000 


40000 


40000 


36500 


50 


38000 


38000 


38000 


33500 


CO 


36000 


36000 


36000 


30500 


70 


34000 


34000 


33750 


27750 


80 


32000 


32000 


31500 


25000 


90 


31000 


30900 


29750 


22750 


100 


30000 


29800 


28000 


20500 


110 


29000 


28050 


26150 


18500 


120 


28000 


26300 


24300 


16500 


130 


26750 


24900 


22650 


14650 


140 


25500 


23500 


21000 


12800 


N 150 


24250 


21750 


18750 


11150 


160 


23000 


20000 


16500 


9500 


170 


21500 


18400 


14650 


8500 


180 


20000 


16800 


12800 


7500 


190 


18750 


15650 


11800 


6750 


200 


17500 


14500 


10800 


6000 


210 


16250 


13600 


9800 


5500 


220 


15000 


12700 


8800 


5000 


230 


14000 


11950 


8150 


4650 


240 


13000 


11200 


7500 


4300 


250 


12000 


10500 


7000 


4050 


260 


11000 


9800 


6500 


3800 


270 


10500 


9150 


6100 


3500 


280 


10000 


8500 


5700 


3200 


290 


9500 


7850 


5350 


3000 


300 


9000 


7200 


5000 


2800 


310 


8500 


6600 


4750 


2650 


320 


8000 


6000 


4500 


2500 


330 


7500 


5550 


4250 


2300 


340 


7000 


5100 


4000 


2100 


350 


6750 


4700 


3750 


2000 


360 


6500 


4300 


3500 


1900 


370 


6150 


3900 


3250 


1800 


380 


5800 


3500 


3000 


1700 


390 


5500 


3250 


2750 


1600 


400 


5200 


3000 


2500 


1500 

















Struts. 




371 


Table No. 2. 


Struts of Wrought=iron or Extreme Soft Steel. 


i Greatest safe load, in pounds per square inch of cross-section, for verti- 


. cal struts. Both ends are supposed to be secured as indicated at the head 


of each column. 


If both ends are not secured alike, take a mean proportional between 


the values given for the classes to which each end belongs. 


If the strut is hinged by any uncertain method, so that the centres of 


-pins and axis of strut may not coincide, or the pins may be relatively 


small and loosely fitted, it is best in such cases to consider the strut as 


"round-ended." 


Length. 


Fixed ends. 


Flat ends. 


Hinged ends. 




Least radius of 


Round ends. 


gyration. 










20 


14380 


14380 


13940 


13330 


30 


13030 


13030 


12460 


11670 


40 


11760 


11760 


11110 


10140 


50 


10860 


10860 


10130 


8930 


60 


10000 


10000 


9230 


7820 


70 


9190 


9190 


8330 


6850 


80 


8420 


8420 


7500 


5950 


90 


7950 


7920 


6840 


5230 


100 


7500 


7450 


6220 


4560 


110 


7070 


6840 


5620 


3980 


120 


6670 


6260 


5060 


3440 


1 130 

I 

1 140 

1 150 


6220 

5800 
5390 


5790 


4580 


2960 

2510 
2120 


5340 
4830 


4120 
3570 


160 


5000 


4350 


3060 


1760 


170 


4570 


3920 


2640 


1530 


180 


4170 


3500 


2250 


1310 


190 


3830 


3190 


2020 


1150 


200 


3500 


2900 


1800 


1000 


210 


3190 


2670 


1590 


890 


220 


2880 


2440 


1400 


790 


230 


2640 


2250 


1260 


720 


240 


2410 


2070 


1140 


650 


250 


2180 


1910 


1040 


600 


1 260 


1960 


1750 


940 


550 


I 270 


1840 


1610 


870 


500 


280 


1720 


1460 


790 


440 


290 


1610 


1330 


730 


410 


300 


1500 


1200 


670 


370 


1 310 


1390 


1080 


620 


350 


320 


1290 


970 


580 


• 320 


330 


1190 


880 


540 


290 


340 


1090 


800 


490 


260 


350 


1040 


720 


450 


240 


360 


980 


650 


420 


230 


370 


920 


580 


380 


210 


380 


850 


510 


340 


200 


390 


800 


470 


310 


80 


400 


740 


430 


280 


70 


! 

i 











372 



Struts. 



Table No. 3. 

Struts of Medium Steel. 

Destructive pressure, in pounds per square inch, for steel of mediuD) 
grade, tensile strength about 70,000 pounds per square inch. 
For extreme soft steel, use Table No. 1. 



Length. 


Fixed ends. 


Flat ends. 


Hinged ends. 




Least radius of 


Bound ends. 


gyration. 










20 


70000 


70000 


70000 


66900 


30 


51000 


51000 


51000 


47700 


40 


46000 


46000 


46000 


41900 


50 


44000 


44000 


44000 


38800 


60 


42000 


42000 


42000 


35600 


70 


40000 


40000 


39700 


32600 


80 


38000 


38000 


37400 


29700 


90 


36100 


36000 


34700 


26500 


100 


34200 


34000 


31900 


23400 


110 


33100 


32000 


29800 


21100 


120 


31900 


30000 


27700 


18800 


130 


30100 


23000 


25500 


16500 


140 


28200 


26000 


23200 


14200 


150 


26800 


24000 


20700 


12300 


160 ' 


25300 


22000 


18100 


10400 


170 


23400 


20000 


15900 


9240 


180 


21400 


18000 


13700 


8030 


190 


19400 


16200 


12200 


6990 


200 


17900 


14800 


11000 


6120 


210 


16200 


13600 


9800 


5500 


220 


15000 


12700 


8800 


5000 


230 


14000 


11950 


8100 


4650 


240 


13000 


11200 


7500 


4300 


250 


12000 


10500 


7000 


4050 


260 


11000 


9800 


6500 


3800 


270 


10500 


9150 


6100 


3500 


280 


10000 


8500 


5700 


3200 


290 


9500 


7850 


5330 


3000 


300 


9000 


7200 


5000 


2800 



Steuts. 



373 



Table No. 4. 

Struts of Medium Steel. 

Greatest safe load for steel of medium grade, tensile strength about 
70,000 pounds. 

For extreme soft steel, use Table No. 2. 

The figures are the working loads, in pounds per square inch, for 
vertical struts. 

Both ends are supposed to be secured as indicated at the head of each 
column. 

If both ends are not secured alike, take a mean proportional between 
the values given for the classes to which each end belongs. 

If the strut is hinged by any uncertain method, so that the centres of 
pins and axis of strut may not coincide, or the pins may be relatively 
small and loosely fitted, it is best in such cases to consider the strut as 
"round-ended." 



Length. 


Fixed ends. 


Flat ends. 


Hinged ends. 




Least radius of 


Round ends. 


gyration. 










20 


21900 


21900 


21200 


20300 


30 


15400 


15400 


14800 


13800 


40 


13500 


13500 


12800 


11600 


50 


12600 


12600 


11700 


10300 


60 


11700 


11700 


10800 


9130 


70 


10800 


10800 


9800 


8050 


80 


10000 


10000 


8900 


7070 


90 


9260 


9230 


7980 


6090 


100 


8550 


8500 


7090 


5200 


110 


8070 


7800 


6410 


4540 


120 


7590 


7140 


5770 


3920 


130 


7000 


6510 


5150 


3330 


140 


6410 


5910 


4550 


2780 


150 


5950 


5330 


3940 


2340 


160 


5500 


4780 


3350 


1920 


170 


4980 


4250 


2860 


1660 


180 


4460 


3750 


2400 


1410 


190 


3960 


3310 


2080 


1190 


200 


3580 


2960 


1830 


1020 


210 


3180 


2670 


1590 


890 


220 


2880 


2440 


1400 


790 


230 


2640 


2250 


1250 


720 


240 


2410 


2070 


1140 


650 


250 


2180 


1910 


1040 


600 


260 


1960 


1750 


940 


550 


270 


1840 


1610 


860 


500 


280 


1720 


1460 


790 


440 


290 


1610 


1330 


720 


410 


300 


1500 


1200 


670 


370 


1 











Stbuts, 







Table No, 5. 






Struts of Hard Steel. 




Destructive pressure, in pounds per square inch, for hard 
strength about 100.000 pounds. 
For softer steel, see Table No, 3. 


steel, tensile 


Length, 


Fixed ends. 


Flat ends. 


Hinged ends. 




Least radius of 


Round ends, 


gyration. 










20 


100000 


100000 


100000 


95600 


30 


74000 


74000 


74000 


09800 


40 


62000 


62000 


62000 


56600 


50 


00000 


60000 00000 


52900 


60 


58000 


58000 


58000 


49100 


70 


55500 


55500 


55100 


45300 


80 


53000 


53000 


52200 


41400 


90 


49000 


49700 


47800 


36600 


LOO 


46800 


46500 


48700 


32000 


110 


44700 


48200 


40400 


28500 


120 


42000 


40000 


80900 


25100 


130 


80400 


80700 


33500 


21000 


140 


36300 


33500 


29900 


18200 


150 


34200 


80700 


26500 


15700 


160 


32200 


28000 


28100 


13300 


170 


20800 


25500 


20300 


11800 


ISO 


27400 


23000 


17500 


io;\\) 


190 


25100 


21000 


15800 


9000 


200 


22900 


19000 


14100 


7800 


•210 


20900 


17200 


12400 


0950 


220 


18300 


15500 


10700 


0100 


230 


10900 


14400 


9820 


5600 


240 


15500 


13400 


8900 


5140 


250 


14200 


12400 


8270 


4780 


200 


12000 


11500 


7630 


4400 


270 


12200 


10000 


7060 


4050 


280 


11400 


9700 


0500 


3650 


290 


10900 


9000 


0180 


8 HO 


800 


10000 


8500 


5890 


3800 



Struts. 



375 



Table No. 6. 
Struts of Hard Steel. 

Greatest safe load for hard steel, tensile strength about 100,000 pounds. 

For soft steel, see Table No. 4. 

The figures are the working loads, in pounds per square inch, for verti- 
cal struts. 

Both ends are supposed to be secured as indicated at the head of each 
column. 

If both ends are not secured alike, take a mean proportional between 
the values given for the classes to which each end belongs. 

If the strut is hinged by any uncertain method, so that the centres of 
pins and axis of strut may not coincide, or the pins may be relatively 
small and loosely fitted, it is best in such cases to consider the strut as 
"round-ended." 



Length. 


Fixed ends. 


Flat ends. 


Hinged ends. 




Least radius of 


Round ends. 


gyration. 










20 


31200 


31200 


30300 


29000 


30 


22400 


22400 


21400 


20100 


40 


18200 


18200 


17200 


15700 


50 


17100 


17100 


16000 


14100 


60 


16100 


16100 


14900 


12600 


70 


15000 


15000 


13600 


11200 


80 


13900 


13900 


124C0 


9860 


90 


12800 


12700 


11000 


8410 


100 


11700 


11600 


9710 


7110 


110 


10900 


10500 


8670 


6130 


120 


10100 


9520 


7690 


5230 


130 


9160 


8530 


6770 


4360 


140 


8250 


7610 


5860 


3570 


150 


7600 


6820 


5050 


2990 


160 


7000 


6090 


4280 


2460 


170 


6340 


5420 


3660 


2130 


180 


5710 


4790 


3070 


1810 


190 


5120 


4280 


2700 


1550 


200 


4580 


3800 


2350 


1310 


210 


3980 


3370 


2020 


1130 


220 


3520 


2980 


1700 


970 


230 


3190 


2720 


1500 


870 


240 


2870 


2480 


1360 


780 


250 


2580 


2250 


1220 


710 


260 


2300 


2050 


1100 


650 


270 


2240 


1860 


1000 


570 


280 


1960 


1670 


900 


510 


290 


1850 


1520 


830 


470 


300 


1800 


1420 


780 


440 



376 



Section Elements. 



Elements of Pencoyd Z=Bars. 



A 



< 



If 



a 

a 


Size, in inches. 


Area, 

in 
square 
inches. 


Weight 

per 

foot, 

in 

pounds. 


Moments of 
inertia. 


Resistance. 


.2 & 

<v 
m 


Axis 

AB. 


Axis 

CD. 


Axis 
EF. 


Axis 
AB. 


Axis 
CD. 


30Z 


2%X3 X 2% X % 


1.94 


6.60 


2.81 


2.61 


0.59 


1.9 


1.0 


31Z 


2i£ X 3ft X 2H X ft 


2.44 


8.29 


3.52 


3.38 


0.74 


2.3 


1.3 


32Z 


2% X 3% X 2% X % 


2.94 


10.00 


4.34 


4.22 


0.92 


2.8 


1.7 


33Z 


2H X 3 X 2|| X ft 


3.25 


11.15 


4.20 


4.24 


0.95 


2.8 


1.7 


34Z 


2ft X 8& X 2ft X if 


3.51 


11.93 


4.54 


4.64 


1.01 


3.0 


1.9 


35Z 


2§i X 3ft X 2ff X K 


3.75 


12.75 


4.88 


5.04 


1.11 


3.2 


2.0 


40Z 


2% X 4 X 2% X H 


2.32 


7.88 


5.95 


3.47 


0.95 


3.0 


1.3 


41Z 


211 X 4ft X 2i| X ft 


2.91 


9.89 


7.52 


4.49 


1.23 


3.7 


1.6 


42Z 


3 X 4y 8 X 3 x % 


3.52 


11.90 


9.14 


5.58 


1.53 


4.4 


2.0 


43Z 


2§* X 4 X 2fi X ft 


3.96 


13.46 


9.40 


6.09 


1.63 


4.7 


2.2 


44Z 


3& X 4ft X 3& X K 


4.56 


15.50 


10.92 


7.21 


1.94 


5.4 


2.6 


45Z 


% X 4^ X 8ft X ft 


5.16 


17.54 


12.40 


8.40 


2.27 


6.0 


3.0 


46Z 


3ft X 4 X 3ft X % 


5.55 


18.80 


12.11 


8.73 


2.32 


6.1 


3.2 


47Z 


3% X 4ft X 3^ X U 


6.14 


20.87 


13.52 


9.95 


2.67 


6.7 


3.6 


48Z 


3ft X 4^ X 3ft X % 


6.75 


22.95 


14.97 


11.24 


3.03 


7.3 


4.0 


50Z 


3ft X 5 X 3ft X ft 


3.36 


11.42 


13.14 


5.81 


1.86 


5.3 


1.9 


51Z 


3M X 5ft X 3% X % 


4.05 


13.77 


15.93 


7.20 


2.28 


6.3 


2.4 


52Z 


3ft X 5>^ X 3ft X ft 


4.75 


16.15 


18.76 


8.67 


2.75 


7.3 


2.8 


53Z 


% x 5 x 3 3 7 2 x y 2 


5.23 


17.78 


19.03 


8.77 


2.76 


7.6 


3.0 


54Z 


3 3 % X 5ft X 3& X 1% 


5.91 


20.09 


21.65 


10.19 


3.20 


8.6 


3.4 


55Z 


3& X b% X 3£* X % 


6.60 


22.44 


24.33 


11.70 


3.73 


9.5 


3.9 


56Z 


3^X5 X 3% X H 


6.96 


23.66 


23.68 


11.37 


3.59 


9.5 


3.9 


57Z 


3ft X 5ft X 3ft X % 


7.64 


25.97 


26.16 


12.83 


4.12 


10.3 


4.4 


60Z 


3^X6 X 33^ X % 


4.59 


15.61 


25.32 


9.11 


3.11 


8.4 


2.8 


61Z 


3ft X 6ft X 3ft X ft 


5.39 


18.32 


29.80 


10.95 


3.74 


9.8 


3.3 


62Z 


3% x ey 8 x 3-% x % 


6.19 


21.05 


34.36 


12.87 


4.37 


11.2 


3.8 


63Z 


3^X6 X3^X ft 


6.68 


22.71 


34.64 


12.59 


4.37 


11.6 


3.9 


64Z 


3ft X 6ft X 3ft X % 


7.46 


25.36 


38.86 


14.42 


4.92 


12.8 


4.4 


65Z 


3 5 ^X6%X3%X H 


8.25 


28.05 


43.18 


16.34 


5.66 


14.1 


5.0 


66Z 


3^X6 X 3% X % 


8.64 


29.37 


42.12 


15.44 


5.61 


14.0 


4.9 


67Z 


3ft X 6ft X 3ft X Jg 


9.38 


31.89 


46.13 


17.27 


6.16 


15.2 


5.5 


68Z 


3% X 6>£ X S% X % 


10.16 


34.54 


50.22 


19.18 


6.85 


16.4 


6.0 



Section Elements. 



377 



Elements of Pencoyd Z-Bars. 



OQ 
I 



c 



f 



J D 



Radius 


of gyration. 


Coefficient, in net tons, 

for greatest 

safe load distributed. 


Coefficient for deflec- 
tion about axis AB. 


Maxi- 
mum 
load, 
in net 
tons. 


a 


Axis 

AB. 


Axis 
CD. 


Least. 
Axis 
EF. 


Fibre stress, 
16,000 
pounds. 


Fibre stress, 
12,000 
pounds. 


Distrib- 
uted. 


Centre. 


a 
o .^ 

+3 © 


1.20 


1.16 


0.55 


100 


7.5 


.000 5694 


.000 9167 


11.0 


30Z 


1.20 


1.18 


0.55 


12.3 


9.2 


.000 4545 


.000 7317 


14.4 


31Z 


1.21 


1.20 


0.56 


14.8 


11.1 


.000 3687 


.000 5937 


18.0 


32Z 


1.13 


1.14 


0.54 


14.9 


11.2 


.000 3809 


.000 6132 


20.4 


33Z 


1.14 


1.15 


0.54 


16.0 


12.0 


000 3524 


.000 5674 


22.2 


34Z 


1.14 


1.16 


0.55 


17.0 


12.8 


.000 3279 


.000 5279 


24.0 


35Z 


1.60 


1.22 


0.64 


15.9 


11.9 


.000 2689 


.000 4329 


13.6 


40Z 


1.61 


1.24 


0.65 


19.7 


14.8 


000 2128 


.000 3426 


18.2 


41Z 


1.62 


1.26 


0.66 


23.6 


17.7 


.000 1750 


.000 2817 


23.0 


42Z 


1.54 


1.24 


0.64 


25.1 


18.8 


.000 1702 


.000 2740 


26.6 


43Z 


1.55 


1.27 


0.65 


28.7 


21.5 


.000 1465 


.000 2359 


31.2 


44Z 


1.55 


1.28 


0.66 


32.1 


24.1 


.000 1290 


.000 2077 


35.8 


45Z 


1.48 


1.26 


0.65 


32.3 


24.2 


.000 1321 


.000 2127 


39.0 


46Z 


1.48 


1.27 


0.66 


35.5 


26.6 


.000 1183 


.000 1905 


43.6 


47Z 


1.49 


1.29 


0.67 


38.7 


29.0 


.000 1069 


.000 1721 


48.6 


48Z 


1.98 


1.32 


0.74 


28.0 


21.0 


.000 1218 


.000 1961 


21.4 


50Z 


1.98 


1.33 


0.75 


33.6 


25.2 


.000 1005 


000 1618 


27.0 


51Z 


1.99 


1.35 


0.76 


39.1 


29.3 


.000 0853 


.000 1373 


32.8 


52Z 


1.91 


1.30 


0.73 


40.6 


30.5 


.000 0841 


.000 1354 


37.6 


53Z 


1.91 


1.31 


0.74 


45.6 


34.2 


.000 0739 


.000 1190 


43.2 


54Z 


1.92 


1.33 


0.75 


50.6 


38.0 


.000 0658 


.000 1059 


49.0 


55Z 


1.84 


1.28 


0.72 


50.5 


37.9 


.000 0676 


.000 1088 


53.2 


56Z 


1.85 


1.30 


0.73 


55.1 


41.3 


.000 0612 


.000 0984 


59.0 


57Z 


2.35 


1.41 


0.82 


45.0 


33.8 


.000 0632 


.000 1017 


30.8 


60Z 


2.35 


1.43 


0.83 


52.4 


39.3 


.000 0537 


.000 0864 


37.6 


61Z 


2.36 


1.44 


0.84 


59.8 


44.9 


.000 0466 


.000 0750 


44.6 


62Z 


2.28 


1.37 


0.81 


61.6 


46.2 


.000 0462 


.000 0744 


50.2 


63Z 


2.28 


1.39 


0.81 


68.4 


51.3 


.000 0412 


.000 0663 


57.0 


64Z 


2.29 


1.41 


0.83 


75.2 


56.4 


.000 0370 


.000 0596 


64.0 


65Z 


2.21 


1.34 


0.81 


74.9 


56.2 


.000 0380 


.000 0612 


69.0 


66Z 


2.22 


1.36 


0.81 


81.2 


60.9 


.000 0347 


.000 0559 


76.0 


67Z 


2.22 


1.37 


0.82 


87.5 


65.6 


.000 0319 


.000 0513 


83.0 


68Z 



378 



Section Elements. 



Elements of Z-Bar Columns. 



I = moment of inertia. 



Y-- 




Y -B = radius of gyration. 



The thicknesses of Web Plate and Z-Bars are the same. 



V web plate. 7%" face to face. 



Size of Z-bar, in 
inches. 



Area 
of4Z- 
bars 
and 1 
plate. 



Axis XX 



I. R. 



Axis YY. 



I. B, 



Area 
of4Z- 
bars 
aud 1 
plate. 



8" web plate. 8}^ 7/ face to face. 
YY. 



Axis XX. 



I. R, 



Axis '. 



R. 



3^X 
3ft X 
3%X 
3KX 
3ft X 
3%X 
3^X 
3ft X 
3%X 



6 X 
6ft X 
6^X 
6 X 
6ft X 
6^X 
6 X 
6ft X 
6^X 



3^X% 
3ft X ft 
3%X^ 
3^Xft 
3ft XVs 
3%X« 
3>^X% 

Q9 Y 13 

3%X% 



3ft X 5 X 3ft X ft 
334X5ftX3^X% 
3ft X5^X 3ft X ft 
3& X 5 X 3^ X % 
3^5 X Ojg X 3 5 9 ^ X ft 

m x 5% x m x % 

3^X5 X 3% X « 
3ft X 5ft X 3ft X % 



2%X4 X 

m x 4ft x 

3 X iV 8 X 
2fi X 4 X 
3^ X 4ft X 
3 3 3 5 X 4% X 
3ft X 4 X 
W% X 4ft X 
3ft X iV 8 X 



2%XM 
2« X ft 
3 X% 
2fi X ft 
3&X^ 
3AXA 
3ft X % 
Z% X U 
3ft X % 



%X3 X 2% X 34 
2HX3ftX2j£Xft 
^X3%X23/ 4 X 3 / 8 
2§£ X 3 X 2§J X ft 
2§§X3ftX2f§X% 



20.99 
24.62 
28.26 
30.66 
34.22 
37.81 
39.81 
43.21 
46.77 



3.55 
3.53 
3.51 
3.45 
3.43 
3.41 
3.36 
3.34 
3.32 



287.85 
346.98 
409.27 
426.34 
489.21 
555.79 
562.39 
628.18 
699.11 



3.70 
3.75 
3.80 
3.73 
3.78 
3.83 
3.76 
3.81 
3.87 



21.36 
25.06 
28.76 
31.22 
34.84 
38.50 
40.56 
44.02 
47.64 



337.09 
391.45 
444.60 
469.13 
518.08 
566.52 
579.76 
622.55 
666.66 



3.97 287.86 
3.95 346.99 
3.93 409.29 
3.88 426.36 
3.86 489.23 
3.83 555.82 
3.78 562.43 
3.76628.23 
3.74J699.17 



V web plate. 7%" face to face. 



8" web plate. 8%" face to 



3.67 
3.72 
3.77 
3.69 
3.75 
3.80 
3.72 
3.78 
3.83 
face. 



15.63 
18.83 
22.06 
24.42 
27.58 
30.78 
32.65 
35.81 



193.88 
231.00 
267.64 
287.66 
321.15 
354.33 
364.87 
395.55 



3.52 
3.50 
3.48 
3.43 
3.41 
3.39 
3.34 
3.32 



147.41 
183.49 
222.06 
234.48 
273.70 
315.67 
320.05 
363.02 



3.07 
3.12 
3.17 
3.10 
3.15 
3.20 
3.13 
3.18 



15.94 
19.20 
22.50 
24.92 
28.14 
31.40 
33.34 
36.56 



248.26 
295.96 
343.27 
370.54 
414.03 
457.20 
472.86 
513.07 



3.95 
3.92 
3.91 
3.86 
3.83 
3.81 
3.77 
3.74 



147.41 
183.50 
222.07 
234.50 
273.72 
315.69 
320.08 
363.05 



6" web plate. 6%" face to face. 



7" web plate. 7%" face to 



3.04 
3.09 
3.14 
3.07 
3.12 
3.17 
3.10 
3.15 
face. 



10.78 
13.52 
16.33 
18.47 
21.24 
24.02 
25.95 
28.69 
31.50 



3.07 
3.05 
3.04 
3.00 
2.98 
2.96 
2.92 
2.91 
2.89 



65.71 
85.80 
107.87 
115.62 
138.66 
163.07 
167.28 
192.77 
220.51 



2.47 
2.52 
2.57 
2.50 
2.55 
2.60 
2.54 
2.59 
2.64 



11.03 
13.83 
16.71 
18.90 
21.74 
24.58 
26.58 
29.37 
32.25 



134.71 
166.94 
199.42 
220.65 
250.89 
280.45 
296.36 
323.88 
351.59 



3.49 
3.47 
3.45 
3.42 
3.40 
3.38 
3.34 
3.32 
3.30 



65.79 
85.80 
107.87 
115.63 
138.67 
163.08 
167.30 
192.80 
220.55 



G" web plate. 6%" face to face. 



7" web plate. 7%"face to 



2.44 
2.49 
2.54 
2.47 
2.52 
2.58 
2.51 
2.56 
2.61 
face. 



9.26 
11.64 
14.01 
15.63 
18.00 



84.78 
105.17 
125.10 
134.64 
153.14 



3.03 
3.01 
2.99 
2.93 
2.92 



31.74 
41.89 
53.41 
55.24 
67.17 



1.85 
1.90 
1.95 
1.88 
1.93 



9.51 
11.95 
14.39 
16.06 
18.50 



112.65 


3.44 


31.74 


139.88 


3.42 


41.89 


166.56 


3.40 


53.42 


180.30 


3.35 


55.25 


205.32 


3.33 


67.18 



1.83 
1.87 
1.93 
1.85 
1.90 



Section Elements. 



379 



Elements of Pencoyd Tees. 

Uneven Legs. 
€ 



-*-« 



I. 


II. 


in. 


IV. 


V. 


VI. 


VII. 


VIII. 


IX. 


X. 


XI. 


u 

CD 

§1 


Size, in 


Area, 
in 


Weight 
per 


Moments 
of inertia. 


Kesistance. 


Kadius of 
gyration. 


Dist., d, 

from 
hase to 


is 


inches. 


square 
inches. 


foot, in 
pounds. 


Axis 
AB. 


Axis 
CD. 


Axis 
AB. 


Axis 
CD. 


Axis 
AB. 


Axis 
CD. 


neutral 
axis. 


66T 


6 X4K 


8.21 


28.2 


14.74 


13.81 


4.71 


4.60 


1.33 


1.29 


1.37 


64T 


6 X4 


4.61 


15.6 


5.82 


8.19 


1.92 


2.73 


1.12 


1.33 


0.97 


1 65T 


6 X5% 


11.58 


39.0 


28.68 


18.75 


8.19 


6.25 


1.57 


1.27 


1.75 


: 53T 


5 X3K 


4.95 


17.0 


5.29 


5.47 


2.17 


2.19 


1.03 


1.05 


1.06 


54T 


5 X4 


4.54 


15.3 


6.16 


5.41 


2.11 


2.16 


1.17 


1.09 


1.08 


42T 


4X2 


1.93 


6.5 


0.53 


1.75 


0.34 


0.87 


0.52 


0.95 


0.46 


43T 


4X3 


.2.67 


9.0 


1.99 


2.10 


0.90 


1.05 


0.87 


0.89 


0.78 


44T 


4X3 


3.05 


10.2 


2.24 


2.44 


1.02 


1.22 


0.85 


0.89 


0.81 


45T 


4 X4^ 


4.29 


14.6 


7.87 


2.80 


2.50 


1.40 


1.37 


0.81 


1.37 


i 46T 


4X4^ 


4.65 


15.8 


4.93 


3.67 


2.05 


1.63 


1.03 


0.89 


1.11 


i 47 T 


3.38 


11.4 


6.31 


2.11 


1.96 


1.06 


1.37 


0.79 


1.28 


38T 


3^X3 


2.11 


7.0 


1.65 


1.18 


0.75 


0.67 


0.88 


0.75 


0.80 


39T 


3^X3 


2.46 


8.5 


1.91 


1.41 


0.88 


0.81 


0.88 


0.75 


0.83 


30T 


3 Xl^ 


1.20 


4.0 


0.18 


0.60 


0.16 


0.40 


0.39 


0.71 


0.36 


i 31T 


3X2% 


1.46 


5.0 


0.78 


0.60 


0.42 


0.40 


0.73 


0.64 


0.66 


i 32T 


3 X2>^ 


1.76 


6.0 


0.93 


0.74 


0.51 


0.49 


0.73 


0.65 


0.68 


j 33T 


3X2^ 


2.06 


7.0 


1.08 


0.89 


0.60 


0.59 


0.72 


0.66 


0.71 


| 34T 


3 X2^ 


2.38 


8.0 


1.32 


0.91 


0.78 


0.61 


0.74 


0.62 


0.80 


35T 


3 X3^ 


2.46 


8.3 


2.82 


0.89 


1.17 


0.59 


1.07 


0.60 


1.08 


1 36T 


3 X3K 


2.81 


9.5 


3.19 


1.04 


1.33 


0.69 


1.07 


0.61 


1.10 


, 28T 


2%X1% 


1.96 


6.6 


0.56 


0.60 


0.50 


0.44 


0.54 


0.56 


0.64 


29T 


2%X2 


2.14 


7.2 


0.82 


0.61 


0.66 


0.44 


0.62 


0.54 


0.75 


25T 


2^X1^ 


0.97 


3.3 


0.10 


0.33 


0.11 


0.26 


0.32 


0.58 


0.31 


26T 


2^X2% 


1.68 


5.7 


1.16 


0.43 


0.60 


0.34 


0.83 


0.51 


0.83 


27T 


23^X3 


1.76 


6.0 


1.48 


0.44 


0.71 


0.35 


0.92 


0.50 


0.93 


24T 


2^X A 


0.66 


2.2 


0.01 


0.24 


0.03 


0.21 


0.14 


0.60 


0.17 


20T 


2 X A 


0.60 


2.0 


0.01 


0.17 


0.03 


0.17 


0.14 


0.53 


0.17 


22T 


2 X 1A 


0.62 


2.0 


0.04 


0.16 


0.05 


0.16 


0.24 


0.51 


0.23 


21T 


2X1 


0.72 


2.5 


0.05 


0.17 


0.07 


0.17 


0.26 


0.49 


0.27 


' 23T 


2 xiy 2 


0.91 


3.0 


0.16 


0.17 


0.15 


0.17 


0.42 


0.44 


0.45 


17T 


1% X 1A 


0.56 


1.9 


0.05 


0.11 


0.06 


0.13 


0.30 


0.45 


0.24 


18T 


mxm 


1.04 


3.5 


0.12 


0.21 


0.14 


0.24 


0.35 


0.45 


0.40 


i 15T 


VAX if 


0.41 


1.4 


0.02 


0.07 


0.03 


0.09 


0.22 


0.41 


0.21 


j 12T 


i^x M 


0.35 


1.2 


0.02 


0.03 


0.03 


0.05 


0.24 


0.30 


0.22 



380 



Section Elements. 



Elements of Pencoyd Angles. 



> 



21 i 



I. 


II. 


in. 


IV. 


V. 


VI. 


VII. 


Section 


Size, in 
inches. 


Thick- 
ness. 


Area, in 
square 
inches. 


Weight 
per foot, 

in 
pounds. 


Moments of inertia. 




Axis AB. 


Axis EF. 


880A 

888A 


8X8 
8^X8^ 


i 


7.75 
15.29 


26.4 

52.8 


48.47 
94.14 


19.60 
39.01 


660A 
669A 


6 X6 

6^X6^ 


% 

15 
16 


4.36 
10.65 


14.8 
35.9 


15.37 
36.69 


6.20 
15.48 


550A 
559A 


5 X5 

5^X5y 4 




3.61 

8.77 


12.3 
29.4 


8.73 
20.72 


3.54 
9.09 


440A 
447A 


4 X4 
4^X4^ 


ft 


2.40 
5.69 


8.2 
18.6 


3.69 

8.71 


1.50 
3.82 


350A 
355A 


3^X3^ 
3^ X 3% 


5 

16" 
% 


2.09 
4.06 


7.1 
13.7 


2.45 
4.60 


0.99 
1.97 


330A 
336A 


3X3 
3ft X 3 T 3 S 




1.44 
3.51 


4.9 
11.5 


1.25 
3.01 


0.50 
1.32 


275A 
279A 


2^X2% 
3 X3 




1.31 
2.70 


4.5 

8.6 


0.95 
2.11 


0.39 
0.90 


250A 
255A 


2^X2^ 
2%X2% 


ft 

y* 


0.90 
2.33 


3.1 

7.8 


0.54 
1.33 


0.22 
0.59 


225A 
228A 


2^X2^ 
2ft X 2& 


ft 


0.81 
1.66 


2.7 
5.4 


0.39 
0.85 


0.16 
0.37 


220A 
223A 


2X2 
2ft X 2 T ^ 


ft 


0.71 
1.47 


2.5 

4.8 


0.27 
0.61 


0.11 
0.26 • 


175A 

178A 


1%X1% 

ill x in 


ft 


0.62 
1.28 


2.1 
4.1 


0.18 
0.39 


0.08 
0.18 


150A 
154A 


iKxi^ 

1% x 1% 


3^ 
% 


0.36 
1.14 


1.2 
3.5 


0.08 
0.29 


0.03 
0.13 


125A 
127A 


Ws X 1% 


y* 


0.30 
0.62 


1.0 
2.0 


0.05 
0.10 


0.02 
0.04 


110A 
112A 


1 XI 

i%Xiy a 




0.23 
0.49 


0.8 
1.5 


0.02 
0.05 


0.01 
0.02 



Jv 



Section Elements. 



381 



Elements of Pencoyd Angles. 



A— 



v ; 



B 



-J-_-L.- 



VIII. 


IX. 


X. 


XI. 


I. 


Radius of gyration. 


Resistance. 


Distance from base 
to neutral axis. 


Section 
number. 


Axis AB. 


Axis EF. 


Axis AB. 


d. 




2.50 


1.59 


8.34 


2.19 


880A 


2.48 


1.60 


16.18 


2.43 


888A 


1.88 


1.19 


3.53 


1.64 


660A 


1.86 


1.21 


8.43 


1.90 " 


669A 


1.56 


0.99 


2.42 


1.39 


550A 


1.54 


1.02 


5.76 


1.65 


559A 


1.24 


0.79 


1.28 


1.12 


440A 


1.24 


0.82 


3.10 


1.34 


447A 


1.08 


0.69 


0.98 


0.99 


350A 


1.06 


0.70 


1.84 


1.13 


355A 


0.93 


0.59 


0.58 


0.84 


330A 


0.93 


0.61 


1.39 


1.02 


336A 


0.85 


0.55 


0.48 


0.78 


275A 


0.88 


0.58 


1.02 


0.93 


279A 


0.77 


0.49 


0.30 


0.70 


250A 


0.76 


0.50 


0.75 


0.84 


255A 


0.69 


0.44 


0.24 


0.63 


225A, 


0.72 


0.47 


0.50 


0.75 


228A 


0.62 


0.39 


0.19 


0.58 


220A 


0.64 


0.42 


0.40 


0.68 


223A 


0.54 


0.36 


0.15 


0.51 


175A 


0.55 


0.38 


0.30 


0.63 


178A 


0.47 


0.28 


0.07 


0.42 


150A 


0.50 


0.34 


0.25 


0.57 


154A 


0.41 


0.26 


0.06 


0.35 


125A 


0.40 


0.25 


0.11 


0.43 


127A 


0.29 


0.21 


0.03 


0.30 


110A 


0.32 


0.20 


0.07 


0.37 


112A 


: 











382 



Section Elements. 



Elements of Pencoyd Angles. 




I. 


II. 


III. 


IV. 


V. 


VI. 


VII. 


VIII. 


Section 


Size, in 
iuches. 


Thick- 
ness. 


Area, in 
square 
inches. 


Weight 
per foot, 

in 
pounds. 


Moments of inertia. 


number. 


Axis 
AB. 


Axis 

CD. 


Axis 
EF. 


860A 


8X6 


A 


6.75 


23.0 


44.38 


21.73 


12.04 


868A 


8M X 6^ . 


1 


13.29 


45.6 


85.34 


41.67 


24.76 


730A 


7 X3K 


A 


5.00 


17.0 


25.29 


4.37 


3.64 


738A 


7^X3% 


1 


9.79 


32.5 


48.59 


8.47 


7.47 


650A 


6^X4 


% 


3.80 


12.9 


16.83 


5.03 


3.29 


659A 


6%X4% 


15 
T6 


9.48 


31.9 


42.40 


12.91 


9.28 


640A 


6X4 


% 


3.61 


12.2 


13.48 


4.91 


3.04 


649A 


6%X4% 


15 
15 


9.01 


29.4 


33.95 


12.47 


8.57 


630A 


6 X3K 


VS 


3.42 


11.6 


12.82 


3.32 


2.39 


639A 


6%X3% 


15 
16 


8.54 


28.6 


32.56 


7.74 


6.50 


500A 


^AXVA 


% 


3.23 


11.0 


10.15 


3.28 


2.14 


504A 


&AX 3% 


% 


5.47 


17.9 


17.62 


5.85 


3.82 


540A 


5X4 


% 


3.23 


11.0 


8.13 


4.65 


2.50 


546A 


5ft X 4ft 


% 


6.35 


21.3 


15.65 


8.74 


4.95 


510A 


5X3^ 


ft 


2.56 


8.7 


6.58 


2.71 


1.65 


517A 


5^X3% 


% 


6.07 


20.0 


15.51 


6.41 


4.17 


530A 


5 X3 


5 
IB 


2.40 


8.2 


6.27 


1.75 


1.20 


537A 


5^X3^ 


% 


5.69 


18.7 


14.75 


4.18 


3.05 


450A 


4^X3 


5 
15 


2.25 


7.7 


4.72 


1.72 


1.10 


457A 


4% X 3M 


% 


5.32 


17.4 


11.04 


4.07 


2.96 


410A 


4 X3^ 


ft 


2.25 


7.7 


3.57 


2.56 


1.18 


417A 


4M X B% 


% 


5.32 


17.4 


8.42 


6.06 


3.08 



Section Elements. 



383 



Elements of Pencoyd Angles. 




IX. 


X. 


XI. 


XII. 


XIII. 


XIV. 


XV. 


I. 


Radii 


is of gyrc 


ition. 


Resistance. 


Distance from base to 
neutral axis. 


Section 


Axis 
AB. 


Axis 
CD. 


Axis 
EF. 


Axis 

AB. 


Axis 
CD. 


d. 


1. 


number. 


2.56 


1.79 


1.34 


8.03 


4.80 


2.47 


1.47 


860A 


2.53 


1.77 


1.37 


15.43 


9.20 


2.72 


1.72 


868A 


2.25 


0.93 


0.85 


5.66 


1.61 


2.53 


0.78 


730A 


2.23 


0.93 


0.87 


10.85 


3.10 


2.77 


1.02 


738A 


2.10 


1.15 


0.93 


3.87 


1.62 


2.15 


0.90 


650A 


2.12 


1.17 


0.99 


9.58 


4.07 


2.45 


1.20 


659A 


1.93 


1.17 


0.92 


3.32 


1.60 


1.94 


0.94 


640A 


1.94 


1.18 


0.98 


8.21 


3.98 


2.24 


1.24 


649A 


1.94 


0.99 


0.84 


3.24 


1.23 


2.04 


0.79 


630A 


1.95 


0.95 


0.87 


8.05 


2.77 


2.33 


1.08 


639A 


1.77 


1.01 


0.81 


2.76 


1.22 


1.82 


0.82 


500A 


1.79 


1.03 


0.84 


4.66 


2.10 


1.97 


0.97 


504A 


1.59 


1.20 


0.88 


2.34 


1.57 


1.53 


1.03 


540A 


1.57 


1.17 


0.88 


4.50 


2.93 


1.71 


1.21 


546A 


1.60 


1.03 


0.80 


1.93 


1.02 


1.59 


0.84 


510A 


1.60 


103 


0.83 


4.51 


2.38 


1.81 


1.06 


517A 


1.62 


0.85 


0.71 


1.89 


0.75 


1.68 


0.68 


530A 


1.61 


0.86 


0.73 


4.40 


1.78 


1.90 


0.90 


537A 


1.45 


0.87 


0.70 


1.55 


0.75 


1.46 


0.71 


450A 


1.44 


0.87 


0.75 


3.61 


1.76 


1.69 


0.94 


457A 


1.26 


1.07 


0.72 


1.27 


1.00 


1.18 


0.93 


410A 


1.26 


1.07 


0.76 


2.95 


2.33 


1.40 


1.15 


417A 



384 



Section Elements. 



Elements of Pencoyd Angles. 

A P 



- F 




I. 


II. 


III. 


IV. 


V. 


YI. 


VII. 


VIII. 


Section 


Size, in 
inches. 


Thick- 
ness. 


Area, in 
square 
inches. 


Weight 
per foot, 

in 
pounds. 


Moments of inertia. 


number. 


Axis 

AB. 


Axis 

CD. 


Axis 
EF. 


430A 


4X3 


& 


2.09 


7.1 


3.38 


1.64 


0.93 


435A 


4^X3^ 


% 


4.06 


13.8 


6.36 


2.59 


1.80 


300A 


3^X3 


?\ 


1.93 


6.6 


2.33 


1.59 


0.80 


305A 


3i§ X 3 T * B 


% 


3.98 


12.9 


5.12 


3.54 


1.88 


310A 


33^X2^ 


% 


1.44 


4.9 


1.81 


0.78 


0.45 


314A 


3%X2% 


% 


2.95 


9.4 


3.93 


1.76 


1.01 


316A 


3^X2 


% 


1.31 


4.5 


1.66 


0.41 


0.30 


318A 


*% X 2% 


% 


1.99 


6.6 


2.55 


0.65 


0.45 


325A 


3 X2% 


% 


1.31 


4.5 


1.15 


0.73 


0.41 


329A 


3MX 2% 


% 


2.70 


8.7 


2.64 


1.71 


0.76 


320A 


3X2 


H 


1.19 


4.1 


1.09 


0.40 


0.24 


324A 


94 X 2% 


% 


2.45 


7.9 


2.41 


0.92 


0.57 


200A 


23^X2 


ft 


0.81 


2.7 


0.51 


0.29 


0.13 


205A 


m x 2 T * 5 


X 


2.26 


7.0 


1.64 


0.97 


0.44 


206A 


Vi x iy 2 


ft 


0.67 


2.3 


0.35 


0.12 


0.08 


209A 


2/« X 1« 


% 


1.38 


4.4 


0.73 


0.29 


0.18 


215A 


2 Xl% 


ft 


0.62 


2.1 


0.25 


0.12 


0.07 


218A 


2& x m 


% 


1.28 


4.3 


0.52 


0.29 


0.15 


210A 


2 X1M 


ft 


0.57 


1.9 


0.23 


0.07 


0.05 


213A 


2ft X l/ 5 


% 


1.19 


3.9 


0.50 


0.17 


0.12 



Section Elements. 



385 



Elements of Pencoyd Angles. 

A P 




IX. 


X. 


XI. 


XII. 


XIII. 


XIV. 


XV. 


I. 


Radii 


is of gyr 


ition. 


Resistance. 


Distance from base to 
neutral axis. 


Section 


Axis 
AB. 


Axis 
CD. 


Axis 
EF. 


Axis 
AB. 


Axis 

CD. 


d. 


1, 


number. 


1.27 


0.89 


0.67 


1.23 


0.73 


1.26 


0.76 


430A 


1.25 


0.80 


0.67 


2.33 


1.16 


1.40 


0.90 


435A 


1.10 


0.91 


0.64 


0.95 


0.73 


1.06 


0.81 


300A 


1.13 

i 


0.95 


0.69 


2.00 


1.53 


1.25 


1.00 


305A 


1.12 


0.74 


0.56 


0.76 


0.41 


1.11 


0.61 


310A 


1.15 


0.77 


0.59 


1.58 


0.88 


1.26 


0.76 


314A 


1.13 


0.56 


0.48 


0.72 


0.27 


1.21 


0.46 


316A 


1.13 


0.57 


0.48 


1.09 


0.41 


1.28 


0.53 


318A 


0.94 


0.75 


0.56 


0.55 


0.40 


0.92 


0.67 


325A 


0.99 


0.80 


0.53 


1.20 


0.88 


1.05 


0.80 


329A 


0.96 


0.58 


0.45 


0.54 


0.26 


0.99 


0.49 


320A 


0.99 


0.61 


0.48 


1.14 


0.57 


1.14 


0.64 


324A 


! 0.79 


0.60 


0.40 


0.29 


0.19 


0.76 


0.51 


200A 


0.85 


0.66 


0.44 


0.88 


0.60 


0.94 


0.69 


205A 


! 0.72 


0.42 


0.35 


0.23 


0.11 


0.74 


0.37 


206A 


0.73 


0.46 


0.36 


0.46 


0.24 


0.86 


0.48 


209A 


0.63 


0.44 


0.34 


0.18 


0.11 


0.64 


0.39 


215A 


0.64 


0.48 


0.34 


0.36 


0.24 


0.76 


0.50 


218A 


0.64 


0.35 


0.30 


0.18 


0.07 


0.69 


0.31 


210A 


j 0.&5 


0.38 


0.32 


0.36 


0.17 


0.80 


0.42 


213A 


1 








25 









386 



Double Angles. 



Radii of Gyration for Two Angles, with Sides 
Parallel. 

The radii of gyration correspond to axes shown. 

T\ Tl V$ 












-TO f 



Tf 






r o -4- 



-*o 



Size, in 


Thick- 
ness. 


Weight per 
foot, in 
pounds. 


d. 




Radius of 


gyration. 


inches. 


r o- 


ri. 


To. 


r 3- 


8 X8 


1 


26.4 

52.8 


2.19 
2.43 


2.50 

2.48 


3.32 
3.47 


3.45 
3.61 


3.58 
3.74 


6X6 
6^X6^ 


% 

15 

T5 


14.8 
35.9 


1.64 
1.90 


1.88 
1.86 


2.49 
2.66 


2.62 
2.80 


2.76 
2.94 


5X5 
5}4X5^ 


if 


12.3 
29.4 


1.39 
1.65 


1.56 
1.54 


2.09 
2.26 


2.22 
2.40 


2.35 
2.54 


4X4 
4^X4^ 


5 


8.2 
" 18.6 


1.12 
1.34 


1.24 
1.24 


1.67 
1.82 


1.80 
1.97 


1.94 
2.12 


33^X3^ 
3^X3% 


5 /£ 


7.1 
13.7 


0.99 
1.13 


1.08 
1.06 


1.46 
1.55 


1.60 
1.69 


1.74 
1.84 


3 X3 
3 T 3 5 X 3 T % 


% 


4.9 
11.5 


0.84 
1.02 


0.93 
0.93 


1.25 
1.38 


1.39 
1.52 


1.53 
1.68 


VAX 2% 
3 X3 




4.5 

8.6 


0.78 
0.93 


0.85 
0.88 


1.15 
1.28 


1.29 
1.42 


1.43 
1.57 


23^X2^ 
2%X2% 


A 


3.1 

7.8 


0.70 
0.84 


0.77 
0.76 


1.04 
1.13 


1.17 

1.28 


1.32 
1.43 


2^X2^ 

7 \y O 7 
Z T5 X ^T« 


h 

% 


2.7 
5.4 


0.63 
0.75 


0.69 
0.72 


0.93 
1.04 


1.07 
1.18 


1.21 

1.34 


2X2 
2& X 2ft 


A 

% 


2.5 

4.8 


0.58 
0.68 


0.62 
0.64 


0.85 
0.93 


0.99 
1.08 


1.14 
1.23 


1%X1% 

lit x in 




2.1 
4.1 


0.51 
0.63 


0.54 
0.55 


0.74 
0.84 


0.88 
0.98 


1.04 
1.14 


m x 1% 


Vs 
Vs 


1.2 
3.5 


0.42 
0.57 


0.47 
0.50 


0.63 
0.76 


0.77 
0.91 


0.92 
1.07 


mxm 

1% x 1% 


Vs 


1.0 
2.0 


0.35 
0.43 


0.41 
0.40 


0.54 
0.59 


0.68 
0.73 


0.83 
0.90 


1 XI 
V/s X 1% 


Vs 

A 


0.8 
1.5 


0.30 
0.37 


0.29 
0.32 


0.42 
0.49 


0.57 
0.64 


0.73 
0.81 



r lt r 2 , and r 3 will also be radii of gyration for star columns. 



Double Angles. 



387 



Radii of Gyration for Two Angles, with Sides 
Parallel. 

The radii of gyration correspond to axes shown. 



w — 


f 3 * l "" 


i 


n 


\ 1 


- ) 






J 








M& 




Size, in 


Thick- 
ness. 


Weight per 
foot, in 
pounds. 


d. 


Kadius of gyratioi 


i. 


inches. 


r o- 


n- 


To. 


r 3 - 


8X6 


V* 


23.0 


2.47 


2.56 


2.32 


2.44 


2.57 


8^X6^ 


1 


45.6 


2.72 


2.53 


2.47 


2.60 


2.74 


7 X3% 


% 


17.0 


2.53 


2.25 


1.21 


1.34 


1.48 


7^X3% 


1 


32.5 


2.77 


2.23 


1.38 


1.51 


1.68 


6^X4 


% 


12.9 


2.15 


2.10 


1.46 


1.58 


1.72 


KVs X 4% 


15 
16 

% 


31.9 
12.2 


2.45 


2.12 


1.68 
1.50 


1.81 
1.62 


1.96 


6 X4 


1.94 


1.93 


1.76 


6 3 / 8 X4% 


H 


29.4 


2.24 


1.94 


1.71 


1.85 


2.00 


6X3% 


Vs 


11.6 


2.04 


1.94 


1.27 


1.39 


1.53 


6%X3% 


M 


28.6 


2.33 


1.95 


1.44 


1.58 


1.74 


5^X3^ 


% 


11.0 


1.82 


1.77 


1.30 


1.43 


1.56 


5%X3% 


% 


17.9 


1.97 


1.79 


1,41 


1.55 


1.69 


5X4 


Vs 


11.0 


1.53 


1.59 


1.58 


1.71 


1.85 


5 T 3 6 X 4& 


% 


21.3 


1.71 


1.57 


1.68 


1.82 


1.97 


5 X3% 


5 
16 


8.7 


1.59 


1.60 


1.33 


1.45 


1.59 


] 5^ X 3% 


% 


20.0 


1.81 


1.60 


1.48 


1.62 


1.77 


5X3 


A 


8.2 


1.68 


1.62 


1.09 


1.21 


1.35 


\h%xm 


3 A 


18.7 


1.90 


1.61 


1.24 


1.39 


1.54 


J 43^X3 


5 


7.7 


1.46 


1.45 


1.12 


1.25 


1.39 


4^X33^ 


% 


17.4 


1.69 


1.44 


1.28 


1.42 


1.58 



388 



Double Angles. 



Radii of Gyration for Two Angles, with Sides 
Parallel. 

The radii of gyration correspond to axes shown. 




■8V 



m 



Size, in 


Thick- 
ness. 


Weight per 
foot, in 
pounds. 


d. 


Radius of 


gyratioE 


. 


inches. 


n> 


n« 


r 2- 


*z- 


4 x*y 2 


T6 


7.7 


1.18 


1.26 


1.42 


1.55 


1.69 


4^X3% 


% 


17.4 


1.40 


1.26 


1.57 


1.71 


1.86 


4X3 


A 


7.1 


1.26 


1.27 


1.17 


1.30 


1.44 


4^X3^ 


% 


13.8 


1.40 


1.25 


1.20 


1.35 


1.50 


3KX3 


5 
T6 


6.6 


1.06 


1.10 


1.22 


1.35 


1.49 


3xi X 3 T \ 


% 


12.9 


1.25 


1.13 


1.38 


1.52 


1.67 


3^X2^ 


A 


4.9 


1.11 


1.12 


0.97 


1.09 


1.23 


3%X2% 


% 


9.4 


1.26 


1.15 


1.08 


1.22 


1.37 


3^X2 


H 


4.5 


1.21 


1.13 


0.72 


0.86 


1.00 


®/s X 2% 


Vs 


6.6 


1.28 


1.13 


0.78 


0.92 


1.07 


3 X2% 


% 


4.5 


0.92 


0.94 


1.00 


1.13 


1.29 


Z X AX2% 


% 


8.7 


1.05 


0.99 


1.13 


1.27 


1.42 


3 X2 


% 


4.1 


0.99 


0.96 


0.76 


0.89 


1.04 


3^X2i^ 


y* 


7.9 


1.14 


0.99 


0.88 


1.03 


1.18 


2^X2 


h 


2.7 


0.76 


0.79 


0.79 


0.92 


1.07 


m x 2& 


V* 


7.0 


0.94 


0.85 


0.95 


1.09 


1.24 


234 x IK 


h 


2.3 


0.74 


0.72 


0.56 


0.70 


0.85 


2&X IH 


Vs 


4.4 


0.86 


0.73 


0.66 


0.81 


0.97 


2 Xl^ 


& 


2.1 


0.64 


0.63 


0.59 


0.73 


0.88 


2 A X 1H 


% 


1.3 


0.76 


0.64 


0.69 


0.84 


1.00 


2 xiM 


ft 


1.9 


0.69 


0.64 


0.47 


0.61 


0.77 


2A X 1$ 


% 


3.9 


0.80 


0.65 


0.57 


0.72 


0.88 



Double Angles. 



389 



Radii of Gyration for Two Angles, with Sides 
Parallel. 

The radii of gyration correspond to axes shown. 
r 2 




Size, in 


Thick- 
ness. 


Weight per 
foot, in 
pounds. 


d. 


Radius of 


gyration. 




inches. 


r 0- 


1- 


r. 2 . 


r 3- 


8 X 6 


y 2 


23.0 


1.47 


1.79 


3.56 


3.69 


3.83 


8^X6^ 


i 


45.6 


1.72 


1.77 


3.71 


3.85 


4.00 


7 X3^ 


k 


17.0 


0.78 


0.93 


3.38 


3.53 


3.67 


7^X3% 


i 


32.5 


1.02 


0.93 


3.56 


3.70 


3.85 


63^X4 


% 


12.9 


0.90 


1.15 


3.00 


3.14 


3.28 


6%X4% 


15 


31.9 


1.20 


1.17 


3.24 


3.38 


3.53 


6X4 


% 


12.2 


0.94 


1.17 


2.74 


2.87 


3.01 


$% X 4% 


11 


29.4 


1.24 


1.18 


2.96 


3.11 


3.26 


6 X3% 


% 


11.6 


0.79 


0.99 


2.81 


2.95 


3.10 


6% X 3% 


15 


28.6 


1.08 


0.95 


3.04 


3.18 


3.33 


5KX3K 


% 


11.0 


0.82 


1.01 


2.54 


2.68 


2.82 


5%X3% 


^ 


17.9 


0.97 


1.03 


2 66 


2.80 


2.95 


5 X 4 


Ys 


11.0 


1.03 


1.20 


2.21 


2.34 


2.48 


5& X 4& 


% 


21.3 


1.21 


1.17 


2.32 


2.46 


2.61 



5 X3^ 


A 


8.7 


0.84 


1.03 


2.25 


2.39 


2.53 


5^/3% 


% 


20.0 


1.06 


1.03 


2.41 


2.56 


2.71 


5 X 3 


A 


8.2 


0.68 


0.85 


2.33 


2.47 


2.62 


m x % 


% 


18.7 


0.90 


0.86 


2.49 


2.64 


2.79 



43^X3 
4%X3^ 


5 

% 1 


7.7 0.71 
7.4 0.94 


0.87 
0.87 


2.06 
2.22 


2.19 
2.37 


2.34 
2.52 



390 



Double Angles. 



Radii of Gyration for Two Angles, with Sides 
Parallel. 

The radii of gyration correspond to axes shown. 



•8^ 



fc 









^% 



Size, in 


Thick- 
ness. 


Weight per 
foot, in 
pounds. 


d. 


Radius of 


gyratioi 


L. 


inches. 


ro. 


n- 


r 2 . 


Hi- 


4 x&A 


A 


7.7 


0.93 


1.07 


1.73 


1.86 


2.00 


434X3% 


% 


17.4 


1.15 


1.07 


1.88 


2.03 


2.18 


4 X3 


i% 


7.1 


0.76 


0.89 


1.79 


1.92 


2.07 


43^X33^ 


% 


13.8 


0.90 


0.80 


1.88 


2.02 


2.17 


33^X3 


5 
IS 


6.6 


0.81 


0.91 


1.53 


1.66 


1.81 


913 V ^5 


H 


12.9 


1.00 


0.95 


1.68 


1.82 


1.98 


33^X2^ 


^ 


4.9 


0.61 


0.74 


1.58 


1.71 


1.86 


3% X 2% 


X 


9.4 


0.76 


0.77 


1.71 


1.85 


2.00 


3^X2 


K 


4.5 


0.46 


0.56 


1.65 


1.80 


1.95 


3% X 2% 


3 /8 


6.6 


0.53 


0.57 


1.71 


1.85 


2.00 


3 X2y 2 


X 


4.5 


0.67 


0.75 


1.31 


1.45 


1.60 


3^ X 2% 


3^ 


8.7 


0.80 


0.80 


1.44 


1.58 


1.73 


3X2 


M 


4.1 


0.49 


0.58 


1.38 


1.52 


1.67 


3^X2^ 


% 


7.9 


0.64 


0.61 


1.51 


1.66 


1.81 


2KX2 


i 3 e 


2.7 


0.51 


0.60 


1.10 


1.23 


1.38 


2*§ X 2^ 


K 


7.0 


0.69 


0.66 


1.27 


1.41 


1.56 


2MX1K 


i 3 e 


2.3 


0.37 


0.42 


1.03 


1.17- 


1.33 


2 A x Ui 


% 


4.4 


0.48 


0.46 


1.13 


1.28 


1.43 


2 XV4 


3 
16 


2.1 


0.39 


0.44 


0.90 


1.04 


1.19 


2A • MA 


% 


4.3 


0.50 


0.48 


0.99 


1.14 


1.30 


2 X1& 


h 


1.9 


0.31 


0.35 


0.94 


1.09 


1.24 


3 A X 1 /r 


% 


3.9 


0.42 


0.38 


1.03 


1.18 


1.34 



Cast iron Columns. 



391 



Safe Loads, in Tons of 2000 Pounds, for Hollow 
Cylindrical Cast=iron Columns. 

Passaic Rolling Mill Company. 
Square ends. Factor of safety of 8. 



© 4> 

a ^ 








Length of column, in 


feet. 






.2 


+* a 
P - 


.5 a 


o 
A Sa 

O » O 

5S.S 




















2 a a 




o a 


8 
47 


10 


12 


14 


16 


18 


20 


22 


24 


Mo « 


6 


% 


41 


36 


31 


27 


24 


21 






12.4 


39 


6 


1 


60 


52 


46 


40 


35 


30 


26 






15.7 


49 


7 


% 


60 


54 


48 


43 


38 


34 


30 


27 


24 


14.7 


46 


7 


1 


76 


69 


62 


55 


49 


43 


38 


34 


30 


18.9 


60 


8 


% 


72 


67 


61 


55 


50 


45 


40 


36 


33 


17.1 


53 


8 


1 


93 


86 


78 


71 


64 


58 


52 


47 


42 


22.0 


69 


8 


*K 


112 


104 


94 


86 


77 


69 


62 


56 


51 


26.5 


83 


9 


% 


85 


80 


74 


68 


62 


57 


52 


47 


43 


19.4 


61 


9 


1 


110 


103 


95 


88 


80 


73 


67 


61 


55 


25.1 


78 


9 


*3 


133 


125 


115 


106 


97 


89 


81 


73 


67 


30.4 


95 


9 


155 


145 


134 


123 


113 


103 


94 


85 


78 


35.3 


110 


10 


1 


127 


120 


112 


105 


97 


89 


82 


76 


69 


28.3 


88 


10 


1^ 

1% 
1 


154 


146 


136 


127 


118 


109 


100 


92 


84 


34.4 


107 


10 


180 


170 


159 


148 


137 


127 


117 


107 


98 


40.1 


125 


10 


203 
144 


192 
137 


180 
129 


168 
122 


155 


143 


132 

100 


121 

91 


111 

85 


45.4 
31.4 


142 


11 


114 


106 


98 


11 


1M 


175 


167 


158 


148 


139 


129 


122 


112 


103 


38.3 


119 


11 


204 


195 


184 


173 


161 


151 


143 


130 


121 


44.8 


140 


11 


232 


221 


209 


197 


184 


172 


162 


148 


137 


50.9 


159 


11 


2 


258 


246 


233 


219 


205 


191 


181 


164 


152 


56.6 


176 


12 


1 


160 


154 


147 


139 


131 


123 


115 


108 


101 


34.6 


108 


12 


1^ 


196 


188 


180 


170 


160 


150 


141 


132 


123 


42.2 


131 


12 


1>| 

■ 1% 


229 


220 


210 


199 


187 


176 


165 


154 


144 


49.5 


154 


12 


261 


251 


239 


226 


213 


201 


188 


176 


164 


56.4 


176 


12 


2 


291 


279 


266 


252 


238 


224 


210 


196 


183 


62.8 


196 


13 


1 


177 


170 


163 


156 


148 


140 


132 


124 


117 


37.7 


118 


13 


Wa 


216 


209 


200 


191 


181 


172 


162 


152 


143 


46.1 


144 


13 


1% 


254 


245 


235 


224 


213 


201 


190 


179 


168 


54.2 


169 


13 


289 


280 


268 


256 


243 


229 


217 


204 


192 


61.9 


193 


13 


2 


324 


312 


300 


286 


272 


257 


242 


228 


214 


69.1 


216 


14 


1 


193 


187 


180 


173 


165 


157 


149 


141 


134 


40.8 


128 


14 


1M 


237 


229 


221 


212 


203 


193 


183 


173 


164 


50.1 


156 


14 


g 


278 


270 


260 


250 


239 


227 


215 


204 


193 


58.9 


184 


14 


318 


308 


297 


285 


273 


260 


246 


233 


220 


67.4 


210 


14 


2 


356 


345 


333 


320 


305 


291 


276 


261 


247 


75.4 


235 


15 


1 


209 


204 


197 


190 


183 


175 


167 


159 


151 


44.0 


137 


15 


1% 


257 


250 


242 


233 


224 


214 


205 


195 


185 


54.0 


168 


15 


1^ 


303 


295 


285 


275 


264 


253 


241 


229 


218 


63.6 


199 


15 


1% 


347 


337 


327 


315 


302 


289 


276 


263 


249 


72.9 


227 


15 


2 


389 


378 


366 


353 


339 


324 


309 


294 


280 


81.7 


255 


16 


1% 


277 


270 


262 


254 


245 


235 


225 


216 


206 


57.8 


180 


16 


1|| 


327 


319 


311 


300 


290 


278 


267 


255 


244 


68.4 


214 


16 


1% 


375 


366 


356 


344 


332 


319 


306 


292 


279 


78.4 


245 


16 


2 


421 


411 


400 


387 


373 


358 


343 


328 


313 


88.0 


275 


16 


2M 


465 


454 


441 


427 


412 


396 


379 


363 


346 


97.2 


304 



392 



Cast-iron Columns. 



Safe Loads, in Tons of 2000 Pounds, for Hollow 
Square Cast=iron Columns. 

Passaic Rolling Mill Company. 
Square ends. Factor of safety of 8. 



1 X 

8* 


<M 






Length of 


column, in 


feet. 








o.2 

Vh CO 


§ g 


O 
x Z 

CO •« 




















£2.5 


u a 

ftB „• 




S3 "-3 a> 
O o o 

2 a .s 


8 


10 


12 


14 


16 


18 


20 


22 


24 




6 


% 


64 


57 


51 


45 


40 


36 


32 






15.8 


49 


6 


1 


81 


73 


65 


58 


51 


45 


40 






20.0 


63 


7 


% 


80 


73 


67 


61 


55 


50 


45 






18.8 


59 


7 


1 


102 


94 


86 


78 


70 


63 


57 






24.0 


75 


8 


% 


96 


90 


83 


77 


71 


65 


59 


54 


49 


21.8 


68 


8 


1 


123 


116 


107 


99 


91 


83 


76 


69 


63 


28.0 


88 


8 


IK 


149 


139 


129 


119 


110 


100 


92 


84 


76 


33.8 


106 


9 


% 


112 


106 


100 


93 


87 


80 


74 


69 


63 


24.8 


77 


9 


1 


144 


137 


129 


121 


112 


104 


96 


89 


82 


32.0 


100 


9 


3 


175 


166 


156 


146 


136 


126 


116 


107 


99 


38.8 


121 


9 


203 


193 


182 


170 


158 


146 


135 


125 


115 


45.0 


141 


10 


l 


166 


159 


151 


142 


134 


125 


117 


109 


101 


36.0 


113 


10 


W 


201 


193 


183 


173 


163 


152 


142 


132 


123 


43.8 


137 


10 




235 


225 


214 


202 


189 


177 


166 


154 


143 


51.0 


159 


10 


266 


254 


242 


228 


215 


201 


188 


175 


162 


57.8 


181 


11 


l 


187 


180 


172 


164 


156 


147 


138 


130 


122 


40.0 


125 


11 


i)l 
1% 


227 


219 


210 


200 


190 


179 


169 


158 


148 


48.8 


152 


11 


266 


256 


246 


234 


222 


209 


197 


185 


174 


57.0 


178 


11 


302 


291 


279 


266 


252 


238 


224 


210 


197 


64.8 


202 


11 


2 


336 


324 


310 


295 


280 


264 


249 


234 


219 


72.0 


225 


12 


1 


208 


201 


194 


186 


177 


169 


160 


151 


143 


44.0 


138 


12 


1 

1% 


254 


246 


237 


227 


217 


206 


196 


185 


174 


53.8 


168 


12 


297 


288 


278 


266 


254 


242 


229 


217 


205 


63.0 


197 


12 


338 


328 


316 


303 


289 


275 


261 


247 


233 


71.8 


224 


12 


2 


377 


366 


352 


338 


323 


307 


291 


275 


260 


80.0 


250 


13 


1 


228 


222 


215 


208 


199 


191 


182 


173 


164 


48.0 


150 


13 


1M 


279 


272 


263 


254 


244 


233 


223 


212 


201 


58.8 


184 


13 


1% 


328 


319 


309 


298 


286 


274 


261 


249 


236 


69.0 


216 


13 


1% 


375 


365 


353 


341 


327 


313 


298 


284 


270 


78.8 


246 


13 


2 


419 


407 


394 


380 


365 


350 


334 


317 


301 


88.0 


275 


14 


1 


249 


243 


236 


229 


221 


213 


204 


195 


186 


52.0 


163 


14 


IK 


305 


298 


290 


281 


271 


261 


250 


239 


228 


63.8 


199 


14 


X H 


359 


851 


341 


330 


319 


307 


294 


281 


268 


75.0 


234 


14 


i% 


411 


401 


390 


378 


365 


351 


336 


322 


307 


85.8 


268 


14 


2 


460 


449 


437 


423 


408 


393 


376 


360 


344 


96.0 


300 


15 


1 


270 


264 


258 


250 


243 


235 


226 


217 


208 


56.0 


175 


15 


IK 


331 


324 


316 


308 


298 


288 


277 


266 


255 


68.8 


215 


15 


i|l 


390 


382 


373 


362 


:•,:>! 


339 


327 


314 


301 


81.0 


253 


15 


1% 


446 


437 


427 


415 


402 


388 


374 


359 


345 


92.8 


289 


15 


2 


501 


490 


479 


465 


451 


436 


420 


403 


386 


104.0 


325 


16 


IK 
IK 
i*K 


357 


350 


343 


334 


325 


315 


305 


294 


286 


73.8 


231 


16 


421 


413 


404 


394 


383 


372 


359 


347 


334 


87.0 


272 


16 


482 


474 


463 


452 


440 


426 


412 


397 


383 


99.8 


312 


16 


2 


541 


532 


520 


507 


493 


ITS 


463 


446 


429 


112.0 


350 


16 


2K 


598 


588 


575 


561 


645 


529 


511 


493 


475 


123.8 


387 



Tohsion. 



393 



Torsion. 

When a prismatic body is subjected to the action of a force tending to 
rotate it about its geometric axis, it opposes to such a force its resistance 
to torsion. This resistance consists of the moments of the fibre stresses in 
the cross-section of the prism, and, until the elastic limit is reached, there 
exists an equilibrium between the external rotating forces on the one 
hand, and the stress moments of the various elements of the section on 
the other hand ; both being taken with regard to the polar axis through 
the centre of gravity of the section, and at right angles to it. 

In computing these relations it is necessary to use the polar moment of 
inertia of the section, which may be indicated' by I p , and determined from 
the two moments of inertia of the section, taken at right angles to each 
other through the centre of gravity of the section. 

If we have I\ and J 2 to be the moments of inertia of the section of the 
prism under consideration, we have 

Polar Moment of Inertia = I p = ii + I2- 

The most usual sections for bodies under torsion are the circle, as in 
shafting, and the square and rectangle, which occur in various examples 
of machine framing. The polar moments of inertia for these are given in 
the annexed table. 

Torsion Sections. 



Number. 



Section. 



Polar moment of 
inertia, I n . 



Polar section modulus, 
Ip 




32 



■& 



16 



■C?3 



II. 




54 

6 



3j/2 



-b—>\ 



III. 




bW 



lx bW 



3"|/ 62 + ffi 



Approximately, 

fr^2 

3(0.46 + 0.96/i) 



We then have the following relation : 

Let M be the statical moment of the external forces at any section of 
the prism ; I pt the polar moment of inertia for that section ; v, the distance 



394 Torsion. 



of the furthest element of the section from the centre of gravity of the 
section ; S, the shearing fibre stress of the material at the distance, a, 
being taken at f the permissible fibre stress for direct tension. 
Then we have 

M — < ? ^ p 
v 

The relative rotation which two sections of a prism at a given distance 
apart make with each other is called the angle of torsion. This may be 
represented by &. 

If we call the distance between two sections x, we have 

dS_ _M 

dx ~ I p G' 

in which G is the modulus of torsion for the material used, and is equal to 
f of the modulus of elasticity, E. 

An example will make the application of the formulas clear. 

Suppose a round shaft of w rough t-iron, 4 inches in diameter and 48 
inches long, is held at one end. A twisting force of 1000 pounds is applied 
at the other end, with a lever arm of 24 inches. 

From the equation, M = £— ^-, we have 

For a circular section, I p = — d 4 , and we have M = 24,000, d = 4, and 
v = y, n = 3.1416. 

Hence, S = -^ . 24,000 = 1909 pounds. 

nd* 

To get the angle of torsion we have, from the torsion table, 

S I 1909 48 



G ' v 11,200,000 ' 2 



= 0.004, 



which is the length of arc of torsion for a radius 1, practically equal to 
the tangent of the angle ; the corresponding angle being 0° 14 r . 

The following tables give the essential elements for all the conditions 
of torsion which are of probable occurrence in practice. 



Torsion. 



395 




Is II 

go o 
o ~»o 



§ 






Sin 



go 



3 



S> Cq 



^ Cq 



o t> 
c"°|I 

la 



H 

rr O O) CC 

aigP?. 
<D SE 3 

a> 3 



80 



a s <t> 
too' 






3 £ 

3^0>- 
J 3 



396 



TOKSION. 






jco 



H 









il 



CO 5> 



>\<o 






^ 
£ 



&2 5> 



3 05 



s 






I 

T 




a u 

s a. 



Internal Pressure. 



397 



Resistance to Internal Pressure. 



Application. 



Pressure, p. 



Thickness, 8. 




i-KV 



i + - 



28 



■) 



r S\ ^ S ) 




p = 2S— 



P 

2S 




-•(■J-)' 



•v* 




-H9" 



8 I 2 p 



p = internal pressure, in pounds per square inch. 
S = fibre stress upon material. 
E = modulus of elasticity. 
8 = thickness of plate, in inches. 



For the deflection,/, we have for Case III., 

^ = -1 

8 6 \ 8 J E' 

ad for Case IV., 

E' 



L = L(jlYj 

8 6 \ 5 / j 



398 Springs. 



Thick Cylinders. 

When the walls of a cylinder are very thick, as in the case of a hy- 
draulic press, the material is not all strained uniformly for a given internal 
stress, the greater strain taking place upon the inner portion. The resist-^ 
ance under such conditions may be found by the formulas of Lame : 



P = S Y I »/, and _ = A /^l^_i, 
(r + 5) 2 + r 2 r \S — p 



in which r is the internal radius, 6 is the thickness, and S is the fibre stress. 

When p reaches the elastic limit of the material the inner fibres will 
begin to yield, regardless of the thickness of the walls. 

For a discussion of the strengthening of cylinders by hooping, see 
Reuleaux's "Constructor." 



Springs. 

The deflection and supporting power of the various forms of springs 
exhibit very fully the laws of bending, torsion, and elasticity ; and the 
data for various forms are given in the following tables, as prepared by 
Reuleaux. 

The quantities in the tables are 

E = modulus of elasticity = 30,000,000 for steel ; 
G = modulus of torsion = § E ; 

5 = fibre stress ; 

6 = angle of torsion = length of arc for radius 1. 

The other data used in the formulas are indicated in the illustrations. 



Springs. 



399 



III. 


1 n 




i 


I. 








| No. 


I^^^^^^B 


I 












^ 














o 


\lllifera/ 












i 


Wills/ 
\\IH 


11 


H 4 1 




t 


B 




il 




11 'T 












/ 1 III 


f 






y< w 


/ 

•< J 


I -J 


Z il 




-L 


", 




T3 


"° ^ 


<~i — Wr 




Compound trian- 
gular spring. 


Simple triangular 
spring. 


Rectangular spring. 


Name. 


II II 








GO 










c 


g C»|Qq- 


* 


^ 






5 ;- 


II 


Ii 




c 


? 48 


o>[^ 


csjcc 




5' 










crs 


a 


1 cy 


lo- 




*d 




e " - tS 


~!^ 




1 


? 








eo 


e+ 










P 










^ 


<-s 


^ 




w 


II 


II 


II 




CD 






s 


c 






o 
o' 


> w 






ag 








.1 


*M 








^K 


H^ 


-K 




g 


II 


II 


Ii 




02 


tq]^ 


M^ 


b^|^ 




5] 




>-| o- 


>-|<^ 






&^.^E • 


£;Z d".a "^2. 

o cr> * a ^ 

1 O <T> <» ££ 


11 If s- 


a 




Soa £ — P M 


^!'° II 


p 
•d 



H 


■ 5 
p 


% ii s-aE 


^8 

05 Pi cj 8 


B 
p 


00 


&«> a p ^ 

IT J** ? ^ 




si 


05 

piva: 


4 


ft 
o 
a 







•a 

3" 

IN 
to 



400 



Springs. 



.5 
'u 

a, 











73 & 


oa'Sa 


5 s£ 






qq 




o 

► O 






o3 

3 




a S3 

>l r-4 OS 


<X> M 2 «*-<73 73 






-!< 




~]< 




H-« 


£ 












"a 


qqN 




sqIk; 




qqN 


'+3 


<N 






<N 




<N 


J 


1! 






II 




II 


S 


H^ 






^Ift? 




^|ft5 














SipS 




Si pi 






Si 


rO 






• 


Ri N 






S! 


^ 




sh 


o 


c^ 






<n 








rH 






tH 






o 


II 






II 




sit 




=b 






<£ 




11 


ft 


II 






11 




II 


(h 














© 














O 
ft 


1H 










%|* 




CQ |«o 






Cq|«d 




{/}|co 


O 


II 






II 




11 


ft 
ft 


Ah 






fti 




fti 


3 














gq 














'ani'BX 


•Sirads x^iids ^.j 


•Siiuds 


•Suuds 
^BOipq putlog 




tL 




Q. 




Dl 




jT 












<^gj|jjjjj 






°^^^_ — ?g=^ 




'Wn21|#^ 






im ^l^ 






a 














u 














o 














fc 


A^ftnRIIIIilll 


k 




H§ 










■M 






HFi^iv"- 








ill 
















ssl 
















OnrTf^ 


"il 








^ a ' ~ J 






^^JJI^'IJ:;. 


|J 


















f 


=' 


"°K 






•AI 






•A 1 




1A 



Springs. 



401 






C3 C3 HH +5 



"T 



^r 



gas a 



cr 


c 


o 




+3 fl 


O CD . 


«♦«. o3 O 


O . 
b0£ 


may als 
bined : 
pound f 


02 








i 






►ss 








+ 


^ 


Cq|Q3 




S 


-O 


II 


^ 






f«o 




^ 









o 


DQ 


T3 


CD 




+" 1 




0)^3 








P< 


+-• 




X 


eS 





o 




CD 




0) 


^ 




e3 


*o 


tx 


be 


o 


W 


CD 


a 


c 


£ 


H 


r 


a> 


ft 

Gfl 



^w 



Sib 












*IS 
!! 



00 IS 
II 



c3 
g 

X - 

&A 



II 



%ta 



•guilds 
uotsia). puhoi 9idmis 



•Suuds * 911 AY 

uotsjo; %i\} oidnixg punoi jo guilds T'gQH^H 




1IA 



'IIIA 

26 



'XI 



402 



Springs. 



to 
C 

7) 





terial 
the 
the 
allel, 
r ob- 
axis. 




s imma 
lether 
adth of 
te is pai 
rmal, o 
lie to the 


■a 
a 

as 

P3 



■^ 9? e3 






V 



A 
* 



ftA 

ft,^ 



^ 



0) 0) r U bd +2 

* .*r2©8eq 

£ M S £ © ^ 
Sjxj ^S^i, en 



^te 



te ^ 
^ ^ 



ft 
ft 



% W. 



MS 



•8JIAV ^13g 

jo guu'ds IgongH 



•ajiAi punoj 
jo Suiids feoiuoo 







-r^° 



>>2.o^^ ©^ C 



M 



V 



^ 









a 
ft/\ 



^ift? 



'Suuds a;n[OA %vi& 



•OK I 



X 



IX 



IIX 



Specifications. 403 



Specifications for Structural Steel. 

Condensed from the Standard Specifications of the Association of Ameri- 
), can Steel Manufacturers. 

Process of Manufacture. 

1. Steel shall be made by either the open hearth or Bessemer process. 

Test Pieces. 

2. All tests and inspections shall be made at place of manufacture prior 
to shipment. 

3. The tensile strength, limit of elasticity, and ductility shall be deter- 
mined from a standard test piece, planed or turned parallel throughout 
its entire length, cut from the finished material. The elongation shall be 
measured on an original length of 8 inches, except when the thickness of 
the finished material is T 5 e of an inch or less, in which case the elongation 
shall be measured in a length equal to sixteen times the thickness; and 
except in rounds of % of an inch or less in diameter, in which case the 
elongation shall be measured in a length equal to eight times the diameter 
of section tested. Two test pieces shall be taken from each heat of finished 
material, one for tension and one for bending. 

4. Every finished piece of steel shall be stamped with the heat number. 
Steel for pins shall have the heat numbers stamped on the ends. Rivet 
and lacing steel, and small pieces for tie plates and stiffeners, may be 
shipped in bundles securely wired together, with the heat number on a 
metal tag attached. 

Finish. 

5. Finished bars must be free from injurious seams, flaws, or cracks, and 
have a workmanlike finish. 

Chemical Properties. 

6. Steel for buildings, train sheds, highway bridges, and similar struct- 
ures shall not contain more than 0.10 per cent, of phosphorus. 

7. Steel for railway bridges shall not contain more than 0.08 per cent, 
of phosphorus. 

Physical Properties. 

8. Structural steel shall be of three grades : rivet steel, soft steel, and 
medium steel. 

Rivet Steel. 

9. Rivet steel shall have an ultimate strength of 48,000 to 58,000 pounds 
per square inch, an elastic limit of not less than one-half the ultimate 

I strength, and an elongation of 26 per cent., and shall bend 180 degrees, 
flat on itself, without fracture on the outside of the bent portion. 

Soft Steel. 

10. Soft steel shall have an ultimate strength of 52,000 to 62,000 pounds 
per square inch, an elastic limit of not less than one-half the ultimate 
Strength, and an elongation of 25 per cent., and shall bend 180 degrees, flat 
on itself, without fracture on the outside of the bent portion. 

Medium Steel. 

I 11. Medium steel shall have an ultimate strength of 60,000 to 70,000 
pounds per square inch, an elastic limit of not less than one-half the ulti- 
mate strength, and an elongation of 22 per cent., and shall bend 180 de- 
crees, around a curve having a diameter equal to the thickness of the 
piece tested, without fracture on the outside of the bent portion. 



404 



Specifications. 



Pin Steel. 

12. Pins made from either of the above-mentioned grades of steel shall, 
on specimen test pieces cut at a depth of 1 inch from the surface of fin- 
ished material, fill the physical requirements of the grade of steel from** 
which they are rolled for ultimate strength, elastic limit, and bending, but 
the required percentage of elongation shall be decreased 5 per cent. 



Eye Bar Steel. 

13. Eye bar material 1%, inches and less in thickness, made of either of 
the above-mentioned grades of steel, shall, on test pieces cut from finished 
material, fill the requirements of the grade of steel from which it is rolled. 
For thicknesses greater than 1% inches there will be allowed a reduction 
in percentage of elongation of 1 per cent, for each % of an inch increase 
in thickness, to a minimum of 20 per cent, for medium steel and 22 per 
cent, for soft steel. 

Full Size Test of Steel Eye Bars. 

14. Full size tests of steel eye bars shall be required to show not less 
than 10 per cent, elongation in the body of the bar, and a tensile strength 
not more than 5000 pounds below the minimum tensile strength required 
in specimen tests of the grade of steel from which the bars are rolled. The 
bars will be required to break in the body ; should a bar break in the head, 
but develop 10 per cent, elongation and the ultimate strength specified, it 
shall not be cause for rejection, provided not more than one-third of the 
total number of bars tested break in the head. 

Variation in Weight. 

15. A variation in cross-section or weight of more than 23^ per cent, 
from that specified will be sufficient cause for rejection, except in the case 
of sheared plates. 

When sheared plates are ordered by weight, the permissible variation 
shall not be more than 2% per cent, from that specified, except for plates 
i^" to A" thick (10.2 to 12.75 pounds per square foot), which, when ordered 
to weight, shall not average a variation greater than 5 per cent, above or 
below the theoretical weight for plates over 75 inches wide. 

When sheared plates are ordered to gauge, the overweight shall not 
exceed the percentages given in the following table : 



Percentages of Allowable Overweights for Sheared Plates when 
ordered to Gauge. 







Width of plate. 




Thickness of plate. 
















Up to 75 inches. 


75 to 100 inches. 


Over 100 inches. 


y± inch 


10 


14 


18 


t 5 b inch 


8 


12 


16 


%inch 


7 


10 


13 


{ Q inch 


6 


8 


10 


y 2 inch 


5. 


7 


9 


T % inch 


±% 


6% 


sy 2 


% inch 


4 


6 


8 


Over % inch 


3K 


5 


V4 



Wood. 



405 



Timber. 

The following data for the strength of wooden posts and beams are 
P based upon tests made at the government arsenal at Watertown and by 
the Forestry Division of the United States Department of Agriculture. 

Thus, tests made on pillars of white and yellow pine at Watertown gave 
the following results for the breaking loads in pounds per square inch : 





Ratio of length to thickness. 




10 


15 


20 


25 


30 


35 


40 


45 


50 


55 


60 


Yellow pine 

White pine 

Hemlock 


4400 

2450 
2200 


4275 
2390 
2150 


4100 
2300 
2050 


3875 
2190 
1950 


3600 
2000 
1850 


3275 
1890 
1700 


2900 
1700 
1530 


2475 
1490 
1340 


2130 
1320 
1190 


1760 
1090 

980 


1480 
910 
820 



The following general facts concerning the physical properties of 
timber are deduced from the experiments of the Forestry Division : 

1. That bleeding (the experiments were made on long leaf yellow pine) 
has no material effect on the strength of timber ; the flexibility is slightly 
increased, but the bled timber will probably endure exposure to the weather 
as well as the other. 

2. That moisture reduces the strength of timber, whether that moisture 
be the sap or water absorbed after seasoning. In general, seasoned timber, 
or with not more than 12 per cent, moisture, is from 75 per cent, to 100 per 
cent, stronger than green timber. 

3. When artificially dried, timber contains a uniform percentage of 
moisture throughout, a condition requiring months or even years to attain 
in air-dried, heavy timber. 

When kiln-dried at usual temperatures, wood shows no loss of strength 
compared with air-dried timber of the same percentage of moisture. The 
effect of very high temperatures and pressures (as used in vulcanizing) is 
lower strengths than when air-dried. 

4. Large timbers are equal in strength per square inch of section, tested 
every way, to small timbers, provided they are equally sound and contain 
the same percentage of moisture. 

5. The tests seem to indicate that the strength of woods of uniform 
structure increases with the specific gravity, irrespective of species,— i.e., 
in general, the heaviest wood is the strongest. Oak seems not to belong to 
the list of woods to which this general remark applies. 

The data on properties of timbers must be used with considerable judg- 
ment and caution. Seasoned wood will gain weight to the extent of 5 to 
15 per cent, if exposed to the weather, and this excess will be reduced if 
the wood is kept a week in a warm, dry place. 

Some of the individual tests made by the United States Forestry Divi- 
sion varied considerably from the mean values given in the table. In the 
case of tension tests, which varied most from the average, a few were as 
low as 25 per cent., while others reached 190 per cent, of the mean. 

The elastic limit given in connection with the data from the United 
States Forestry Division is the relative elastic limit suggested by Professor 
Johnson, as there is no definite " elastic limit" in timber similar to that in 
some metals. This relative elastic limit is taken where the rate of de- 
flection is 50 per cent, more than it is under initial loads. 

Modulus of ultimate bending is extreme fibre stress on beam at rupture. 
The modulus of elastic bending is the fibre stress when the rate of de- 
flection is increased 50 per cent. The modulus of elasticity is derived from 
transverse tests. 



400 



Wood. 



Physical Properties of Wood. 

Seasoned timber, moisture 12 per cent, and under. 
Stresses given in pounds per square inch. . 



Name of material. 



Ash (American) . 

Birch 

Box 



Cedar (white) 

Cedar (American red) 

Chestnut 

Cottonwood (see poplar) 

Douglas spruce (Oregon pine) . . . 

Fir 

Gum 

Hemlock 

Hickory (American average) . . . 

Lignum vitse 

Mahogany (Spanish) 

Maple 

Oregon pine (see Douglas spruce) 

Oak (red) 

Oak (black or yellow) 

Oak ( white) 

Oak (live) 

long- 



Pine (Southern yellow, 
leafed ) 

Pine (Cuban) 

Pine (loblolly) 

Pine (white) 

Poplar 



17000 

15000 
20000 



10800 
11500 



13000 
13000 



8700 
19600 

11800 

14900 
11150 



10250 
10000 
13600 



Spruce (Northern) 

Spruce pine (pinus glabra of 
Southern States) 



Walnut (black) 



13000 
13000 
13000 
10000 
7000 

11000 

12000 
10500 






7200 

8000 
10300 

5200 
6000 
5300 



5700 



7100 

5700 
9500 

9900 

8200 
7150 



7200 

7300 

8500 

10400 

8000 
8700 
7400 
5400 
5000 

6000 

7300 
7500 



tC c 2 



1900 



700 



• rn CO • 

2 O ~ 



<s> 



a ^ . 
g- .2 



1100 



400 



800 



1400 



2700 



500 

1300 

800 

400 
1100 



1800 



500 



2300 
1800 
2200 



1260 

1200 

1150 

700 



1100 
1100 
1000 



835 
770 
800 
400 



1200 
2500 



400 
800 



Wood. 



407 



Physical Properties of Wood. 

Seasoned timber, moisture 12 per cent, and under. 
Stresses given in pounds per square inch. 



Elastic 


Modulus of 
elasticity. 


Modulus 

of 
ultimate 
bending. 


Modulus 

of 

elastic 

bending. 


Ordinary working stress. 


of 


limit. 


Tension. 


Compres- 
sion. 


Trans- 
verse. 


j 3 


7900 


1 640 000 
1 645 000 


10800 
11700 


7900 


2000 

2000 
2500 

1200 
1400 
1400 


1000 

1000 
1200 

600 
700 
600 


1200 

1200 
1500 

800 
900 
900 


39 
33 








5800 


910 000 


6300 
7200 
8100 


5800 


23 




1 140 000 




41 








6400 


1 680 000 
1 530 000 
1 700 000 


7900 


6400 


1400 


700 


1000 


32 


7800 


9500 

7100 
16000 

11700 

9550 
10000 


7800 


1200 


900 


900 

750 
1800 

1500 

1500 


37 








25 


11200 


2 390 000 


11000 


2000 
1500 
1500 


1200 
1200 
1200 


50 

83 




1 255 000 




53 
49 
















9200 
8100 
9600 
9040 


1 970 000 

1 740 000 

2 090 000 

1 851 500 

2 070 000 

2 370 000 
2 050 000 
1 390 000 


11400 
10800 
13100 
11300 

12600 

13600 

11300 

7900 

6500 

8000 

10000 
8000 


9200 
8100 
9600 


1400 
1400 
1700 


900 

900 

1000 


1200 
1200 
1500 


45 
45 

50 


10000 
11100 


9500 

10640 

9400 

6400 


1600 


1000 


1500 


38 


9200 
6400 


1600 

1200 

900 

1200 

1200 
1000 


900 
700 
600 

700 

700 
1000 


1200 
900 
750 

900 

900 
900 


33 
24 




1 400 000 

1 640 000 
1 306 000 




26 


8400 
5700 


8400 


30 
38 









408 



Wooden Beams. 



Greatest Safe Load, Uniformly Distributed, for Rectan = 
gular Wooden Beams One Inch Thick. 



Kind of timber. 



Length of span, in feet. 



8 10 12 14 16 18 20 22 



Safe loads, in pounds, per inch of thickness. 



Hemlock 

Spruce or pine. 
Oak 

Yellow pine . . , 



Hemlock 

Spruce or pine . 
Oak 

Yellow pine. . , 



Hemlock 

Spruce or pine. 

Oak 

Yellow pine. . . 



Hemlock 

8 i Spruce or pine. 

| Oak 

Yellow pine. . . 



Hemlock 

Spruce or pine. 

Oak 

Yellow pine.. . 



Hemlock 

Spruce or pine. 

Oak 

Yellow pine.. . 



Hemlock 

Spruce or pine 

Oak 

Yellow pine. . 



Hemlock 

Spruce or pine. 

Oak 

Yellow pine.. . 



Hemlock 

Spruce or pine . 

Oak 

Yellow pine. . , 



Hemlock 

Spruce or pine. 

Oak 

Yellow pine.. . 



520 


350 


260 


210 


170 


150 


130 


120 


100 


620 


420 


310 


250 


210 


180 


160 


140 


120 


830 


550 


420 


330 


280 


240 


210 


190 


170 


*800 


700 


520 


420 


350 


300 


260 


230 


210 


*640 


500 


380 


300 


250 


210 


190 


170 


150 


900 


600 


450 


360 


300 


260 


220 


200 


180 


1200 


800 


600 


480 


400 


340 


300 


270 


240 


*960 


*960 


750 


600 


500 


430 


370 


330 


300 


*750 


680 


510 


410 


340 


290 


260 


230 


200 


*1120 


820 


610 


490 


410 


350 


310 


270 


240 


1630 


1090 


820 


650 


540 


470 


410 


360 


330 


*1120 


*1120 


1020 


820 


680 


580 


510 


450 


410 


*850 


*850 


670 


530 


440 


380 


330 


300 


270 


*1280 


1060 


800 


640 


530 


460 


400 


360 


320 


*2130 


1420 


1070 


850 


710 


610 


530 


470 


430 


*1280 


*1280 


*1280 


1070 


890 


760 


670 


590 


530 


*960 


*960 


840 


670 


560 


480 


420 


370 


340 


*1440 


1350 


1010 


810 


670 


580 


510 


450 


400 


♦2400 


1800 


1350 


1080 


900 


770 


670 


600 


540 


*1440 


*1440 


*1440 


1350 


1120 


960 


840 


750 


670 


*1070 


*1070 


1040 


830 


690 


590 


520 


460 


420 


*1600 


*1600 


1250 


1000 


830 


710 


620 


560 


500 


*2670 


2220 


1670 


1330 


1110 


950 


830 


740 


670 


*1600 


*1600 


*1600 


*1600 


1390 


1190 


1040 


930 


830 


*1280 


*1280 


*1280 


1200 


1000 


860 


750 


670 


600 


♦1920 


*1920 


1800 


1440 


1200 


1030 


900 


800 


720 


*3200 


♦3200 


2400 


1920 


1600 


1370 


1200 


1070 


960 


♦1920 


*1920 


*1920 


♦1920 


♦1920 


1710 


1500 


1330 


1200 


*1490 


*1490 


*1490 


*1490 


1360 


1170 


1020 


900 


820 


*2240 


*2240 


♦2240 


1960 


1630 


1400 


1220 


1090 


980 


*3730 


*3730 


3270 


2610 


2180 


1870 


1630 


1450 


1310 


*2240 


♦2240 


*2240 


*2240 


*2240 


*2240 


2040 


1810 


1630 


♦1710 


♦1710 


*1710 


*1710 


♦1710 


1520 


1330 


1180 


1070 


♦2560 


*2560 


*2560 


2550 


2130 


1830 


1600 


1420 


1280 


*4270 


*4270 


*4270 


3410 


2840 


2440 


2130 


1900 


1710 


*2560 


*2560 


*2560 


*2560 


*2560 


*2560 


*2560 


2370 


2130 


♦1920 


♦1920 


♦1920 


*1920 


♦1920 


*1920 


1690 


1500 


1350 


*2880 


*2880 


*2880 


*2880 


2700 


2310 


2030 


1800 


1620 


*4800 


♦4800 


*4800 


4320 


3600 


3090 


2700 


2400 


2160 


♦2880 


*2880 


*2880 


*2880 


*2880 


*2880 


♦2880 


♦2880 


2700 



The short lengths, marked with a star, are computed to resist longi- 
tudinal shearing. 



Wooden Beams. 



409 



Greatest Safe Central Loads for Rectangular Wooden 
Beams One Inch Thick. 



Kind of timber. 



Length of span, in feet. 



10 | 12 14 16 



Safe loads, in pounds, per inch of thickness. 



Hemlock 

Spruce or pine . 
Oak 

Yellow pine . . . 

Hemlock 

Spruce or pine . 
Oak 

Yellow pine . . . 

Hemlock 

Spruce or pine . 

Oak 

Yellow pine . . . 

Hemlock 

Spruce or pine. 
Oak 

Yellow pine . . . 

Hemlock 

Spruce or pine . 
Oak 

Yellow pine . . . 



260 
310 
420 
520 



170 
210 
280 
350 

250 

300 
400 
500 

340 



450 
600 
750 

510 

610 410 

820 ! 540 

1020, 680 

670 440 
800; 530 



1070 
1330 

840 

1010 

1350 

*1440 



Hemlock 1040 

Spruce or pine . 1250 

Oak 1670 

Yellow pine . . . *1600 



Hemlock , 

Spruce or pine . 
Oak 

Yellow pine . . . 



Hemlock 

Spruce or pine . 
Oak 

Yellow pine . . . 



Hemlock 

Spruce or pine . 

Oak 

Yellow pine . . . 

Hemlock 

Spruce or pine. 

Oak 

Yellow pine . . . 



*1280 
1800 
2400 

*1920 

*1490 

*2240 

3270 

*2240 

*1710 
*2560 

*4270 

*2560 

*1920 
1*2880 
*4800 

!*2880 



710 

890 

560 

670 

900 

1120 

690 

830 

1110 

1390 

1000 

1200 

1600 

*1920 



130 
160 
210 
260 

190 
220 
300 
370 

260 
310 
410 
510 

330 

400 
530 
670 



100 
120 
170 
210 

150 
180 
240 
300 

200 
240 
330 
410 

270 
320 
430 
530 



420 340 

510 400 

670 540 

840 670 



520 

620 

830 

1040 

750 

900 

1200 

1500 



1360 1020 

1630 1220 

2180 1630 

*2240 2040 

*1710l 1330 

2130 1600 

2840 2130 

*2560 *2560 

*1920! 1690 
2700 1 2030 
3600 2700 

*2880*2880 



420 
500 
670 
830 

600 

720 

960 

1200 

820 

980 
1310 
1630 

1070 
1280 
1710 
2130 

1350 
1620 
2160 
2700 



90 
100 
140 
170 

120 
150 
200 
250 

170 
200 
270 
340 

220 
270 

360 
440 

280 
340 
450 
560 

350 
410 
550 
690 

500 ! 

600 ! 

800 

1000 

680 

810 

1090 

1360) 

89o! 
1060 
1420 

1780 

1120 
1350 
1800 
2250 



J 



90 
120 
150 



651 60 

80, 70 

100 95 

130 120 



110 95 85 

130 110 100 

1701 150 130 

210 190 170 



150 
170 
230 
290 



130 110 

150 j 140 

200 I 180 

260 230 



190 170 150 

230 200 180 

300! 270 240 

380 330 300 

240 ' 210 190 

290 | 250 ! 220 

390 i 340 1 300 

480 420 370 



300 260 

360 310 

480 420 

590 520 

430 370 

510 450 

690 , 600 

860 750 

580 510 ' 

700 610; 

930 820 ! 

1170 1020 



760 

910 

1220 

1520 



230 
280 
370 
460 



330 300 270 

400 360 330 

530 480 ' 440 

670 600: 540 



450 
540 
730 
910 



660t 590 
800 I 710 
1060! 950 

1330 1180 



960 840 750 670 610 

1160 1010! 900 810 740 

1540 1350 1 1200 [ 1080 980 

1930 1690 1500 13501 1230 



The short lengths, marked with a star, are computed to resist longi- 
tudinal shearing. 



410 



Wooden Pillars. 



Total Safe Load, in Net Tons, for Square Pillars. 

For Hemlock Pillars take A of Load for White Pine. 





Kind of timber. 


Side of square pillar, in inches. 




4 


5 


6 


7 


8 


9 


10 


12 


14 


16 


'S3 

w 


Total safe load, in net tons. 


6 


Yellow pine . . . 
White pine 


7.3 
4.5 


11.8 
7.2 


17.3 
10.5 


23.8 
14.4 


31.3 
18.9 


39.8 
24.0 


49.3 
29.7 


71.3 
42.9 


97.3 
58.5 


127.3 
76.5 


7 


Yellow pine . . . 
White pine 


7.1 
4.4 


11.6 
7.1 


17.1 
10.3 


23.6 
14.3 


31.1 

18.7 


39.6 
23.6 


49.1 
29.5 


71.1 

42.8 


97.1 

58.4 


127.1 
76.4 


8 


Yellow pine . . . 
White pine 


6.8 
4.2 


11.3 
6.9 


16.8 
10.2 


23.3 
14.1 


30.8 
18.6 


39.3 
23.7 


48.8 
29.4 


70.8 
42.6 


96.8 
58.2 


126.8 
76.2 


9 


Yellow pine . . . 
White pine 


6.5 
4.1 


11.0 

6.8 


16.5 
10.1 


23.0 
14.0 


30.5 
18.5 


39.0 
23.6 


48.5 
29.3 


70.5 
42.5 


96.5 
58.1 


126.5 
76.1 


10 


Yellow pine . . . 
White pine 


6.2 
3.9 


10.7 
6.6 


16.2 
9.9 


22.7 
13.8 


30.2 
18.3 


38.7 
23.4 


48.2 
29.1 


70.2 
42.3 


96.2 
57.9 


126.2 
75.9 


11 


Yellow pine . . . 
White pine 


5.8 
3.7 


10.3 
6.4 


15.8 
9.7 


22.3 
13.6 


29.8 
18.1 


38.3 
23.2 


47.8 
28.9 


69.8 
42.1 


95.8 
57.7 


125.8 
75.7 


12 


Yellow pine . . . 
White pine 


5.4 
3.5 


9.9 
6.2 


15.4 
9.5 


21.9 
13.4 


29.4 
17.9 


37.9 
23.0 


47.4 

28.7 


69.4 
41.9 


95.4 
57.5 


125.4 
75.5 


13 


Yellow pine . . . 
White pine — 


5.0 
3.3 


9.5 
6.0 


15.0 
9.3 


21.5 
13.2 


29.0 
17.7 


37.5 

22.8 


47.0 

28.5 


69.0 
41.7 


95.0 
57.3 


125.0 
75.3 


14 


Yellow pine . . . 
White pine 


4.5 
3.0 


9.0 
5.7 


14.5 

9.0 


21.0 
12.9 


28.5 
17.4 


37.0 
22.5 


46.5 

28.2 


68.5 
41.4 


94.5 
57.0 


124.5 
75.0 


15 


Yellow pine . . . 
White pine 


3.9 

2.8 


8.4 
5.5 


13.9 

8.8 


20.5 
12.7 


27.9 
17.2 


36.4 
22.3 


45.9 
28.0 


67.9 
41.2 


93.9 
56.8 


123.9 

74.8 


16 


Yellow pine . . . 
White pine 


3.4 
2.5 


7.9 
5.2 


13.4 
8.5 


19.9 
12.4 


27.4 
16.9 


35.9 
22.0 


45.4 
27.7 


67.4 
40.9 


93.4 
56.5 


123.4 
74.5 


17 


Yellow pine . . . 
White pine .... 


2.5 
1.7 


7.3 
4.9 


12.8 

8.2 


19.3 
12.1 


26.8 
16.6 


35.3 
21.7 


44.8 
27.4 


66.8 
40.6 


92.8 
56.2 


122.8 
74.2 


18 


Yellow pine . . . 
White pine 


2.3 
1.5 


6.7 
4.6 


12.2 

7.9 


18.7 
11.8 


26.2 
16.3 


34.7 
21.4 


44.2 
27.1 


66.2 
40.3 


92.2 
55.9 


122.2 
73.9 


19 


Yellow pine . . . 
White pine 


2.1 
1.4 


6.0 
4.2 


11.5 
7.6 


18.0 
11.4 


25.5 
15.9 


34.0 
21.0 


43.5 
26.7 


65.5 
39.9 


91.5 
55.6 


121.5 
73.5 


20 


Yellow pine . . . 
White pine — 


1.9 
1.2 


5.3 
3.9 


10.8 
7.2 


17.3 
11.1 


24.8 
15.6 


33.3 
20.7 


42.8 
26.4 


64.8 
39.6 


90.8 
55.2 


120.8 
73.2 


21 


Yellow pine . . . 
White pine 


1.7 
1.1 


2.6 
1.7 


10.1 
6.8 


16.6 
10.7 


24.1 
15.2 


32.6 
20.3 


42.1 
26.0 


64.0 
39.2 


90.1 
54.8 


120.1 

72.8 


22 


Yellow pine . . . 
White pine 


1.5 
1.0 


2.4 
1.6 


9.3 
6.4 


15.8 
10.3 


23.3 
14.8 


31.8 
19.9 


41.3 
25.6 


63.3 

38.8 


89.3 
54.4 


119.3 

72.4 



Strength of Materials. 411 



Average Strengths of Materials. 

In the foregoing discussion of the strength of materials it has been 
assumed that the elastic limit, ultimate strength, modulus of elasticity, 
and similar data concerning the materials to be used are known for the 
especial case under consideration, and attention has mainly been given to 
the distribution of stresses and strains. In all important works the mate- 
rial should be tested and its properties ascertained, and during the con- 
duct of the work frequent tests should be made by competent persons, 
using reliable testing machines ; the test pieces being selected with care to 
represent the actual material employed. 

In the absence of specific data concerning the actual materials to be 
used, the values in the following tables may be taken as representing 
fairly average results. 

The tables which have been given of the strength of standard rolled 
sections represent experimental results made by the makers, and may be 
accepted, also, as closely corresponding to the similar sections of other 
mills. 

Data concerning the strength and proportions of various machine parts 
will be discussed in connection with the subject of machine design. 

In the use of materials of construction judgment should be used in 
connection with the results of tests, since it is manifestly absurd to take 
the resistance of the material to the pound when the load may be known 
only to the nearest ton. Care should also be taken to use records of 
strength of materials in the same general sense in which the original 
experiments were made, so far as can be ascertained, otherwise there can 
be no certainty that the conditions under which the resistance was ascer- 
tained are reproduced in the case in point. No records of experimental 
work can take the place of sound judgment on the part of the engineer, 
and he should always be liberal in his allowances for unforeseen stresses 
and shocks. 

In many cases it must be remembered that strength is not the only ele- 
ment to be taken into account, but that stability and massiveness may 
sometimes demand far more material than the mere stresses would indi- 
cate. The effects of impact may require masses of metal for their recep- 
tion, while in other cases the section may depend upon the amount of 
heat to be carried away. When it is realized that the actual strength is 
but one of several elements involved in engineering design, it will be 
understood that the main thing is to be on the right side, and that an 
extreme apparent precision may be far from representing true accuracy. 



412 



Strength of Materials. 



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Strength of Materials. 



413 



c 


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d i 

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d h 

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5 

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d ° 
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GO OS 
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414 



Strength of Materials. 



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Strength of Materials. 



415 



Average Ultimate Strengths of Materials. 

Pounds per Square Inch. 



Material. 



Compression. 



Tension. 



Modulus 
of rupture. 



Building Stones. 

Bluestone 

Granite, average 

Granite, Connecticut 

Granite, New Hampshire 

Granite, Massachusetts 

Granite, New York 

Limestone, average 

Limestone, Hudson River, New York 

Limestone, Ohio 

Marble, average 

Marble, Vermont 

Sandstone, average 

Sandstone, New Jersey 

Sandstone, New York 

Sandstone, Ohio 

Slate 

Stonework 

Bricks. 

Bricks, light red 

Bricks, good common 

Bricks, best hard 

Bricks, Philadelphia pressed 

Brickwork, common (lime mortar) 

Brickwork, good (cement and lime 

mortar) 

Brickwork, best (cement mortar) 

Terra-cotta 

Terra-cotta work 

Cements, etc. 

Cement, Rosendale, one month old 

Cement, Portland, one month old 

Cement, Rosendale, one year old 

Cement, Portland, one year old 

Mortar, lime, one year old 

Mortar, lime and Rosendale, one year old 
Mortar, Rosendale cement, one year old . 
Mortar, Portland cement, one year old . . 

Concrete, Portland, one month old 

Concrete, Rosendale. one month old 

Concrete, Portland, one year old 

Concrete, Rosendale, one year old 



13500 

15000 

12000 

15000 

16000 

15000 

7000 

17000 

12000 

8000 

8000 

5000 

12000 

10000 

9000 

10000 



1400 
600 



1000 



2700 
1800 



1500 
1800 



1500 



700 
700 
150 



100 
10000 



1500 

1200 
1200 

650 
1700 

700 
5000 



(1% strength of stone) 



1000 

10000 

12000 

6000 

1000 

1500 
2000 
5000 
2000 

1200 

2000 

2000 

3000 

400 

600 

1000 

2000 

1000 

500 

2000 

1000 



600 
800 
600 



200 
400 
400 
800 
100 
200 
300 
600 
100 

50 
150 

75 



Safe strengths of stone, brick, and cement, ^ to ^ of ultimate. 



416 Machine Design. 



MACHINE DESIGN. 

"A machine," according to the definition of Professor Reuleaux, is 
"a combination of resistant bodies so arranged that by their means the 
mechanical forces of nature can be compelled to do work accompanied by- 
certain determinate motions." 

In designing a machine, therefore, it is essential to consider the resist- 
ance or strength of the bodies of which it is composed, also the work 
which it is to perform and the determinate motions to be made. 

The resistance of the parts of a machine includes the proportions of 
the main framing, as well as the various shafts, gear-wheels, pulleys, con- 
necting rods, and other parts, in most of which the actual strength is of 
less importance than the stiffness, or rigidity, and the mass necessary to 
furnish satisfactory solidity to absorb vibrations and shocks. The resist- 
ance also includes the proportions of the various fastenings, such as bolts, 
rivets, keys, pins, etc. 

The work which the machine is compelled to do is the basis upon which 
the dimensions and form of the resistant parts are determined, and this is 
usually taken from the resistance opposed to the motion by the material 
to be cut, the weight to be lifted or propelled, or, in general, the opposing 
forces to be overcome. 

The determinate motions involve some of the most intricate problems 
in machine design, and properly form the subject of a distinct science, — 
that of Kinematics, the science of controlled motion, considered apart 
from the magnitude and character of the forces involved. 

It is impossible to do more here than to give the results of accepted 
modern practice with regard to these various elements of machine design, 
with such suggestions as experience may indicate for use with the special 
requirements of each case. 

Usually, the determinate motions demand the first attention. The 
form of the work to be done must be considered at the one end of the 
system, and the nature of the motion from which it is to be effected at the 
other, the machine standing between them and effecting the transforma- 
tion. Thus, in the case of the steam engine, the flowing steam, passing 
through a pipe, is converted into the rotary motion of the shaft, fly-wheel, 
and pulley. In like manner, the rotary motion communicated to the 
shaft of a Jacquard loom is transformed into all the complicated sequence 
of intermittent, yet determinate movements, which, through the medium 
of cards, heddles, shuttles, beams, etc., produce the elaborate woven 
fabric. 

Having laid out the movements, and thus determined the positions of 
the various centres and connections, the forces to be transmitted must be 
considered and the dimensions of the actual pieces computed. 

In many portions of a machine the dimensions of the parts may be 
based upon the direct knowledge of the strength of the materials, but in 
other cases empirical rules, based upon the results of experience, must be 
employed. Both methods will here be given, according to current prac- 
tice, and whenever a rational method, used the direct strength of the 
material, is practicable, it will be given. 

FASTENINGS. 
Riveting. 

Rivets are used to secure structures of sheet metal, and may be em- 
ployed solely for strength, as in structural iron or steel work ; or for 
strength and tightness combined, as in tank and boiler construction. 

For strength alone the following method may be used for proportioning 
riveted connections :* 

For any given thickness, 5, of plate it is impracticable to make the 
riveted joint the same strength as the plate itself, but the ratio between 
the strength of the plate and the strength of the joint can be made a max- 
imum. This will best be attained, with the assumption of a sufficient 
margin, when the strength of the rivets and the strength of the remainder 
of the metal between the rivet-holes are equal to each other,— i.e., when 

* Reuleaux, " Constructor," § 55, et seq. 



Riveting. 417 



they reach their limit of elasticity at the same time. If the rivets and 
plate are of the same material, we have the stress in the cross-section of 
the rivets as 0.8 that of the plate. From this we derive the following 
"ormulse, in which the friction of the joint is neglected as being of uncer- 
* tain value : 
Let 

8 = the thickness of the plate, in inches ; 

d = the diameter of rivet, in inches ; 

a = the pitch of rivets,— i.e., the distance from centre to centre of 

adjacent rivets, in inches ; 
n = the number of rows of rivets ; 

<£ = the efficiency of the joint, being the ratio of the resistance of 
the joint to that of the full plate ; 
then the highest efficiency will be attained when we have, for lap-joint 
riveting, 

a 7T / d\ 2 , d 

which gives 

* i d 1 

T = lTX7ATA' 
n it ' d 
or for butt-joint riveting, 

a _ 7t-/ d \2 d 

x =27l ir(x) +T- 

which gives 

♦.-.i-4 



a 15 8' 

+ In ' 7T ' d 

The overlap of the plate is subjected both to shearing and bending. 
For the former conditions call the lap &', and for the latter 6", measuring 
in both cases from the centre of the rivets to the edge of the joint. To 
obtain the same resistance in the lap as in the perforated portion of the 
plate, we have, for lap-joint riveting, 



S /H n8 
i^ = (o.5 + 0.56 



for butt-joint riveting, 



~T^ % —n-8 



-«■*)■■ 

~ 4 W r 



(«■+ 0.7^4)4- 



In both cases a good value of b, in practice, giving sufficient room for 
rivet-heads, will be secured by making 

b = 1.5a, or -|- = L&4-. 

o 5 

A point of interest is the superficial pressure, p, which exists between 
the body of the rivet and the cylindrical surface of the rivet-hole. If S 2 is 
the stress in the punched plate, we have, for lap-riveted joints, 

for butt-riveted joints, 

P n a d 

^ = - 47r T-' 

The following table will serve to reduce the numerical labor of these 
calculations : 

27 



418 



Riveting. 



Proportions for Riveted Joints. 



1.5 



2.5 



ir 



ft 



.°, 



1.63 
0.39 
1.06 
0.39 
0.63 

2.26 
0.79 
1.29 
0.56 
1.26 



2.22 
0.39 
1.06 
0.55 
0.63 

3.52 
0.79 
1.29 



2.92 

0.88 

1.78 
0.49 
0.94 

4.33 
0.96 

2,20 



0.7210.65 
1.261. 



4.33 
0.88 
1.78 
0.65 
0.94 

7.15 
0.96 
2.20 
0.79 

1 



4.52 
1.57 
2.58 
0.56 
1.26 

7.04 
3.14 
3.24 
0.72 
2.51 



7.04 

1.57 

2.58 
0.72 
1.26 

12.05 
3.14 
3.24 
0.83 
2.51 



6.43 
2.54 
3.46 
0.61 
1.57 



10.37 
2.54 
3.46 
0.76 
1.57 



10. 37 i 18.21 



4.91! 4.91 



4.37 
0.76 
3.14 



4.37 
0.86 
3.14 



I 



8.67 
3.53 
4.31 
0.65 

1. 

14.33 
7.07 
5.60 
0.79 
3.77 



14.33 
3.53 
4.31 
0.79 

1.88 

25.61 
7.07 
5.60 
0.90 
3.77 



14.07 
6.28 
6.48 
0.72 
2.51 

24.14 

12.56 

8.32 

0.83 

5.03 



24.14 

6.28 
6.48 
0.83 
2.51 

44.21 

12.56 

8.32 

0.94 

5.03 



An examination of the preceding table shows that the higher efficien- 
cies require the use of inconveniently large rivets. This may be avoided 
if more than two rows of rivets can be used, since they may then be dis- 



1 °te 


* 

2'fl 


iofo 




Q 




io?o 




9- . 




: O ^ 




|0;0 













i^ 





o -m- o 



o o 

o <? o 

O o 

o o 



; r i zhjL i i i II ( 

! d) I ad) ! |.4< 

If ntfl 

<j> T «> * <j> i o 

"""nTo r 



posed in groups. In this arrangement each row in a group has one less 
rivet than the preceding row, as shown in the illustration. 



Riveting. 



419 



Thus, the rivets are arranged according to a certain pitch, a, for the 
middle row of a joint, and 2, 3, 4, or 5 rivets are selected as the base of a 
group. The next row on each side will have a wider pitch, and the next 
.still wider, and so on. 
If, as "before, we take 

8 = thickness of plate, in inches ; 

d = diameter of rivet, in inches ; 

a = pitch of rivets, in inches ; 

4> = efficiency of joint; and 

m = number of rivets in the middle row of each group ; 



we have 



Table for Group Riveting. 



m = 


2 


3 


4 


5 


d 

8 ~ 


1.6 


1.6 


1.6 


1.6 


a 
~d~ 


2.5 


3.33 


4.25 


5.2 


a 
1T~ 


4.0 


5.32 


6.80 


8.32 


</> = 


.8 


.90 


.94 


.96 



The rivet-holes, in all cases, are made T V larger than the diameter of 
the rivet. 

Boiler Riveting. 

The necessity for making a tight joint has necessitated modifications 
of the proportions of joints based solely upon considerations of strength. 
The following tables represent standard practice : 

Table of Proportions for Riveted Joints with Iron Plates and 
Rivets. 



Thickness of plate 

Diameter of rivet 

Diameter of rivet-hole 

Pitch— single riveting 

Pitch— double riveting 

Efficiency— single riveting. . 
Efficiency — double riveting. 



K" 


5 // 
16 


W 


7 // 
16 


W 


11// 
T8 


%" 


h" 


16 


%" 


tgt/r 

16 


y 8 " 


2" 


2tV 


2%" 


2tV 


3" 


3%" 


3M" 


w 


.66$ 


.64$ 


.62$ 


.60$ 


.77$ 


.76$ 


.75$ 


.74$ 



%" 

.'4*" 

.58^ 



Table of Proportions for Riveted Joints in Steel Plates with Iron 

Rivets. 



Thickness of plate 

i Diameter of rivet , 

Diameter of rivet-hole . , 
Pitch— single riveting . 

I Pitch— double riveting 



Inch. 


Inch. 


Inch. 


Inch. 


% 


T 5 6 


% 


* 


xi 




Vs 


Vs 

15 
16 


% 


2 


2A 


Vi 


2 T 3 6 


3 

1 


3^ 


3M 


3% 



Inch. 



1 
2^ 



420 Bolt Fastenings. 



Bolts. 

The dimensions of standard bolts and nuts will be found on pages 304 
and 305, the United States standard being used in America and the Whit- 
worth standard in Great Britain and on the Continent. 

Bolts are usually made of wrought-iron or mild steel. Fibre stresses of 
10,000 to 15,000 pounds per square inch might be permitted under normal 
conditions of loading, but very often a heavy initial stress is put upon a 
bolt by reason of the tension applied when the nut is screwed up. A 
source* of weakness is also found in the varying cross-section of the bolt 
at different points, thus preventing uniform stretching. The result is fre- 
quent breakages at the root of the thread, especially at the point where 
the thread merges into the full body of the bolt, since the change in sec- 
tion at this point localizes the stretch. By drilling a central hole from the 
head to the beginning of the thread, and thus making the cross-section of 
the main body of the bolt the same as at the bottom of the thread, the 
stretch may be distributed. For ordinary joints the fibre stress on bolts 
may be taken as 8000 pounds for iron and 11,000 pounds for mild steel. 

Bolted flange-joints to resist steam-, air-, or water-pressure must be de- 
signed for tightness rather than for strength. Here it is the initial tension 
upon the bolts which holds the faces of the joint together. The sum of 
the initial tensions of all the bolts in a flange-joint must be greater than 
the force acting to separate the joint, or it will open and leakage will 
occur. Under such conditions the maximum stress which should be put 
upon the bolts is about 6000 pounds per square inch, and lower stresses, 
down to 3000 pounds, are preferable. Instead of using larger bolts, the 
stresses should be reduced by using more of them. Recommended spacings 
for bolts on pipe flanges will be found in the table of standard flanges on 
page 329. 

For pipe flanges the number of bolts is always made a multiple of four, 
in order to permit any member to be rotated 90° in making connections. 
For other bolted work the distance between bolt-centres for steam-tight 
work should not be less than 6d, in which d is the diameter of the bolt, 
while for heavy pressures a spacing of 4d is to be preferred. 

For steam cylinders, or similar situations, the number of bolts may be 
determined by the following formula : 



2400 V d ) ' 



in which 



N = number of bolts ; 

D — diameter of cylinder, in inches ; 

d = diameter of bolts ; 

p = pressure, in pounds per square inch. 

Thus, for a cylinder 36 inches in diameter, with a pressure of 100 
pounds, we have, for 1%-inch bolts, 



._ 100 / 36 \ 2 oc , .. 
*= 2460(1^5) = 35bolts - 



For moderate pressures, say under 50 pounds, the number of bolts may 
be taken as 

the diameter of the bolts then being chosen so as to keep the fibre stress 
within the predetermined limit. 

For all general purposes the proportions of bolts are made according to 
the standard sizes, the United States standard, page 304, being used in 
America, and the Whitworth standard in Great Britain and on the Conti- 
nent of Europe. For some purposes, however, special threads are ad- 
visable. In such cases the proportions should be directly designed for the 



Bolt Fastenings. 



421 



existing conditions. The forms most generally used are the square thread, 
suited to receive pressure in either direction, and the trapezoidal thread, 
flat on one face and inclined on the other, to sustain heavy pressures in 
. one direction, as in screw-presses and similar work. 





i_.i 



Referring to the figures, 



d = outside diameter of screw ; 
d l = bottom diameter of thread ; 
s = pitch ; 
P = total load on screw. 



For a fibre stress of 3000 pounds the diameter, di, at the bottom of the 
thread is obtained from 

d x = 0.02>/ P~ ; P = 2360^2 ; 
or for a fibre stress of 6000 pounds per square inch, 
di = 0.0145 \/~¥ ; P = 4720dx 2 . 
The depth of thread, both for square and trapezoidal threads, is 



and for square threads 



and for trapezoidal threads 







d 


di 


t 




10 ~~ 


8 ' 






d 


d, 


s 




5 


4» 






2 . 


<f, 


s 




Jg* - 


6 



For such threads the nut should be made deeper than for ordinary 
bolts ; from 1% to 2 times the outside diameter of the screw being a pro- 
portion found m practice. This insures a sufficient number of threads in 
the nut and provides for wear. 

In important structures, or where much vibration is expected, some 
form of nut-lock is used to prevent the bolt from working loose. 



422 



Bolt Fastenings. 



One of the oldest and most useful forms is the jam-nut shown at A. 
Both nuts should he truly faced, so that they will bear fairly upon each 
other. The thin nut is frequently placed under the thicker one, but this 
is immaterial, since a nut of a thickness of 0.45 to OAd is as strong as the 
bolt thread. At B is shown a split pin, often used in connection with a 
jam-nut. At C is shown an arrangement with a key upon the nut, making 




a very convenient and secure combination. In all three cases the action 
is such as to tighten the nut upon the thread. 

Numerous patented devices have been made to secure bolts and nuts, 
but the design necessarily varies according to the conditions. In some 
cases a washer is arranged to be bent up against the face of the nut, or a 
wedge is driven between the nut and some portion of the structure ; these 
methods being used in rail- joints. In other cases set-screws are used, or 
special nuts with ratchets are employed, as in heavy steam-engine work. 
These forms are modified in many ways, according to the ingenuity of the 
designer. Numerous examples will be found in Reuleaux's "Constructor" 
and in Unwin's "Machine Design." 



Wrenches. 



0.4 D 




For most purposes wrenches to fit the standard nuts are to be had as a 
regular article of trade, being drop-forged to standard sizes. If special 



Keyed Fastenings. 



423 



wrenches are desired, the following proportions may be used, the unit 
being the diameter, D, of the hexagon nut across the flat. 





~8D- 



Keyed Fastenings. 

For many purposes, where the amount of movement is slight, and where 
the parts may be required to be readily disconnected, keys or cotters may 
be used. The principal proportions of such cot- 
ters have been determined empirically. The 
depth of a cotter is made equal to the diameter 
of the rod to be secured, and the thickness is 
made one-fourth of the depth. The taper varies 
from 1 in 30 to 1 in 100. For many purposes % 
inch in the foot is taken = 1 in 96. A greater 
taper than 1 in 30 is apt to cause the cotter to 
fly back. The general proportions of a gib and 
cotter connection are shown in the figure. 

The use of gibs, as shown in the figure, in- 
creases the bearing surface of the cotter, and 
such gibs should always be used when the parts 
are to be frequently disconnected. 



If 



we have 



d = diameter of rod ; 
h = mean depth of cotter ; 
t = thickness of cotter ; 



h = d; t = 0.25d 




The tip of the cotter should not be less than 
%d. 

Keys are used to secure the hubs of pulleys, wheels, levers, etc., to 
shafts, to prevent rotation of one piece upon another. 

If the shaft to which a hub is to be keyed is proportioned to stand a 
certain load, the dimensions of the key may be based upon the diameter 
of the shaft. 
Let 

d = diameter of shaft ; 
s = breadth of key ; 
si = depth of key. 



Then, according to Reuleaux, 



d 



= — + 0.16 inch ; si = 



10 



+ 0.16 inch. 



Feathers or splines are keys upon which a sleeve or collar may slide in 
a direction parallel to the axis of the shaft, while compelled 'to rotate 
with it. The proportions of a feather may be taken as a key placed on 
edge,— i.e., with the greater dimension of the cross-section upon the radius 
of the shaft. 



424 



Keys. — Journals. 



The following table, while giving dimensions differing slightly from 
those determined by the formula of Reuleaux, correspond with American 
machine shop practice. 



Standard Keys, Splines, Etc. 



Diameter 



Key. 



Spline. 



Double spline. 



01 

shaft. 


Wide. 


Deep. 


Wide. 


Deep. 


Wide. 


Deep. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


% 


& 


5 
32 


5 
37 


ft 


% 


5 
32 


1 


M 


& 


T% 


% 


% 


5 
'35 


VA 


5 
15 


H 


% 


5 
IF 


A 


A 


1% 


% 


T% 


5 
16" 


Vs 


A 


X 


2 


7 


Vs 


Vs 


T 7 6 


M 


5 
16 


2^ 


i% 


7 
16 


7 
T6" 


9 
T6 


A 


% 


3 


% 


V* 


X A 


% 


% 


A 


3^ 


% 


9 
T6" 


& 


% 


To 


A 


4 


Vs 


% 


Vb 


Vs 


3^ 


% 


5 


1 


% 


% 


1 


A 


% 


6 


1% 


Vs 


Vs 


1% 


% 


% 



Spline 




acter of the bearing. 



Double splines are set opposite to each other, and their sizes are taken 
from the last two columns of the table. For sizes of shafts not tabulated 
take the sizes of keys for shafts of the next 
hJ£>i smaller size. Thus, for a 4%-inch shaft take sizes 

fiu for 4-inch shaft. 

JOURNALS. 

The most important form of journal is the 
overhung form shown in the illustration, and 
from its computed dimensions other forms may 
be proportioned. The ratio of length, I, to diame- 
ter, d, varies according to the service and the char- 
For rigid bearings, such as pillow-blocks, with the 

pressure constant in one direction, — = 1.5 to 2, while for crank pins and 

similar locations, in which the pressure is alternating in direction, 

4- = 1 to 1.3. 
a 

When ball and socket bearings are used, as in shafting hangers, etc., 

— = 4 in general practice. 

Referring to the figure, let 

d = diameter of journal ; 

I = length of journal ; 

e = shoulder of collar ; 

p = pressure per square inch 

of projected area ; 
P = total load on journal ; 
S = fibre stress on material ; 
n = revolutions per minute. 




Journals. 



425 



We then have 



*-VStt)r^- 



For speeds up to 150 revolutions per minute the following values may 
be used : 

Constant Pressure. 





Wrought-iron. 


Cast-iron. 


Steel. 


p = 


700 


360 


700 


s - 


8500 


4260 


14000 


I 

d ~ 


1.5 


1.5 


2 


d = 


0.03]/ P^ 


0.043i/ P 


0.027l/P~ 




Intermittent Pressure. 






Wrought-iron. 


Cast-iron. 


Steel. 


V = 


1400 


700 


1400 


S = 


7000 


3500 


12000 


I 
d ~ 


1 


1 


1.3 


d = 


0.027 \/~P 


0.037 V P 


0.024 i/p" 



When the speeds become higher than 150 revolutions per minute, the 
ratio, -r, should be determined from the speed, according to the following 





Constant Pressure. 






Wrought-iron. 


Steel. 


5 = 


8500 


14000 


I 

d 


0.13|/ti 


0.17i/w" 


d = 


0.0244-J-^-i/p 


wna^iy 




Intermittent Pressure. 




Wrought-iron. 


Steel. 


S = 


7000 


12000 


I 

d ~ 


0.08 j/ n 


0.10j/^ 


d = 


0.0273a/^-i/p 


Wt^ 



The value of -=■ is first computed, and then substituted in the following 

formula to find the value of d. 

The depth of shoulder, e, is obtained from the diameter of the journal. 

e = 0.07d + y & inch. 

The following table gives the diameters of journals for various 
pressures for speeds not exceeding 150 revolutions. For higher speeds 
the formulas should be used. 



426 



Journals. 



Table of Journal Proportions. 

Total pressure, P, pounds. 



Diameter 


Direction of pressure, constant. 


Direction of pressure, alternating. 


of 
journal, 


Wrought-iron. 


Steel. 


Wrought-iron. 


Steel. 


d. 


4-="- 

d 


4— 


4--- 


4- 1A 

d 


Inch. 


Lb. 


Lb. 


Lb. 


Lb. 


1.00 


1100 


1400 


1400 


1800 


1.25 


1700 


2 200 


2 200 


2 200 


1.50 


2 500 


3 200 


3 200 


4 100 


1.75 


3 400 


4 300 


4 300 


5 200 


2.00 


4 500 


5 700 


5 700 


7 300 


2.25 


5 700 


6 800 


6 800 


9 300 


2.50 


7 000 


8 900 


8 900 


11400 


2.75 


8 500 


10 700 


10 700 


13 800 


3.00 


10 000 


13 000 


13 000 


16 500 


3.25 


11800 


15 000 


15 000 


19 300 


3.50 


13 700 


17 300 


17 300 


22 400 


3.75 


15 800 


19 800 


19 800 


25 000 


4.00 


17 900 


22 700 


22 700 


29 300 


4.25 


20 000 


25 600 


25 600 


33 100 


4.50 


23 000 


28 700 


28 700 


37 100 


4.75 


25 000 


32 000 


32 000 


41300 


5.0 


28 000 


35 500 


35 500 


45 800 


5.5 


34 000 


43 000 


43 000 


55 400 


6.0 


40 000 


51000 


51000 


66 000 


6.5 


47 000 


60 000 


60 000 


79 200 


7.0 


55 000 


69 500 


69 500 


89 800 


7.5 


63 000 


80 000 


80 000 


103 000 


8.0 


72 000 


91000 


91000 


117 000 


8.5 


81000 


102 000 


102 000 


132 000 


9.0 


91000 


115 000 


115 000 


148 000 


9.5 


101 000 


128 000 


128 000 


165 000 


10.0 


112 000 


142 000 


142 000 


183 000 


10.5 


124 000 


156 000 


156 000 


202 000 


11.0 


135 000 


172 000 


172 000 


222 000 


11.5 


148 000 


188 000 


188 000 


242 000 


12.0 


160 000 


204 000 


204 000 


264 000 



The use of the table is apparent. 

When the diameter of the journal is given, the load which may be put 
upon it is found. When a given load is to be put upon a journal, the 
nearest value in the proper column is found and the corresponding diam- 
eter of journal taken. 

Necked journals, formed in the body of a shaft, are naturally stronger 
than overhung journals, but the diameter in this ease is determined by 
the duty to be performed by the shaft, which will generally make it larger 
than would be required for an overhung journal. 



Pivots. 



427 



k 



PIVOTS. 

The bearing end of a vertical shaft or spindle is termed a pivot. Such 
pivot bearings are usually made with a recess in the middle, with cross oil 
Channels. Taking the diameter of the recess as % the diameter of the 
shaft, we may make the oil channels T V the diameter in width. 

Let 

P = total vertical pressure on pivot, in pounds ; 
p = pressure, in pounds, per square inch ; 
d = diameter of pivot, in inches ; 
n = number of revolutions per minute. 

We then have the following relations, according to 
Reuleaux : 



Formulas for Pivots. 



Wrought-iron 

or steel on 

bronze. 



Cast-iron 

on 
bronze. 



Slow-moving (p = 1422 700 

pivots. ...| d = 



I n = or < 150 



n >150 



,035 V P 0.05 /P 

700 __ 350 __ 

0.05 ]/ P 0.07 V P 



-- 0.004 V Pn 



Iron or steel 

on 
lignum vitae. 



1422 

0.035 1/ P 

p = 1422 _ 

d = 0.035 j/P 




d = 


0.035]/ P 


0.05 j/ P 


0.07/ P 


\ 


P 


P 


P 


1.00 
1.25 
1.50 
1.75 


816 
1275 
1836 
2500 


398 

622 

895 

1219 


204 
319 
459 
625 


2.00 
2.25 
2.50 
2.75 


3265 
4132 
5102 
6173 


1592 
2016 

2488 
3011 


816 
1033 
1275 
1543 


3.00 
3.25 
3.50 
3.75 


7347 

8622 
10000 
11479 


3494 

4205 
4877 
5599 


1836 
2155 
2500 
2869 


4.00 
4.25 
4.50 
4.75 


13061 
14745 
16530 

18418 


6370 
7192 
8063 
8983 


3265 
3686 
4132 
4604 


5.00 
5.25 
5.50 
5 . 75 


20498 
22140 
24694 
26990 


9954 
10974 
12044 
13164 


5102 
5535 
6673 

6747 


6.00 
6.25 
6.50 
6.75 


29388 
31890 
34490 
37190 


14334 
15630 
16900 
18220 


7344 

7972 
8623 
9298 


7.00 


41690 , 


19600 


10000 



428 



Shafting. 



The three columns headed P give the total pressures permissible for 
wrought-iron or steel on bronze, cast-iron on bronze, and wrought-iron or 
steel on lignum vitse, respectively. If the load is given, find the nearest 
value in the proper column and take the corresponding diameter. 

The frictional resistance of a flat pivot bearing may be determined amP 
follows : 

Let 

F = the tangential frictional resistance, in pounds, 

at the periphery of the pivot ; 
r = the radius of shaft, in inches ; 
r x — the radius of recess, in inches ; 
P = total load on shaft, in pounds ; 
/ = coefficient of friction. 



Then 



■H^r 



If we take, as indicated above, r\ = %r Q , 




F=%fP. 

These formulas apply also to collar bearings of the 
form here shown. 

For very heavy pressures, as in the thrust bearings 
of screw-propeller shafts, the thrust is taken upon a 
number of collars. Good practice limits the pressure 
upon such collars from 40 to 80 pounds per square inch. 

If 

n = the number of collars ; 
d = diameter of shaft ; 
D — outside diameter of collars ; 
P = total thrust ; 

we have, according to Seaton, 



D- 



M 



& + 



P 

47n' 



This provides for a pressure of 60 pounds per square inch on the 
collars. The thickness of each collar is made = 0.4(Z> — d), and the 
space between the collars may be 0.75(2) — d). 



SHAFTING. 



In determining the dimensions of shafting there are two principal ele- 
ments to be considered : the strength and the stiffness. Generally, the 
load acting upon the shaft is given in either one of two forms, — as horse- 
power to be transmitted at a given number of revolutions per minute, or 
as a twisting moment, or torque, expressed in a certain force acting at the 
end of a lever of a given length. In the latter case, the torque is here 
considered to be in inch-pounds. Thus, a belt pulling 100 nounds over a 
20-inch pulley would give 100 pounds at a lever arm of 10 inches, or 1000 
inch-pounds, etc. 

In order that satisfactory results may be secured, a shaft should be so 
proportioned that it may not be subjected to a fibre stress at the circum- 
ference greater than the predetermined limit ; and also that it may not be | 
twisted through a greater angle than has been established as satisfactory. 
It is, therefore, necessary to compute the diameter by two methods, one 
for strength and the other for stiffness, and use the result which gives the 
greatest size. 



Shafting. 429 



In the formulas the following symbols are 

p == the force acting to rotate the shaft ; 
R = the lever arm at which it acts ; 
jV = the horse-power transmitted ; 
n = the number of revolutions per minute ; 
d = the diameter of the shaft ; 
L = the length of shaft, in feet ; 
& = the angle of torsion, in degrees ; 
S = the fibre stress at the circumference ; 

Q — the modulus of torsion of the material = § of the modulus 
of elasticity. 

We then have, for strength, 



'-V3** 



and for stiffness, 



d— y. 



32 12 . L 360 
irG' &> ' ~2n FK ' 



Taking the fibre stress, S = 8500 pounds, we have for wrought-iron 
shafts, for strength, 

d = 0.091 \/PR = 3.33-*/—. 

r \ n 

I 

In taking the torsion of shafting into consideration, the greatest allow- 
j able twist in degrees should not be over 0.075° per foot in length of shaft- 
ing,— that is, #° = 0.075L, which gives for stiffness, against torsion, 



d = 3^/ PK = 4.7-1/—. 
\ n 



N 
The quotient of effect, — , is obtained from the relation to the statical 

; moment, PR, as follows : 

pR = 330M )X 12 .K =m26 *. 

2n n n 

From these formulas the following table for round wrought-iron shafts 
has been calculated. An inspection of the table will show that it is quite 
possible for a shaft to be strong enough to resist permanent deformation 
and yet be so light as to be liable to spring under its load. For example, a 
shaft 26 feet long, with a twisting force of 220 pounds applied at one end, 
and acting with a lever arm of 20 inches, gives a turning moment, PR = 
4400 inch-pounds, which would require a shaft only 1% inches diameter 
(see column 2). This, however, would permit far too much torsion, and 
in order that the angular deflection should not exceed the limit of 0.075° 
per foot, a corresponding value of PR, in column 4, must be found, and 
against it, in column 1, will be given the diameter,— in this case about 2% 

1 inches,— which, by comparison with column 2, gives about five-fold 
strength. 

For short shafts this examination of angular deflection is unnecessary, 
as, for example, in the short lengths between two gear-wheels, for here 

t the value of & will be small enough in any case. With longer shafts, and 

] in all special constructions, it is important to consider the angular deflec- 

I tion and keep it within the given limit. 

For steel shafts, whose modulus of resistance is f greater than wrought- 



430 



Shafting. 



iron, the diameters in both cases may be taken as ^0.6,— that is, 0.84 
times that of correspondingly-loaded wrought-iron shafts. 

Shafting which is subjected to sudden and violent shocks, as in rolling 
mills, etc., must be made much stronger than the preceding formulas re^ 
quire, and these must be classed with the special cases which occur iri 1 
every branch of construction. 



Wrought=iron Shafting. 





For strength. 


For stiffness 


(torsional). 


d. 


PR. 


N 
n 


PR. 


N 


Inch. 










1 


1327 


.021 


123 


.0019 


IK 


2 591 


.052 


301 


.0048 


IX 


4 479 


.071 


625 


.0099 


m 


7 112 


.114 


1157 


.0183 


2 


10 616 


.168 


1975 


.0313 


2M 


15 115 


.239 


3164 


.0502 


?A 


20 730 


.329 


4 822 


.0765 


2% 


27 600 


.438 


7 061 


.1120 


3 


35 830 


.568 


10 000 


.1587 


3^ 


56 890 


.902 


18 520 


.2941 


4 


84 930 


1.347 


31600 


.5015 


±A 


120 900 


1.919 


50 620 


.8032 


5 


165 800 


2.632 


77 160 


1.2240 


&A 


220 800 


3.503 


111 000 


1.7920 


6 


286 600 


4.548 


160 000 


2.5390 


&A 


364 400 


5.784 


220 300 


3.4960 


7 


455 200 


7.222 


296 400 


4.7040 


^A 


559 800 


8.883 


390 600 


6.2000 


8 


679 400 


10.780 


505 700 


8.0240 


*A 


815 000 


12.930 


644 400 


10.2300 


9 


967 400 


15.350 


810 000 


12.8600 


$A 


1 138 000 


18.050 


982 700 


15.6000 


10 


1 327 000 


21.050 


1 230 000 


19.5900 


10% 


1 536 000 


24.380 


1 501 000 


23.8100 


11 


1 766 000 


28.020 


1 808 000 


28.6800 


UK 


2 018 000 


32.020 


2 159 000 


34.2600 


12 


2 293 000 


36.390 


2 560 000 


40.6200 



d = diameter of shaft, in inches ; 

R = lever arm of torque, in inches (as radius of pulley or gear-wheel) ; 

P — force on lever arm, in pounds ; 

N = actual horse-power transmitted ; 

n = revolutions per minute. ^ 



Find the nearest value for PR or 



N 



and take the largest diameter of shaft corresponding, 
multiply this diameter by 0.84. 



both for strength and for stiffness, 
For steel shafts 



Shafting. 431 



For any given shaft the angle of torsion, &, for a given statical moment, 
PR, may be found from the following formulas : 



d- = 0.00062^ 



d* 



= 0.0001208>s4-> 
a 

in which L is the length of shaft, in feet, and d the diameter, in inches, — 
S being the fibre stress at the point of application on the shaft. 

When the force is applied at one end of the shaft and taken off at the 
other, L is the whole length of the shaft. When the twisting forces are 
applied over the whole length of the shaft uniformly, L may be taken as 
one-half the length of the shaft; and when the twisting forces diminish 
uniformly from one end to the other, iis taken as one-third the length. 

For a number of twisting forces applied at various points along the 
shaft, multiply the horse-power at each point by its distance from the end 
of the shaft, add the several products together, and divide by the total 
horse-power transmitted. The quotient may be used as the mean value of 
L in the formula. 

Since the modulus of elasticity is practically the same for iron and 
steel, these formulas are good for either material. 
N 

Since PR = 63025 — -, the above formulas can easily be used when the 
n 
load is given in horse-power for a given number of revolutions instead of 
torque. 

Thus, suppose a shaft 164 feet long transmitting 70 horse-power at 100 
revolutions, the power being taken off by machines uniformly distributed 
along its length. The effective length, L, may then be taken as ^f- 4 = 82 

N 70 
feet. We also have — = -— - = 0.7, and from the preceding table, under 

n 100 
the column for torsional stiffness, we find the values 0.5015, corresponding 
to 4 inches diameter, and 0.8032, corresponding to 4% inches diameter, so 
that we make the shaft 4% inches diameter. We have, also, 

PR = 63025— = 63025 X 0.7 = 44117. 
n 

The angular deflection will then be 

* = 0Wig^? = 6.88, 
or 6° 53'. 

Hollow Shafts. 

Since the metal close to the axis of a shaft is of much less value in 
resisting stresses than the portion near the perimeter, there is a manifest 
advantage in using hollow or tubular shafting. Such shafts are very gen- 
erally used for screw-propeller engines. 
If . ' 

d = the diameter of a solid shaft ; 

d' = the outside diameter of a hollow shaft of equal strength ; 
d = diameter of hole through hollow shaft ; 

dr, 
^ = ~- = ratio of external to internal diameter of hollow shaft ; 
di 
then 

7 'I d3 

^ = \'W4- 

For d = unity we have, for the following values of i//, the corresponding 
values of d\ : 



^-=0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


di = 1.002 


1.000 


1.021 


1.047 


1.096 


1.192 


1.427 



432 



Shafting. 



For any diameter solid shaft, therefore, we have simply to multiply by 
the value for di, for the chosen ratio of external to internal diameters, to 
get the external diameter of a hollow shaft of equal strength. 

In marine practice the ratio, \f/, of diameter to bore is generally 0.5, the 
bore being one-half the external diameter. The external diameter will 
then be 1.2 times that of an equivalent solid shaft, or only 2 per cent, 
greater, and, at the same time, the shaft will be 25 per cent, lighter. 

The power which a hollow shaft will transmit may be obtained by 
finding the capacity of the corresponding solid shaft. 

Thus, for 

vi/ = 0.3 0.4 0.5 6 0.7 0.8 0.9 

d = 0.998 0.992 0.979 0.955 0.912 0.839 0.7 

Thus, a shaft 10 inches in diameter, with a 5-inch hole through it, will 
transmit as much power as a solid shaft 10 X 979 = 9.79 inches diameter. 



Deflection of Shafts. 

In proportioning a shaft to carry a given load the practical method is to 
determine the pressures upon the journals and proportion them according 
to the methods already given. The rest of the shaft can then be propor- 
tioned according to the statical moments at various points, determined 
graphically, as follows : 

Draw the line, A-C, equal in length to the distance between centres of 
journals, and upon it construct any triangle, ABC, whose apex lies on the 
line of the load, Q. Draw AS normal to A-C, making AS = Q; draw 3-0 
parallel to B-C, and 2-0 parallel to A-C; then A-2 = P u 2-3 = P 2 . 
The two journals may then be proportioned for these pressures. 
By dropping the perpendiculars from the ends of the hub-seat we may 
divide Q into two forces, Q\ and Q 2 , shown in the force polygon by O-b, 
parallel to B\B 2 , giving A-b = Q lt 6-3 = Q 2 . The vertical ordinate, t, at 
any point of the surface of moments is proportional to the statical 
moment, M XJ , at its point of intersection with the axis, as. for example, 

the ordinate, fa at the base of the 
journal for P^ We have, in any 
~ case, 

V2 . 




32 



irS 



Y M y , d?- 



32 

irS 



■M lt 



S being the fibre stress ; and hence 



<*> Vltf' 



^V 



from which y can readily be ob- 
tained. 

A similar diagram may be 
drawn for any loading, and the 
relative proportions of the various parts of the shaft or axle determined. 
The graphical method has the further advantage of showing the distribu- 
tion of stresses on the axle at one time ; and even if a straight axle is used 
the points of greatest stress are thus clearly seen. 

The practice of mounting heavy fly-wheels, or the rotor members of 
large electric generators, upon engine shafts, renders it necessary to con- 
sider the influence of bending and twisting moments combined. This 
may be done by uniting the two moments into an equivalent or "ideal" 
bending moment, such that the proportions of the shaft may be computed 
from it directly. 
Let 

M d = the twisting moment for a given shaft section ; 
M b = the bending moment for the same section. 
Then the ideal moment, combining them both, will be 
M = %M b + %VM b * + M d K 
The application of this formula is best seen by an example. 



Shafting. 



433 



If we have a wheel subjected to a force of 6000 pounds, at a radius of 
12 inches, on a shaft 100 inches long, being 20 inches from one end and 80 
inches from the other, we have a bending moment, M b l , of 



^ 



6000 X ^ X 20 = 



96000 inch-pounds. 



We have, also, the twisting moment, 

M d = PR = 6000 X 12 = 72000 inch-pounds. 
The combined moment will then be 



M = % . 96000 + %y 720002 + 96000 2 
= 36000 + 75000 = 111 000 inch-pounds ; 
and the corresponding shaft diameter from the table, page 430, is about 4% 
inches. For the twisting moment of 72,000 inch-pounds alone the diameter 
would have been about 3% inches. 

The graphical method may be applied to this problem very effectively. 




Referring to the figure, first construct the link polygon, abc, for the 
bending moments, and the force polygon, alO, giving the forces, P x and A. 
and also ace' , the surface of moments for the shank, AC. 

The moment, M d , is yet to be determined. In the force polygon, with a 
I distance, R, from the pole, O, draw a vertical ordinate ; this will be M d . 
| Lay its value off at c'c x and 6& lt and five-eighths of these values give c%b b 
for the parallelogram of torsion for the shank, CB. 

The combination of the bending and twisting moments may then be 
made according to the formula. Make cc^ = %cc', and join cb; then at 

28 



434 



Bearings. 



any point of the polygon, as, for example, at /, the distance, // 2 = %ff. 
Now transfer c f c to ab, at c'c p' ; then will the hy pothenuse of the" triangle, 
Coc'cq', divided by C2C0' = 1/ \%cc') 2 +. {%Ci&) 2 , and the sum, cc% + c^ = 
cc 2 + C0C3, the desired moment, (M h ) it for the point, C. In the sam^ 
manner we obtain // 2 +/2/0' =fh +A/& the moment, (if&)i, for the point^' 
F. The line, c 3 / 3 bo, is a curve (hyperbola), which may be taken approxi- 
mately with sufficient accuracy as a straight line, C3&02. The various 
dimensions may be obtained from the polygon, acbb^c' , in a similar 
manner, as shown in the discussion of axles. 



BEARINGS. 

The form and shape of bearings in which journals are carried vary 
much with the service demanded. For line shafting the most satisfactory 
results are obtained with cast-ijon boxes, usually made four diameters in 
length, and supported on spherical seats in adjustable screw-plugs, per- 
mitting a limited adjustment and enabling the box to align itself to the 
shaft. Such boxes are used in drop-hangers, pillow-blocks, and wall- 




brackets, the whole being a general article of manufacture. When the 
bearing is made rigid, the length is rarely more than one to two diameters, 
as it is difficult to maintain good alignment for greater lengths, and heat- 
ing and cutting are apt to follow. 

The proportions of a standard pillow-block bearing are given in the 
illustration, this being a form used in heavy mill shafting, in the outboard 
bearings of steam engines, and similar work. 

The proportions of the pillow-block are in terms of a modulus, d x =^ 
l.lfid + 0.4 inches, in which d is the diameter of the journal to run in the 
bearing. 

The dimensions of the brasses are in terms of the modulus, e ■— 0.07d + 
0.125 inch. 

The main bearings of steam engines are similar in design, but are gen- 



Hangers. 



435 



erally made with side brasses and adjustments to take up wear both hori- 
zontally and vertically, the casting of the pillow-block forming a portion 
of the engine bed-plate. For various designs of such bearings see Reu- 
leaux's "Constructor" and Unwin's "Machine Design." 
f* The proportions of hangers and pillow-blocks for shafting are given in 
the following illustrations. 



i £-- — - J j^H" - ^j_ _ 



j }--lT6-|-*i 




The dimensions of the hanger are in terms of the modulus, d' = lAd + 
0.25 inches. The drop, a, varies according to local conditions. The screw- 




_.t_l/ 



blugs are of cast-iron, as are also the boxes. Lateral adjustment is made 
i>y providing slotted bolt-holes in the base of the hanger. 



436 



Bearings. — Couplings. 



The wall-bracket is based upon the same modulus as the hanger, d x *= 
lAd + 0.25 inches. 

The pillow-block differs from the hangers in having no vertical adjust- 
ment, the spherical sockets being cast in the base and cap, as shown ir< 




the illustration. The modulus is d 1 = lAd + 0.25 inches. These designs 
are originally due to Wm. Sellers & Co., of Philadelphia. 



COUPLINGS. 

The simplest form of coupling is the plain cylindrical muff coupling 
shown in the illustration. 

The thickness 5 = — + 0.25 inch, d being the diameter of the shaft; 

the length being 85. This coupling is cheap, and serves for light work. 



? 8(5 *j 




For heavier shafts the plate coupling is used. 

The thickness of the hub 8 = -~ + 0.25 inch, and the length of hul^ 

on each plate = 46. The other dimensions are based on the modulus, d' = 

0.125d -t-^inch, this being the diameter of the bolts. The number of 

16 
bolts, N = 2 + O.Sd. 



Couplings. 



437 



The two halves of a plate coupling should be forced on their respective 
shafts and keyed fast as well, and should then be turned up and faced off 

fw 8;75d', 




on the same centres as were used in turning the shafts. All pulleys, gears, 
etc., should then be put on the shaft in halves, the plates of the coupling 
not being removed. 



i 

r 







For screw-propeller shafts the plates are forged on the shafts, as shown 
in the illustration. 

According to Seaton, the diameter, d\ of the bolts in such couplings 
should be : 

For 4 bolts, 6! = 0.32d; 
For 5 bolts, d' = 0.28d ; 
For 6 bolts, d' = 0.25d; 
For 8 bolts, d' = 0.20d; 
For 10 bolts, d f = 0.18d; 

Thickness of plate = — = 0.3<2; 

Diameter of bolt circle = 1.6<3; 
Outside diameter, D = 1.6d + 2%d'. 

Number of bolts, one for every two inches diameter of shaft. 
For ordinary line shafting the double-cone coupling of Sellers' has been 
very extensively used. As shown in the illustration, the tightening of the 




olts draws the cones together and clamps them upon the shafts. The 

iimensions given are in terms of the modulus, 5 = — + 0.25 inch, d 

eing the diameter of the shaft. This coupling permits a slight variation 



438 



Couplings. 



in the diameters of the shafts, and, unlike the plate coupling, it may be 
removed and replaced satisfactorily to permit the placing of pulleys upon 
the shaft. 

When it is desired to disconnect portions of a transmission, various 

forms of clutch couplings are used. 
!*-2.-o»i A great variety of these have been 

designed, and, as an example, the 
cone clutch is given. 

The general proportions of a 
cone clutch are given in the illus- 
tration, based on the modulus, 

5 = — + 0.25 inch. The angle of 

bevel, a, is made not less than 10°, 
or it is found difficult to disengage 
the parts. If PE is the turning 
moment, or torque, to be trans- 
mitted, the axial pressure, Q, to 
hold the parts in engagement will 
be 

. PE ( sin a \ 

Q = — ^-y- + COSaj, 

in which r is the radius of the 
cone bearing and / the coefficient 
of friction. For iron on iron, / 
may be taken at 0.15, and r should 
not be made less than 3d,— preferably greater. 

For connecting shafts which are placed at an angle with each other the 
universal joint is employed, shown in skeleton in the illustration. While 
this is convenient, it must be remembered that the angular velocity trans- 
mitted is not uniform. If a is the 
angle between the shafts, and to and 
(o l the angular movements of the two 
shafts, respectively, then 

tan (*>! = tan <o cos a. 

The variation thus has a period of 
180°. 

The following table gives the values of <ai for successive values of w, 
for various angles, a : 





to 


a. = 


10° 


20° 


30° 


40° 


Deg. 


Deg. 


Min. 


Deg. Min. 


Deg. Min. 


Deg. Min. 


30 


29 


38 


28 29 


26 34 


23 51 


45 


44 


34 


43 12 


40 54 


37 27 


60 


59 


34 


58 26 


56 22 


53 4 


90 


90 




90 


90 


90 


120 


120 


26 


121 34 


123 38 


126 56 


135 


135 


26 


136 48 


139 6 


142 33 


150 


150 


22 


151 31 


153 26 


156 1 


180 


180 




180 


180 


180 . <. m 



Where this variation is injurious it may be avoided by using two 
universal joints which correct each other. 



Lever Arms. 



439 



LEVERS. 

The proportions of the various forms of lever arms used in machine 
I design are dependent upon various considerations. Thus, the ends are 
** determined by the diameters of the pins to be inserted. 






0.64 , . 




In the above forms the proportions are given in terms of the diameter 
of the pin. These dimensions are for wrought-iron ; for cast-iron they 
should be doubled. 

The calculations of the dimensions of simple lever arms of rectangular 
section are made upon the assumption that the force, P, acts in a plane, 
passing through the middle of the arm and in a direction normal to the 
arm. 

If we let 

h = width of the arm at the axis ; 

b = thickness of the arm at the axis ; 

S = the maximum permissible fibre stress ; 

Taking S for wrought-iron = 8500, and for cast-iron = 4250, we have, 

pj? PR 

for wrought-iron, b = 0.00072— — ; for cast-iron, 0.00144- 



W 



h* 



These formulas are adapted for the determination of b when h has been 
selected, the latter being most con- 
veniently chosen with regard to the 
other condition. ^ P 

Example. Let P = 4400 pounds, 
R = 24 inches for a lever arm of 
wrought-iron, and h = 7% inches, we 
have 



b = 0.00072 



,4400 X 24 
(7.125)2 



1% inches. 




If b is kept constant for the whole 
length of the arm, the width at the 

small end may be Q.bh, while if a constant ratio of 6 : Ms kept, the small 
end = %h. 

If, as often occurs, the force, P, does not act in the middle plane, then 
there must exist a combined bending and twisting stress on the arm. We 
may then derive a combined stress whose bending moment will give an 
ideal arm, R'. 

If the plane in which the force, P, acts is distant from the middle of 
the arm by an amount, a, we may make, approximately, 

B' = y*R + %V Rz + cfi. 

Thus, if the lever in the preceding example was acted upon by the 
same force with an overhang, a, of 15 inches, we have 



whence 



' = % . 24 + %V 242 + 152 
= 9 + % . 28.3 = 26.7 ; 



= 0.00072 



4400 X 26.7 
(7.125)2 



= 1.66 inches. 



440 



Lever Arms. 



Sometimes it is desired to make a lever arm of ribbed or I-section to 
secure lightness or economy of material. The dimensions may then best 
be obtained by computing the rectangular section and transforming this 
into the section desired. 

Let h be the width and b the thickness of the rectangular section as « 
found in the preceding method, and let h and b be the corresponding di- 
mensions in the section selected from among those given in the illustration. 




Then we have 



in which 



(f-0 [•*--(*)■]• 



These formulas permit a choice of the ratios, — and 



which may be 

left to the judgment of the designer. 

In the structural built-up sections the value of the angle-irons has been 
neglected, as it may be considered as making up for the weakening effect 
of the rivet-holes. 

The following table of values of - — - — enables the transformation to be 



readily effected. 



1 + a 



Table for Transforming Arm Sections. 











Values of 


1 










h 








1 + a 










c 


4-« 


3 


3.5 


4 


4.5 


5 


6 


7 


8 


10 


6 


.50 


.43 


.38 


.33 


.30 


.27 


.23 


.20 


.18 


.14 


7 


.52 


.45 


.40 


.35 


.32 


.29 


.25 


.21 


.19 


.15 


8 


.54 


.47 


.42 


.37 


.34 


.31 


.26 


.23 


.20 


.16 


9 


.56 


.49 


.44 


.39 


.36 


.33 


.28 


.24 


.22 


.18 


10 


.58 


.51 


.46 


.41 


.37 


.34 


.29 


.26 


.23 


.19 


11 


.60 


.53 


.48 


.43 


.39 


.36 


.31 


.27 


.24 


.20 


12 


.62 


.55 


.50 


.44 


.41 


.37 


.32 


.29 


.26 


.21 


14 


.64 


.58 


.52 


.47 


.44 


.40 


.35 


.31 


.28 


.23 


16 


.67 


.60 


.55 


.50 


.47 


.43 


.38 


.34 


.30 


.25 


18 


.69 


.63 


.57 


.52 


.49 


.46 


.40 


.36 


.33 


.27 


20 


.71 


.65 


.60 


.55 


.52 


.48 


.42 


.38 


.34 


,29 


22 


.73 


.67 


.62 


.57 


.53 


.50 


.45 


.40 


.37 


.31 


24 


.75 


.68 


.64 


.59 


.56 


.52 


.47 


.42 


.38 


.33 


27 


.76 


.71 


.66 


.62 


.58 


.^ 


.50 


45 


.41 


.35 


30 


.78 


.73 


.68 


.64 


.61 


.57 


.52 


.47 


.43 


.37 


33 


.79 


.75 


.70 


.66 


.63 


.60 


.54 


.50 


.45 


.39 


36 


.81 


.76 


.72 


.68 


.65 


.61 


.56 


.52 


.48 


.41 


40 


.83 


.78 


.74 


.70 


.67 


.64 


.58 


.,54 


.50 


.44 


45 


.84 


.80 


.76 


.72 


.69 


.66 


.61 


.57 


.53 


.47 


50 


.85 


.81 


.78 


.74 


.71 


.68 


.63 


.59 


.56 


.49 



Cranks. 441 



Example. A lever arm has a length, R = 78.75 inches, and the journal 
pressure at the end = JP = 5500 pounds. It is to be of cast-iron of double 
T-section with a height, ho = 12% inches. We have, for a rectangular 
section, 

With this the I-section may be compared. Here we may take c : h = 

1 : 12, B : b = 4, and we get from the table = 0.44 and b = 0.44b = 1.71 

1 + a 
inches, and the flange breadth, B = 1.71 X 4 = 6.84 inches, the flange 

1 1^ 625 

thickness = c = — h = — -^ — = 1.05 inches, all of which are practical 

dimensions. It may be found desirable to have c = b or any reasonable 
ratio, or B : b, and c : hbe chosen. * 

Example. A wrought-iron arm has been found to require b = 2% inches, 

h = 12% inches. It is desired to make -=— = 0.25, and in column 10 we find 

0o 

0.25 opposite — = 16 ; hence, b = 0.57 inch and B = 10 X 0.59 = 5.90 

. , _ 12.625 _ _ . , 

inches, and c = — =^ — = 0.8 inch, 
lb 
This table may be used for transforming sections for many other pur- 
poses, such as beams, crane booms, struts, etc. 

CRANKS. 

The general proportions of engine cranks are obtained from the methods 
already given. The diameter and length of pin are found, as are those of 
a journal subjected to alternating stresses, according to the table on page 
425, and the shaft is determined by the values of the twisting and bending 
moments upon it. The thickness of metal about the hub and eye of the 
crank is then proportioned according to the diameters of the shaft and 
pin, as shown on page 439. 

For important structures it is desirable to make a graphical analysis of 
all the stresses upon the crank and its shaft. Starting with a pin propor- 
tioned to resist the maximum effort of the connecting rod upon it, the 
following graphostatic analysis will enable all the parts to be equal in 
strength to the pin, according to Reuleaux : 

The Crank Axle.— Having calculated d and I, draw the skeleton dia- 
gram of the crank,— that is, the neutral axis, ABODE, in which BC repre- 
sents the axis of the crank arm, which in this case lies normal to the axis 
of the shaft, and is placed in its proportional distance from the centre of 
the crank pin, A, and from the bearing, D. Then lay off the force, P, 
from a, normal to Ea, choose the pole, 0, of the force polygon (this being 
best placed upon a line passing through the end of P and parallel to the 
axis, Ea), draw the ray, adO, and line, dE, also the ray, OP\, parallel to 
dE ' then adE will represent the cord polygon for the bending which P 
proauces upon the axle, aCE, and PP\ represents the force upon the jour- 
nal, E, and P^ the force upon the journal, D. Also make aPequal to the 
crank radius, R, and draw EG ; this latter will be the twisting moment 
which P exerts upon the axis. This moment, M d , may be combined with 
the bending moment, M b , to give for each point an ideal bending moment, 



Mi = %M b + %\/Mb* + Mi, 

from which the polygon curve, dd'e' , and surface of moments, Cdd'e'E, 
are obtained. From the latter, in combination with the pin diameter, d, 
and ordinate, t, of the base of the pin, the diameter of the shaft may be 
obtained according to formula 



<h. \< 



442 



Connecting Rods. 



The Crank Arm.— Prolong Ea to a , and transfer the cord polygon, 
Dad, to the base line, BC,— that is, make the angle a^BC = the angle Dad, 
and then will BoqC be, with horizontal ordinates, the surface of moments 

for the bending of 

P 1 t the crank arm due to 

the force, P. Also 
make Ceo = Bb = Cc, 
then will the hori- 
zontal ordinates of 
the torsion rectan- 
gle, Bb c C, be the 
moments with which 
P acts to twist the 
crank arm about the 
axis, BC. This mo- 
ment may again be 
combined with the 
bending moment to 
give an ideal mo- 
ment, as before; 
(a a f = %a d C, draw 
Ba f , make at any 
point, H, the space, Hi = %Bb , and make Hh = hji' + h'i), which gives 
the surface of moments, Bb'hFC, for the crank arm. From this and from 
the diameter, d, and ordinate, t, we can construct the conoidal form of the 
arm, IKLM, according to formula 




JL 



■M\- 



From this, again, the profile, STUV, of an arm of rectangular section 
may be derived, the width, h, being assumed for any point, and the corre- 
sponding thickness, b, obtained from the value, y, of the conoid, according 
to the formula 



7 -"(*)• 



If the position of the axis, BC, does not give satisfactory results, the 
operation must be repeated with a better relation of parts. By proceeding 
in this manner the dimensions of a crank and axle may be so determined 
that they will be equal in strength to the pin upon which the power is 
exerted. 

For a similar treatment of other forms of cranks and cranked axles see 
Reuleaux's "Constructor." 



CONNECTING RODS. 

The body of a connecting rod may be made of wrought-iron, cast-iron, 
steel, or even of wood. In the latter case it is usually only subject to 
tension. 

If the rod is of circular cross-section of diameter, D, and the force of 
tension be P, we have the following relations : 



wrought-iron, 
steel, 



D 

l/P 

D 

Vp 



= 0.015 ; 



0.012; 



cast-iron, 
oak, 



D 

Vp 

D 

VI? 



= 0.03 ; 



= 0.06. 



These give stresses of 5600, 9500, 2800, and 400 pounds, respectively, or about 
two-thirds the value given for ordinary conditions. 



Connecting Rods. 443 



For short connecting rods the formulas cited are all right for com- 
pression as well as for tension, but for long rods a greater diameter should 
be used to provide against buckling. Owing to the great variety of condi- 
tions, a factor of safety, m, must be introduced, and we have the following 
formulas, in which D is the diameter of the round rod ; L, its length, in 
inches ; and P, the total pressure, in pounds : 



wrought-iron or steel, D = 0.01641/ m "\ L\/ P ; 



cast-iron, D = 0.0195 y 7 m *\ LV P \ 



wood, D = 0.034 \/ m \ LV P . 

We have for 

7/1 = 1.5 2.0 3.0 4.0 6.0 8.0 10.0 15.0 20.0 25.0 30.0 40.0 50.0 60.0 
y / ~m = 1.11 1.19 1.32 1.41 1.56 1.68 1.78 1.97 2.11 2.24 2.34 2.51 2.66 2.78 

For various services the following values of m may be taken : locomo- 
tive engines, m = 2 to 5 ; high-speed stationary engines, m = 10 ; ordinary 
stationary engines, m = 20 to 25 ; marine engines, ra = 30 to 40. 

The above dimensions are for the middle of the rod. When the rod is 
tapered both ways, it is made 0.8D at the crank ends and 0.7Z> at the cross- 
head end. For high-speed engines the size is usually made greatest at the 
crank end, being about 1.7D, the cross-head end being 0.7D. 

For rods of a rectangular cross-section, in which the depth of cross- 
section = h and thickness = b, we have, for any given ratio of h to b, 



h = 0.0144 y 7 m a/ ' (y) \ L\/ P. 



In order to simplify the use of this formula, the following table will be 
of use : 

-^- = 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 
b 



A'U) 



1.36 1.42 1.49 1.55 1.62 1.68 1.74 1.80 1.87 1.93 1.99 

h 



Example. Let P = 30,000 pounds, L = 72 inches, -r- = 2.5. Taking 
m = 2 for a locomotive engine, we have 

y~m = 1.19, 



and h = 0.0144 X 1.19 X 1.99 \72y 30000 == 3.8 inches, 



3 8 
and b = - 1 - = 1.52 inches. 



Connecting=rod Ends. 

The general proportions of a strap end for a connecting rod are given 
in the illustration. The dimensions are in terms of the modulus, 

d l = d + 0.2 inch, 

d being the diameter of the journal or crank pin. The dimensions of the 
brasses are in terms of the unit, 

e = 0.07d + 0.125 inch. 



444 



Connecting Rods. 



The modulus, d\, is assumed on the basis of an ordinary overhung 
crank pin. For a cranked axle or an eccentric, however, the increased 
diameter would give unsuitable dimensions, and in such cases the modulus 
becomes 



d'i 



di + 



lb . /# 



in which d\ is the modulus for an overhung crank pin for the same press- 
ure as the one under consideration ; b and d being the corresponding 
values for the overhung pin and b l and d> those selected for the new one. 



fr b<=0.M>i 



2e 0.-33 




For heavy service the marine type of rod end is used, one form being 
shown in the illustration. Here the end of the rod is forged into a T. and 
the brasses, cored out as shown, form the bearing and the rod end, the 
bolts and steel cap forming the resistance to the driving stresses. 




- 



The dimensions here are based upon the diameter of the bolts at the 
bottom of the thread. The bolts are turned down, as shown, in order to 
distribute the strain and avoid breakage at the base of the thread. 



Eccentrics. 



445 



Taking fibre stresses of 5000 pounds for wrought-iron and 6600 pounds 
for steel, we have, according to Unwin, for the diameter of the bolts at the 
bottom of the thread, 

8 = 0.02 j/p for wrought-iron, 

= 0.018 i/P for steel; 

the other dimensions being in terms- of 8 ; P being the one-half maximum 
pressure upon the piston, or the pressure upon one bolt. 

For rods of moderate size, where a closed end is permissible, the follow- 
ing type is convenient, compact, and inexpensive. 




The proportions of the above type of stub end are based on the 
modulus, 

8 = Q.lbd + 0.2 inch. 

The brasses are based on t = 0.08d + % inch. 

The diameter of the bolts is 0.02<2 = 0.25 inch, taking the nearest even 
size. 

The taper of the wedge is made >g incn to the inch. 

The square end brass is made with a small lip on each side on the end 
only, to prevent lateral movement; the collar on the pin prevents the 
lateral movement of the bevelled brass. 

The brasses should not be left open upon the pin, but either fitted close, 
and filed off when wear is to be 
taken up, or else a number of 
sheets of copper foil placed be- 
tween them before boring the 
hole, forming slivers which can 
be taken out one at a time when 
necessary. 

A variety of connecting-rod 
ends will be found in Reuleaux's 
"Constructor" and Unwin's 
"Machine Design." 



ECCENTRICS. 




nal 



Eccentrics may be considered 
as cranks in which the diameter 
of the pin is greater than the sum of the crank circle plus the shaft 
diameter. 

The breadth of the eccentric (properly the length of pin, I) is the same 
as that of the equivalent overhung journal subjected to the same pressure ; 
for the depth of flange, a, we have 

a = 1.5c = 0.07 £ + 0.2, 

from which the other dimensions can be determined as in the illustrations. 



446 



Eccentrics. 



For some forms of shafts with multiple cranks or other obstructions the 
eccentrics cannot be made as shown before, but must be in halves, bolted 
together. 

The eccentric straps may be proportioned as in the illustrations, the 
modulus being derived from that for the equivalent overhung journal, as 




already described. The diameter of the bolts, 6, is found from the two 
moduli, as follows : 

5 = 0.33d! -f 0.06d'i. 

The form on the left is intended to be made in cast-iron, and that on 
the right in wrought-iron. The most important feature in the operation 
of eccentrics is the maintenance of complete lubrication, as otherwise the 
high lineal velocity of the rubbing surfaces is apt to produce heating. 

In the form shown in the illustration the strap is made wider than the 
eccentric, and the lip is bevelled as shown, thus forming a circular 
channel on each side, in which the oil collects and is distributed over the 




rubbing surfaces. When this form of strap is to be used in the horizontal 
position the strap should be divided at an angle of 45° with the horizontal, 
otherwise the oil will run out at the joint, when standing. In practice, 
cast-iron on cast-iron is found to give excellent results as to wear and 
smooth running. 



Cross-heads. 



447 



CROSS-HEADS. 



For a simple T cross-head the dimensions shown in the illustration 
may be used. If Pis the maximum load upon the rod the journals are to 



& 



i JH-M).5d 

— A: f*| i 





- l79d- 



be computed for a load of %P, and the depth, h, in the middle may be 
made = 2.5d + &A. The thickness, 6, may be determined from 

PA 
b = 0.00035-r-, 

which corresponds to a 

fibre stress of 8500 pounds 

for wrought-iron. 

The arrangement of 

cross-heads for use in 

connection with guide 

bars depends very much 

upon the form of guides 

employed. Some ex- 
amples will serve to Cotter 

indicate the general pro- di* o.2±di | 

portions. 

For four-bar guides 

the following design may 

be employed, the propor- 
tions being those given 

by Unwin. The length and diameter of the pin are determined by 

the pressure upon the rod, the diameter usually being about 0.8 the size 

for the crank pin, and the 
length equal to the 
diameter. The dimensions 
of the other parts are in 
terms of the pin diameter. 
d. The cross-head itself is 
of wrought-iron, with cast- 
iron slide blocks. 

When but two guide bars 
are used the form of cross- 
head shown here may be 
employed. 

The dimensions are in 
terms of the pin diameter. 

In these cross-heads the 
length and width, A and £, 
of the slides should be so 
proportioned that the press- 
ure should not be more than 
40 to 60 pounds per square 
inch. For large engines the 
slides may be fitted with 
bronze shoes and with set- 
screws or wedges for ad- 
justment. If, however, the 
area is made large, the wear 

is very slight, and no such provision is necessary. 

In some engines of moderate size the guides are cast on the frame and 




448 



Gearing. 



bored out in line with the cylinder. The cross-head shown below is an 
example designed for use with such guides. Numerous special forms of 
cross-heads will be found in Reuleaux's " Constructor'' and Unwin's 
"Machine Design." 

The pressure on the guides depends upon the total pressure on the 
piston-rod and upon the maximum angle which the connecting rod makes 





with the line of the guides. Assuming that the greatest pressure occurs 
when the position of the crank is at right angles with the line of the 
guides, we have 

1/ n? — 1 

in which P 1 is the pressure on the guide ; P, the total pressure on the 
piston-rod ; and n, the ratio of the length of the connecting rod to the 
radius of the crank. Thus, if the connecting rod be made 5 cranks in 
length, we have 

Pi = 1 „_„. P = 0.204P. 

1/25 = 1 

If the pressure on the piston is 10,000 pounds, the greatest pressure on 
the guides will be 2040 pounds ; and at 40 jKmnds per square inch a single 
slide-block should have an area of 51 square inches. 



GEARING. 



In the transmission of motion by toothed gearing it is necessary to 
know the number of teeth and their shape, as well as the dimensions of 
the cylinders, cones, or other figures upon which they are formed. 

In all cases toothed gear-wheels are substitutes for smooth, rolling sur- 
faces, the teeth being employed merely to obviate the slipping which 
might otherwise take place. 

If we consider two spnr-gears in engagement with each other, we can 
imagine the teeth being made smaller and smaller in size and, at the same 
time, greater and greater in number, until they become indefinitely small 
and the surfaces become practically smooth. Such rolling surfaces con- 
stitute the pitch surfaces of the gear-wheels, and the aim of toothed-gear- 
ing design is to shape the teeth so that the rolling action of the pitch 
surfaces may be maintained and, at the same time, forces of determinate 
magnitude transmitted without slip. In discussing gear-teeth, therefore, 
the pitch circles, of which the rolling action is to be reproduced, are the 
basis upon which the teeth are constructed. 

Let 



P = radius of pitch circle ; 
t = distance from centre to centre of adjacent teeth - 

pitch ; 
Z number of teeth. 



•■ circumferential 



We then have 



(i l.v.nr.z. 

In 



Toothed Gearing. 



440 



When the gear-wheels are of large size and to be cast, made from 
wooden patterns, it is desirable to work to definite and convenient lineal 
pitch distances, in which case the pitch, /, is selected, and the correspond- 
! Gig radius, A', found for the given number of teeth, thus, 

R = 0.159162*. 

Thus, for a wheel of 75 teeth and 2% inches pitch, we have 

R = 0.15916 X 187.5 = 29.85 inches, 

and the pitch diameter 27v = 59.70 inches. 

In order to abridge the work of computation the following table may 
be used. It is only necessary to take out the number corresponding to the 
number of teeth and multiply it by the pitch to obtain the radius of the 
corresponding pitch circle. The pitch may be taken in any unit, inches, 
sixteenths of an inch, millimetres, etc., and the radius will be in the same 
unit. 

Thus, for 75 teeth, we fmd opposite 70 and under 5 the number 11.94; 
and 11.94 X 2.5 -= 29.85 inches, the same as before. 



Table of Radii of Gear=wheels. 

Multiply tabular number for given number of teeth by the circumfer- 
ential pitch to obtain radius. 



z 





1 


2 


3 


4 


5 


6 


7 


8 


9 







.159 


.318 


.477 


.637 


.796 


.955 


1.114 


1.273 


1.432 


10 


1.59 


1.75 


1.91 


2.07 


2.23 


2.39 


2.55 


2.71 


2.86 


3.02 


20 


3.18 


3.34 


3.50 


3.66 


3.82 


3.98 


4.14 


4.30 


4.46 


4.62 


30 


4.77 


4.93 


5.09 


5.25 


5. 1 1 


5.57 


5.73 


5.89 


6.05 


6.21 


40 


6.37 


6.53 


.6.68 


6.84 


7.00 


7.16 


7.32 


7.48 


7.61 


7.80 


50 


7.96 


8.12 


8.28 


8.44 


*.'» 


8.75 


8.91 


9.07 


9.23 


9.39 


60 


9.55 


9.71 


9.87 


10.03 


10.19 


10.35 


10.50 


10.6(5 


10.82 


10.98 


70 


11.14 


L1.30 


11.46 


11.62 


11.78 


11.91 


12.10 


12 25 


12.41 


12.57 


80 


12.73 


12.89 


13.05 


13.21 


13.37 


13.53 


13.69 


13.85 


14.01 


14.16 


90 


14.32 


14.48 


14.64 


14.80 


14.96 


15.12 


L5.28 


15.44 


15.60 


15.76 


100 


15.92 


16.07 


16.23 


16.39 


16.55 


16.71 


16.87 


17.03 


17.19 


17.35 


110 


17.51 


17.67 


17.83 


17.98 


18.14 


18.30 


18.46 


18.62 


18.78 


18.94 


120 


19.10 


19.26 


19.42 


19.58 


L9.73 


19.: SO 


20.05 


20.21 


20.37 


20.53 


130 


20.69 


20.85 


21.01 


21.17 


21.33 


21.49 


21.05 


21.80 


21.96 


22.12 


140 


22.28 


22.44 


22.60 


22.76 


22.92 


23.08 


23.21 


23.40 


23.55 


23.71 


150 


23.87 


24.03 


24.19 


24.35 


21.51 


24.67 


21 83 


24.99 


25.15 


25.31 


160 


25.46 


25.62 


25.78 


25.91 


26.10 


26.26 


26.42 


26.58 


26.74 


20.90 


170 


27.06 


27.21 


27.37 


27.53 


27.69 


27.85 


28.01 


28.17 


28.33 


28.49 


180 


28.65 


28.81 


2S.97 


29.13 


29.28 


29.44 


29.60 


29.76 


29.92 


30.08 


190 


30.24 


30.40 


30.56 


30.72 


30. SS 


31.04 


31.19 


31.35 


31.51 


31.67 


200 


31.83 


31.99 


32.15 


32.31 


32.47 


32.63 


32.79 


32.95 


33.10 


33.26 


210 


33.42 


33.58 


33.74 


33.90 


34.06 


34.22 


34.38 


34.54 


34.70 


34.85 


220 


35.01 


35.17 


35.33 


35.49 


35.65 


35.81 


35.97 


36.13 


36.29 


36.45 


230 


36.01 


36.76 


36.92 


37.08 


37.21 


37.40 


37.56 


37.72 


37. SS 


38.04 


240 


38.20 


38.36 


38.51 


38.67 


38.83 


38.99 


39.15 


39.31 


39.47 


39.63 


250 


39.79 


39.95 


40.11 


40.27 


40.42 


40.58 


40.74 


40.90 


41.06 


41.22 


260 


41.38 


41.54 


41.70 


41.86 


42.02 


42.18 


42.34 


42.49 


42.65 


42.81 


270 


42.97 


43.13 


43.29 


43.45 


43.61 


43.77 


43.93 


44.09 


44.25 


44.40 


280 


44.56 


44.72 


44.88 


45.04 


45.20 


45.36 


15.52 


45.68 


45.84 


46.00 


290 


46.15 


46.31 


46.47 


46.63 


46.79 


46.95 


47.11 


47.27 


47.43 


47.59 



29 



450 



Toothed Gearing. 



For small pitches, especially for cut gearing, the so-called Diametral 
Pitch is much used. 

Thus, we have, as before, 



whence 



A 
t 



Z 2R 

7t— ; or— - 



2R 



Or, the number of teeth is equal to the pitch diameter of the gear mul- 
tiplied by n, divided by the circumferential pitch. By making the circum- 
ferential pitch an aliquot part of n, the relation of the number of teeth to 
the diameter may be very simply expressed. Thus, instead of making a 
gear of %-inch pitch, the pitch may be made equal to 



3.1416 



: 0.5236 inch, 



and we have 



t 



-. 6 and Z = 2R X 6, 



so that for every wheel we have only to choose the diameter and multiply 
by 6 to obtain the number of teeth ; or select the number of teeth and 
divide by 6 to obtain the diameter. In like manner we may choose pitches 
of 77, %t, i^ i^v etc., or, as they are commonly called, one pitch, two 
pitch, three pitch, etc., these really meaning the number of teeth corre- 
sponding to each inch in diameter of the wheel ; hence, the name, Diame- 
tral Pitch. 

Since such gears are cut with standard cutters, the fractional circum- 
ferential pitch is provided for in the making of the cutter, and need not 
be further considered. 

For convenience in selecting the approximate size of tooth required, the 
following table, showing the lineal value of diametrai pitches, is given : 



Diametral 
pitch. 


Circumferential 
pitch. 


Diametral 
pitch. 


Circumferential 

pitch. 




Inch. 




Inch. 


1 


3.1416 


6 


.5236 


2 


1.5708 


7 


.4488 


3 


1.0472 


8 


.3927 


4 


.7854 


9 


.3491 


5 


.6283 


10 


.3142 



Thus, 3 diametral pitch is about equal to 1 inch circumferential pitch. 4 
diametral pitch is a little more than % inch, and so on. The diametral 
system simply throws the inconvenient fraction into the size of the gear- 
cutter, and thus simplifies all the succeeding work. 

The form of gear teeth is a subject to which much study has been given. 
Formerly, when each establishment made its own gear-cutters and de- 
signed its own tooth outlines, the question was of more practical impor- 
tance than at the present time, when accurately-formed cutters are regular 
articles of merchandise, and when it is only necessary to indicate the 
diametral pitch and the number of teeth to enable the proper cutter to be 
selected. 

Two systems are in general use, the epicycloidal and the involute, and 
their respective merits have been actively discussed. Practice has shown, 
however, that there is little real difference between them, but the facility 
with which the involute system adapts itself to the design of machines 
for automatically generating tooth outlines gives it practical advantages. 



Toothed Gearing. 



451 



Epicycloidal teeth are generated in the following manner : 
External Teeth (Fig. a).— Given the number of teeth, Z, and pitch, t, 
or ratio, — , of the wheel. Make OP = R = — = %Z( — J , and the radius, 

| IT 2lT \ TT J 

r„, of the rolling circle, W= O.S7bt, or = 2.75— ; draw the outside circle of 

IT 

the teeth, K, with a radius = R + O.St, and the inside circle, F, with 
radius = R — OAt, and make the thickness of tooth = ±%t. Arc Sb = arc 




ab ; arc Sc = arc ic. Sa, the face curve, is generated by the rolling of W 
upon T; Si, the flank curve, by the rolling of W in T. For pinions of 
eleven teeth, Si becomes a straight line and radial. Pinions with as few 
as seven teeth can be made to work on this system, for although the flanks 
are undercut, they are still within the limits of the theoretical flank 
profile, as shown in the following illustration, where a 7-tooth pinion is 
shown with a rack tooth. The backlash is T y. 

Internal Teeth (Fig. b). — The generation of internal teeth is similar 
to the preceding. The radius of base circle is — R, and the length of 
tooth above and below the pitch circle is O.St and OAt, as before ; r = 

t 19 

O.Slbt = 2.75 -, and the thickness of tooth = — t. The flank, Sa, is generated 

7T 40 

by rolling TTupon T, and the face. Si, by rolling W inside of T. 

In the case of a rack, R = oo , Sa and Si then become similar portions 
of the common cycloid. 

In teeth of this form the line of action coincides with the rolling 
circles, the portion included being = arc ba + the corresponding arc, b\0\, 
of the opposing wheel, when both are external gears, and + the arc, ci, 
for an internal gear working with a spur gear. The duration of action, e, 
varies between 1.22 and 1.60. 

Involute teeth are generated as follows : 

The curve is developed by unwrapping a line from a base circle, which 
is concentric with and bears a definite relation to the pitch circle. 

External and Internal Teeth.— Given the number of teeth, Z, and 
t Zt 

pitch, t, or ratio, — , for the required wheel. Make OP = R = -^- = 

IT LIT 

Y^zi — \ , and draw the outer and inner circles, giving the distances, / = 

OAt, k = O.St, above and below the pitch circle ; also make the thickness 
of the tooth = ±%t. 



452 



Toothed Gearing. 



Draw the line, NPN^ at an angle of 75° with OP, and it will be tangent 
to the base circle, G, the radius of which = r = 0.966P == 0.154Z£ = 




Limiting case of epicycloidal gear teeth, showing 7-tooth pinion engaged with 

rack. 

0.483Z( j . If, now, we unwrap the line, NP, upon the circle, G, from P 
outward to a and inward to g, the path, aPg, of the point, P, will be the 




required tooth outline, which for wheels of fewer than 55 teeth may be 
prolonged by a radial line to reach the bottom circle. 



Toothed Gearing. 



453 



For two equal wheels of 14 teeth, e is only a little greater than unity ; 
it varies between 1 and 2.5. 




Rack Teeth.— The profile, aPi, is straight and makes an angle of 75° 
with the pitch line, T. The angle 75° can readily he laid off by using the 
drawing triangles of 45° and 30° together. 

For low-numbered pinions on the involute system care must be taken 
to avoid interference. Thus, in the illustration below, in which a 12-tooth 




pinion is shown engaged with a rack, it will be seen that the radial 
flanks of the pinion are crossed by the dotted line of action of the rack 
teeth, as at Ag. This may be remedied by reducing the length of the rack 
teeth, or by rounding off their points. 



454 Toothed Gearing. 



Diametral Pitch Formulas. 

Brown & Sharpe Manufacturing Company. 



Let 

P = the diametral pitch ; 
jy = the pitch diameter ; 
D = the outside diameter 



:1 



Gear. 



- These wheels 
run together. 



N = the number of teeth ; I 
V — the velocity ratio ; J 
d' = the pitch diameter ; ^ 
d = the outside diameter ; I . 
n = the number of teeth ; t 
v = the velocity ratio ; J 

a = distance between the centres of the two wheels ; 
b = the number of teeth in both wheels. 



6 


— 


2aP; 


n 


= 


bV 


v+ V 


N 


= 


nv 


n 


= 


NV 

V 


V 




bv 



v r V 



r 

n = 


Drmuias. 

PD f V 

V 


F = 


nv 
~N~} 


f = 


NV 
n ' 


?> = 


PD'V 
n 


f) = 


2a(N + 2) 



d _ 2a(n + 2) 



5 

2P ; 



77= 2at ' 



v+ F' 


2a F 


t» + F' 


2/ + d' 



Toothed Gearing. 



455 



Circular Pitch. 

With its Equivalent in Diametral Pitch, Depth of Space, and Thick- 
ness of Tooth. 



Circular 
pitch. 


Diametral 
pitch. 


Thickness of 

tooth on pitch 

line. 


Depth to he cut 
in gear. 


Addendum. 


Inch. 




Inch. 


Inch. 


Inch. 


G 


.5236 


3.0000 


4.1196 


1.9098 


5 


.6283 


2.5000 


3.4330 


1.5915 


4 


.7854 


2.0000 


2.7464 


1.2732 


3% 


.8976 


1.7500 


2.4031 


1.1140 


3 


1.0472 


1.5000 


2.0598 


.9550 


2% 


1.1424 


1.3750 


1.8882 


.8754 


2K 


1.2566 


1.2500 


1.7165 


.7958 


2% 


1.3963 


1.1250 


1.5449 


.7162 


2 


1.5708 


1.0000 


1.3732 


.6366 


Ws 


1.6755 


.9375 


1.2874 


.5968 


1% 


1.7952 


.8750 


1.2016 


.5570 


m 


1.9333 


.8125 


1 .1158 


.5173 


ik 


2.0944 


.7500 


1.0299 


.4775 


i% 


2.2848 


.6875 


.9441 


.4377 


i% 


2.5133 


.6250 


.8583 


.3979 


v& 


2.7925 


.5625 


.7724 


.3581 


i 


3.1416 


.5000 


.6866 


.3183 


15 

16 


3.3510 


.4687 


.6437 


.2984 


Vs 


3.5904 


.4375 


.6007 


.2785 


« 


3.8666 


.4062 


.5579 


.2586 


% 


4.1888 


.3750 


.5150 


.2387 


H 


4.5696 


.3437 


.4720 


.2189 


5 /8 


5.0265 


.3125 


.4291 


.1989 


_9_ 
16 


5.5851 


.2812 


.3862 


.1790 


K 


6.2832 


.2500 


.3433 


.1592 


A 


7.1808 


.2187 


.3003 


.1393 


^ 


8.3776 


.1875 


.2575 


.1194 


A 


10.0531 


.1562 


.2146 


.0995 


M 


12.5664 


.1250 


.1716 


.0796 


% 


25.1327 


.0625 


.0858 


.0398 


i 

TB 


50.2655 


.0312 


.0429 


.0199 



456 



Toothed Gearing. 



Diametral Pitch. 

With its Equivalent in Circular Pitch, Depth of Space, and Thickness 
of Tooth. 



Diametral 
pitch. 


Circular pitch. 


Thickness of 

tooth on pitch 

line. 


Depth to be cut 
in gear, 


Addendum. 




Inch. 


Inch. 


Inch. 


Inch. 


% 


6.2832 


3.1416 


4.3142 


2.0000 


% 


4.1888 


2.0944 


2.8761 


1.3333 


1 


3.1416 


1.5708 


2.1571 


1.0000 


IK 


2.5133 


1.2566 


1.7257 


.8000 


IX 


2.0944 


1.0472 


1.4381 


.6666 


m 


1.7952 


.8976 


1.2326 


.5714 


2 


1.5708 


.7854 


1.0785 


.5000 


2^ 


1.3963 


.6981 


.9587 


.4444 


2K 


1.2566 


.6283 


.8628 


.4000 


2% 


1.1424 


.5712' 


.7844 


.3636 


3 


1.0472 


.5236 


.7190 


.3333 


3K 


.8976 


.4488 


.6163 


.2857 


4 


.7854 


.3927 


.5393 


.2500 


5 


.6283 


.3142 


.4314 


.2000 


6 


.5236 


.2618 


.3595 


.1666 


7 


.4488 


.2244 


.3081 


.1429 


8 


.3927 


.1963 


.2696 


.1250 


9 


.3491 


.1745 


.2397 


.1111 


10 


.3142 


.1571 


.2157 


.1000 


11 


.2856 


.1428 


.1961 


.0909 


12 


.2618 


.1309 


.1798 


.0833 


14 


.2244 


.1122 


.1541 


.0714 


16 


.1963 


.0982 


.1348 


.0625 


18 


.1745 


.0873 


.1198 


.0555 


20 


.1571 


.0785 


.1079 


.0500 


22 


.1428 


.0714 


.0980 


.0455 


24 


.1309 


.0654 


.0898 


.0417 


26 


.1208 


.0604 


.0829 


.0385 


28 


.1122 


.0561 


.0770 


.0357 


30 


.1047 


.0524 


.0719 


.0333 


32 


.0982 


.0491 


.0674 


.0312 


36 


.0873 


.0436 


.0599 


.0278 


40 


.0785 


.0393 


.0539 


.0250 


48 


.0664 


.0327 


.0449 


.0208 



Toothed Gearing. 



457 



Strength of Gear Teeth. 

(Lewis.) 

W = load transmitted, in pounds ; 
p = circular pitch ; 
/ = face ; 

y = factor for different number and forms of teeth ; 
s = safe working stress of material. 

W = spfy. 





Value of factor, y. 


Number 


Value of factor, y. 


Number 




^ 






_" 




of teeth. 


a> 


-2 £ 


00 


of teeth. 


<v 


48 -8 






> © 

H 


3 O 

o o o 


1^ 




Involu 

20°. 


£ S 

oo o 
> <o >-> 

fl r-l O 
1— 1 


3 -a 


12 


.078 


.067 


.052 


27 


.111 


.100 


.064 


13 


.083 


.070 


.053 


30 


.114 


.102 


.065 


14 


.088 


.072 


.054 


34 


.118 


.104 


.066 


15 


.092 


.075 


.055. 


38 


.122 


.107 


.067 


16 


.094 


.077 


.056 


43 


.126 


.110 


.068 


17 


.096 


.080 


.057 


50 


.130 


.112 


.069 


18 


.098 


.083 


.058 


60 


.134 


.114 


.070 


19 


.100 


.087 


.059 


75 


.138 


.116 


.071 


20 


.102 


.090 


.060 


100 


.142 


.118 


.072 


21 


.104 


.092 


.061 


150 


.146 


.120 


.073 


23 


.106 


.094 


.062 


300 


.150 


.122 


.074 


25 


.108 


.097 


.063 


Rack 


.154 


.124 


.075 



Safe Working Stress, s, for Different Speeds. 





Speed of teeth, in feet, per minute. 


Material. 


100 

or less. 


200 


300 


600 


900 


1200 


1800 


2400 


Cast-iron 

Steel 


8000 
20000 


6000 
15000 


4800 
12000 


4000 
10000 


3000 
7500 


2400 
6000 


2000 
5000 


1700 
4300 











When great strength is required, and the pressure is always in one direc- 
tion only, the teeth may be shaped with a much greater angle of curvature 
on the back than on the working faces, it being only necessary that the 
back outlines clear each other properly. This may be done by making the 
working faces of the teeth according to the epicycloidal form, as already 
described, and the backs of the teeth of the involute curve, using an angle 
of 53° instead of 75°, as in the usual method. This is equivalent to the use 
of a generating circle for the involute of a diameter of 0.8 times the pitch 
diameter of the gear-wheel. The so-called "thumb-shaped" teeth thus 
derived are sharp on the point and thick at the base, and are much 
stronger than the ordinary form of teeth. 



458 



Toothed Gearing. 



There has been much controversy as to the relative merits of epicy- 
cloidal and involute teeth, but in actual practice there is little difference. 
With wheels properly proportioned to their work, and especially with the 
correct relations of the axes firmly maintained, either form answers all 
practical requirements fully. The greater convenience with which invo- 
lute teeth may he made, especially in the machines in which the tooth 
profile is automatically generated, gives it advantages in construction 
which in most cases far outweigh any points of superiority which have 
been advanced for the epicycloidal system. 



:., 



Bevel Gears. 

When the axes between which motion is to be transmitted are not 
parallel, but intersect each other, the gear teeth must be formed upon 
conical surfaces. Such gears are broadly called bevel gears, and when the 
shafts form a right angle with each other and the wheels are equal to each 
other in diameter they are called mitre gears. 

The geometrical figures, which are formed by one cone rolling upon 
another, require that both cones should have a common apex. The sur- 
face thus developed is called a spher- 
ical cycloid. Of these there are five 
particular forms, as with the plane 
cycloids, the latter being really those 
for a cone with an apex angle of 180°. 
The spherical cycloid is very similar 
in form to the plane cycloid, as are 
also the corresponding evolutes. 

The use of the spherical cycloid 
for the formation of bevel gear teeth 
would involve many difficulties. In 
order to construct such teeth it is, 
therefore, common to use the method 
(first devised by Tredgold) of aux- 
iliary circles, based upon the supple- 
mentary cones, and enabling the 
teeth to be laid out in a similar 
manner to those of spur gears. The 
auxiliary circles for the bevel gears, 
R and i2 lt are those of the spur gears 
having the same pitch, their radii being respectively r and n, the elements 
BS and CS of the supplementary cones. 

For any given angle, a, between the axes, the radius, r, and the number 
of teeth, 3, for the auxiliary circle can be determined from the radii, R 
and Ri, and tooth numbers, Zand Z h by the following formula : 

JL — ^ R2 + -fr 2 + 2RR i cos a 
R ~~ Ri + R cos a 

_Z_ _ V Z°~ + Zr 2 + 1ZZ X COS a 
Z ~ Zi + Z COS a 

If the axes are at right angles we have 

r _ l/ ig~M- R\ 2 
R " 




Vz* + z? 



Hi 



z 



Zi 



n \ n J ' 



Example. A pair of bevel gears have 30 and 50 teeth, and an angle 
between axes a = 60° ; hence, cos a = %> an ^ we have for the auxiliary 
circle of the 30-tooth gear 



'-» ^T + + ^- M -" -^-»*«»» 



For the 50-tooth gear we have, also, 



60; 



l/4900 



30 + 50 . 0.5 



64. 



Toothed Gearing. 



459 



From these numbers and the given pitch the auxiliary circles can be 
laid off and the teeth drawn, 

Low tooth numbers are not available for bevel gears, since the errors 
k which are involved in the method of auxiliary circles become dispropor- 
tionately great. By using not fewer than 24 teeth for the bevel gear a 
minimum of 28 for the auxiliary circle is obtained, and the evolute system 
can be used to advantage. This form of tooth is best adapted for this 
purpose, on account of its simplicity of form, notwithstanding the minor 
defects which have already been noticed. 

Owing to the fact that the form and shape of teeth on bevel gears vary 
along the face of the tooth, such gears cannot be cut theoretically correct 
by rotating cutters. When such cutters are used an approximate form is 
obtained, and filing must be resorted to in order to correct the shape of the 
teeth. At the present time, large bevel gears are usually made by planing 
the teeth, the tool being guided by a former, while small gears are cut on 
machines in which the tooth outlines are generated by the movement of 
the gear blank under the cutter, according to the method first devised by 
Professor Hermann, of Aix-la-Chapelle, in 1877, and employed in the inge- 
nious machines of Bilgram and of Warren. 

The whole subject of the form and action of gear teeth is thoroughly 
discussed in Grant's " Handbook on the Teeth of Gears," Beale's " Practical 
Treatise on the Teeth of Gears," Reuleaux's " Constructor," Unwin's " Ma- 
chine Design," and numerous other standard works. 



Spiral Gears. 

When the axes between which motion is to be transmitted are not 
parallel to each other, and yet do not intersect, gears with spiral teeth are 
usually employed. 

o 




There are a number of useful variations of spiral gears. In the illus- 
tration is shown a pair of wheels, i and £, both with left-hand spirals 
and corresponding tooth profiles. The pitch angles, y and yi , are so chosen 
that at the point of contact the pitch cylinders have a common tangent, 
so that if a be the angle of inclination of the axes, y + 71 + a = 180°. If 



460 Toothed Gearing. 



we indicate by v and i\ the circumferential velocity in the direction of the 
tangent and normal, respectively, we have 

i'i sin v , ni R sin 7 Z 

— - = — — -, whence — - = ^ — : — — = -— . ^ 

v sin y x n Ei sin y x Z Y 

The normal pitches, r = t sin 7 and r x = t\ sin y t , must be equal to each 

other, whence - = . > 
t\ sm 7 
As indicated by the components of velocity, v' and vi', there is an end, 
long-sliding action of the teeth upon each other, with a velocity 

& — i? + i\' = c(cot 7 + cot 71). 

This sliding consumes power and causes wear, and will be at a mini- 
mum when v' and v\' are equally great,— that is, when 7 = y x . 

With regard to the choice of 7 and yi, the conditions may be so taken 
that the position of the coinciding tangents of the two spirals shall be 
slightly before or slightly after the actual line of contact, but as close as 
may be possible. The position of the line of contact may be stated as 
follows : 



it _ cot 7 
as also 



— + COS a 



Ei cot 71 n 

1- COS a 



fh 



cot 7 = 



+ COS o. 



For a = 90° we have cot 7 = — . Such spiral wheels, when the teeth 
n 
are well made, transmit motion very smoothly, but the surface of working 
contact is very small. One of the most important applications is that of 
the worm and worm-wheel. In this case a = 90° and Z— 1, the teeth of 
the wheel, E x , being inclined at an angle, 7, with the edge of the wheel ; 

whence tan 7 = — — = 0.15916 _, r . The velocity ratio of transmission is 

ni : n = Z : Z\. 

The subject of spiral gears is extensively discussed in Reuleaux's " Con- 
structor" and in Halsey's "Spiral Gearing." See, also, "Transactions of 
the American Society of Mechanical Engineers," 1886, Vol. VII., p. 273. 

Proportions of Gear=wheels. 

Gear-wheels may be divided into two classes : 

Hoisting Gears, such as are used in cranes and similar machinery, 

and 
Transmission Gears, used to transmit power continuously at a 
determinate velocity. 
We may include under the term Hoisting Gears all those having a linear 
velocity at the pitch circle of not more than 100 feet per minute, and under 
Transmission Gears all those running at a higher velocity. 

For a pitch, /, face, b, length of teeth, I, and base thickness of tooth, h, 
we have for a tooth pressure, P, corresponding to a stress, S, the general 
formula : 

and for a length of 0.7^ and a thickness of Q.&t we have- 

bt = 16.8-^. 



Toothed Gearing. 



461 



This assumes that the resistance of the teeth is proportional to their 
cross-section, which is also equally true for those which have the same 
ratio of b to t to each other, a condition which is often of much service in 
Dractice. 
^■J* For a hoisting gear of cast-iron let 

(PR) = the statical moment of the driving force ; 
Z = the number of teeth ; 

R = its previously-determined pitch radius, in inches ; 
t = the pitch. 

We have for the given dimensions 



= 0.230- 



V 



(PB) 



t = 0.045 
the face, b, being made 



A^ 



= 0.0730-1/ 
= 0.0145a I 



(PR) 



(PR) . 
R ' 



b = 2L 



These are intended to give a fibre stress, S, of about 4200 pounds. The 
actual stress is properly somewhat less, because the thickness of the tooth 
at the base is usually more than %t. 
PE 

Since the value of — =— is the same as the pressure, P, we can use the 

Jx 

above formulas in cases in which Ponly is given, as for rack teeth. 

In proportioning transmission gears, in which the velocity is greater 
than 100 feet per minute, the greater liability to shock with increased speed 
renders it desirable to assume a lower working fibre stress, S, as the cir- 
cumferential velocity, v, increases. 

For cast-iron we mav take 

9 600 000 

k _ v + 2164 ' 

in which v is the lineal velocity, in feet, per minute. For steel, £ may be 
taken 3% times, and for wood, T % times the value thus obtained. For 



Material . 


v= 100 


200 


400 


600 


800 


1000 


1500 


2000 


2500 


Cast-iron 

Steel 


S= 4240 
5 = 14112 
S= 2544 


4060 

13520 

2436 


3744 

12467 

2246 


3473 
11565 

2083 


3238 

10782 

1943 


3034 

10103 

1820 


2620 
8725 
1572 


2302 
7665 
1381 


2068 

6886 


Wood 


1240 



The velocity, v, may be obtained when n and R (the latter in inches) 
are given, by the following formula : 



2ttRu 

~ 12 



= 0.5236ifri. 



It is also found that the breadth of face, b, should increase with the in- 

p 
crease of P. Tredgold states that the pressure per inch of face, — , should 

not exceed 400 pounds. This, however, is not to be followed implicitly, 
since pressures as high as 1400 pounds have been successfully used in prac- 
tice. It is better, however, to consider the question of wear from the 

P 

product of -r- into n, which should not exceed a predetermined maximum. 

P 
It is found that if — x n exceeds 67,000, the wear becomes excessive. In a 
o 



462 



Toothed Gearing. 



pair of wheels where the teeth of both are made of iron, the greatest wear 
comes upon the teeth of the smaller wheel. In this case we may make 



Pn 

~b~~ 



■■ not more than 28,000, 



and, if possible, it should be taken at less than this value. For smaller 
forces this constant, which we may call the coefficient of wear and desig- 
nate as A, may readily be made as low as 12,000, and even 6000, without 
obtaining inconvenient dimensions. When the teeth are of wood and iron 
the wear upon the iron may be neglected, as the wear comes almost en- 
tirely upon the wooden teeth. For wooden teeth the value of A should 
not exceed 28,000, and is better made about 15,000 to 20.000. 

It must be remembered that the different values of A do not appreciably 
affect the strength, but rather control the rapidity of wear. When sufficient 
space is available, and a low value can be given to the coefficient of wear, 
it is advisable to do so ; if this cannot be done, the coefficient which is se- 
lected will give an indication of the proportional amount of wear which 
may be expected. 

In cases where a number of wheels gear into one other wheel it is better 
to take, instead of the number of revolutions of the common wheel, the 
number of tooth contacts.— that is, the product of the revolutions and 
number of wheels in the group. 

If E is given, as is often the case with water-wheels, fly-wheels, etc., P 
is also known ; and since A can be chosen, we have, taking JV to be the 
horse-power transmitted, 

_ Pn^ __ 63000 _N_ . 
~A~~~^ 22 ' 

hence, 

t = 16.8P 16.8^4 



Sb 



Sn 



If, however, as occurs in many cases, R is not previously determined, 
the choice of the number of teeth, Z, is unrestricted. In such cases we 
have for the width of face, b, 



b = 



396 000 



N 
Zt 



For transmission gears the minimum number of teeth should not be 
fewer than 20. in order that the unavoidable errors of construction shall 
not cause excessive wear ; for quick-running gears it is desirable to have 
still more teeth. The gear-wheels on high-speed turbines seldom have 
fewer than 40, and often as many as 80 teeth. When wood and iron teeth 
are used the least wear is produced when the wooden teeth are on the 
driver, because the action begins at the base of the tooth and passes 
towards the point, while on the driven gear the action is reversed. 



Proportions of Gear = wheel Parts, 



i-— *---- i 



%6j;~di: 








*■- 


6-4 






1.5.5] 6L U| 


y 



The Rim.— The ring of metal upon which the teeth of a gear-wheel are 
placed is called the rim. For cast-iron spur gears the thickness of the rim 
is given by the formula 

S — OAt + 0.125 inch. 



Toothed Gearing. 



463 



The rim is thickened in the middle, or at one edge, to f 8, and also stif- 
fened by a rib, and for gears of tine pitch the section of the rim is curved, 
which harmonizes well with arms of oval section. Accordingly, a pitch 
of 1 inch would give a rim thickness 8 = 0.4 inch + 0.125 inch = 0.525 
s#*inch, or a little over % inch ; and for a pitch of % inch, 8 = 0.325 inch. 




<%6 




3%d 




For bevel gears of cast-iron the rim is made f S thick at the outer edge, 
and of the various forms shown in the illustrations. 

For wooden teeth it is necessary to have a deeper and stronger rim, the 
dimensions being dependent somewhat upon the method of inserting the 




teeth. The proportions are shown in the illustrations. For very wide 
faces the wooden teeth are made in two pieces and a stay bar cast in the 
mortise. 





Small pinions are often cast solid, and when subjected to heavy press- 
ures are strengthened by shrouding, and sometimes this shrouding is 
turned down to the pitch line. 



464 



Toothed Gearing. 



Wheel Arms. — The arms of gear-wheels are made according to the 
following forms, dependent upon the kind of rim used. 

Ribbed sections are made sometimes as shown in the dotted lines, as 
may be most convenient in moulding. Oval sections have the thickness, 









0.8d 

0, of the arm generally made one-half the width, h. A good proportion 
for the arms is obtained when their number, A, is made as follows : 

A = 0.55]/ ZV t . 
From these we obtain the following : 



A =3 


4 


5 


6 


7 


8 


10 


12 


zy t = 30 


53 


83 


119 


162 


211 


330 


475 



Example. Jfor a gear-wheel of 50 teeth and 2-inch pitch we have 
Z-[/ t = 50 1/ 2 = 50 X 1.414 = 70, and this lies between 53 and 83 ; being 
nearer the latter, we give the wheel five_arms. I f th e pitch had been % 
inch, and the same number of teeth, Z\/ t == 50 j/ 0.75 = 50 X 0.866 = 43.3, 
or between three and four arms, the latter number being used in practice. 

The width of arm, h, in the plane of the wheel is somewhat a matter of 
judgment, but may suitably be made according to the ratio, h = 2 to 2.bt, 
when the thickness, /3, may be obtained from the following formula : 



= 0.07- 



,h) ' 



Should this formula give a thickness either too great or too small for 
convenience in casting, another value for — must be taken and the calcu- 
lation repeated. The following table will assist in this operation. 

Table of Gear-wheel Arms. 



h 








Value of 


— , when 

6 










t 


i-' 


9 


12 


16 


20 


25 


30 


35 


40 


1.50 


.20 


.28 


.37 


.50 


.62 


.78 


.93 


1.08 


1.24 


1.75 


.16 


.21 


.27 


.37 


.46 


.57 


.69 


.80 


.91 


2.00 


.12 


.16 


.21 


.28 


.35 


.44 


.53 


.61 


.70 


2.25 


.10 


.12 


.17 


.22 


.28 


.35 


.41 


.48 


.55 


2.50 


.08 


.10 


.13 


.18 


.22 


.28 


.34 


.39 


.45 


2.75 


.06 


.08 


.11 


.15 


.18 


.23 


.28 


.32 


.37 


3.00 


.05 


.07 


.09 


.12 


.16 


.19 


.23 


.27 


.31 



Toothed Geaking. 



465 



The taper of the arms may be made as follows : the ribs at the rim are 

made slightly narrower than the breadth of face, b, and at the hub equal 

to, or slightly greater, than b. For arms of oval section, h may be made 

equal It at the centre of the wheel, tapering to two- thirds this width at 

«uhe rim. 

Hub.— The thickness, w, of the hub may be made 

iv = 0.4ft + 0.4 inch. 
The above proportions are those recommended by Reuleaux. 



Efficiency of Gearing. 

The efficiency of spur gearing depends upon the lineal speed at the 
pitch line, while for spiral and worm gearing the angle of the teeth must 
also be taken into account. 



Velocity at pitch line in feet per minute. 
4 5 6 7 8 9 10 12 15 20 30 40 50 60 70S0 100120150 200 




85 

HO 

S 75 

ho 

>> 

a 

B 
W 
60 

55 

50 
45 
40 



;2: 



m 



z 



^ 



m 



2 



Spur pinion 
Spiral piDion 45 
Sp.Pn.30° pitch 
Sp.Pn.20° 
Sp.Pn.l5° 

<-^ Sp.Pn. or o 

Worm 10° 

5^t~ Sp.Pn. or 

Worm 7° 

-Sp.Pn. or 

Worm 5° 



The accompanying diagram, from experiments by William Sellers & Co., 
Incorporated, gives the efficiencies for practical cases. 

For all ordinary calculations the following efficiencies may be used : 

Cut spur gears 0.96 

Cast spur gears 0.94 

Cut bevel gears 0.95 

Cast bevel gears 0.92 



30 



466 



Toothed Geakixg. 



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Toothed Gearing. 



467 



Proportions of Gear=wheels. 

The following proportions are those of the Yale & Towne Manufac- 
^ turing Company : 




See table opposite. 



BELTS AND PULLEYS. 



Where an exact velocity ratio of transmission is not essential, and when 
the distance between shafts is too great for a positive means of trans- 
mission, belting and pulleys are extensively employed. The question of 
the transmitting capacity of belting is one upon which many discussions 
have been held, and the differences of opinion which have been found 
serve to emphasize the fact that the conditions under which belts are used 
are too varying to permit absolute rules and formulas to be employed. 
For a full discussion of the elements which enter into the problems of 
belt transmission reference must be had to such works as Reu]eaux's 
11 Constructor," Unwin's "Machine Design," and especially to the valuable 
practical paper of Mr. J. W. Taylor in the "Transactions of the American 
Society of Mechanical Engineers," Vol. XV., p. 204. We shall here give 
the general working principles, which will serve to guide practical installa- 
tions. 

The power which can be transmitted by a belt is measured by the pull 
and by the lineal velocity at which the belt travels. The pull is limited 
by the strength of the belt and by the friction upon the pulleys, while the 
lineal velocity is dependent upon the revolving speed of the pulleys and 
upon their diameter. If it is attempted to increase the strength by in- 
creasing the thickness, it is possible that the stiffness of the belt will 
prevent it from wrapping closely about the pulley, and hence the friction 



468 Belting. 



will be reduced. If the speed is made too high the centrifugal force will 
act to throw the belt out of close contact with the pulley, and the friction 
will again be reduced. There are, therefore, several practical limits 
within which satisfactory belt transmissions should be kept. 

The tension which can be maintained in actual practice ranges from 1 
about 30 to 60 pounds per inch of width. If a high tension is put upon a 
belt transmission when it is installed it will gradually diminish, owing to 
stretch, and, unless some tightening device is employed, the belt will, 
before long, slacken until the stress upon it becomes low enough to check 
further stretching. If this tension is sufficient to transmit the power the 
transmission will run well and give but little trouble, while if the load is 
too heavy the belt will slip, and it must either be tightened or a change 
made in width or speed. 

If the power to be transmitted is given in horse-power, we have 33,000 
foot-pounds per minute to consider. If the belt tension is to be 30 pounds 
per inch of width, we must, therefore, have a speed of 1100 feet per 
minute. If the speed is one-half as much, the width must be twice as 
great, and so the given elements must be taken and the others found. 
Usually, the speed and the power are given and the width required. 

w = width, in inches ; 

s = speed, in feet, per minute ; 
N = horse-power ; 

t *= tension, per inch width of belt ; 



:i 



we have 



„__ t ws 
~ 33000 



33000iV 



33000iV 

s = 



tw 

Or, if we have given the width, speed, and horse-power, the minimum 
tension which can be reached before slipping will occur is 

_ 33000JV 
ws 

Thus, if a belt 10 inches wide, running at 4000 feet per minute, is trans- 

33000 V 50 
mitting 50 horse-power, the tension is 1 A = 41.25 pounds. 

The tension available for transmitting power is really the difference 
between the tensions of the tight and slack sides, since there must always 
be tension enough on the slack side to secure sufficient friction on the 
pulley to keep the belt from slipping. 
If we take the formula 

*-- *SL, 

33000 ' 
and write it 

the last term will represent square feet per minute passing a given point. 
By substituting any value for t, and making N= 1, we can thus find how 
many square feet per minute will transmit a horse-power. Good, practical 
belting rules are: For single belts, 60 square feet per minute equals 1 horse- 
power ; and for double belts, 40 square feet per minute equals 1 horse- 
power. These correspond to 45 pounds and 68 pounds tension per inch of 
width, respectively, — tensions which are readily maintained in practice. 

These values are based on the assumption that the belt embraces 180° 
of each pulley. If the arc of contact is less, the power transmitted may be 
taken in the following proportions : 



Pulleys. 



469 



Percentage of Efficiency for Various Arcs of Contact. 



90° 
0.65 



100° 
0.70 



110° 
0.75 



120° 
0.79 



130° 
0.83 



140° 
0.87 



150° 
0.91 



160° 
0.94 



170° 
0.97 



180° 
1.00 



The power for 180° is to be multiplied by the percentage coefficient for 
other arcs. Thus, for 130°, only 83 per cent, as much power is transmitted 
as with 180°. 



Pulleys. 

The function of a pulley is to enable the rotary motion of the shaft on 
which it is mounted to be translated into the lineal motion of the belt, 
and vice versa. This is accomplished by the frictional contact of the 
wrapping connection — be it belt, rope, or wire cable — with the perimeter 
of the pulley. The following general discussion, from Reuleaux, will 
enable special computations to be made for any given conditions : 

When a tension organ, which is loaded at both ends, is passed over a 
curved surface there is produced between the tension organ and the sur- 
face a very considerable sliding friction. The curved surface over which 
the cord is passed is the pulley, and the motion of the cord takes place in 
the plane of the pulley. If the tension, T, on the driving side of the cord 
is to overcome the cord friction, F, as well as the tension, t, of the driven 
side, we have, for the value of the friction, F = T—t. It is dependent 
upon the magnitude of the angle of contact, a, and upon the coefficient of 
friction,/, but is independent of the radius, E, of the pulley ; it is also de- 
pendent upon the influence of centrifugal force. For these conditions we 
have 



T=tef al *- Z) , 
F=t ( e / a( i- 2 



-1). 



In these e is the base of the natural system of logarithms = 2.71828, and 



:12 



yV 1 



v being the velocity of the tension organ, in feet, per second; 



S, the stress in its cross-section ; y, the weight of a cubic inch of the ma- 
terial ; and g, the acceleration of gravity = 32.2. 

The influence of centrifugal force becomes important at high speeds 
and when the tension organ is under small stress. For hemp or cotton 
rope, or for leather belting, we may take y = 0.035, and for wire rope ^bout 
9 times as great. 

The value of S in the formula, z = 12-^—, is properly considered a lunc- 

gS 
tion of a, and we may therefore assume a constant value for the arc, a, and 
thus calculate the following table for the values of 1 — z. 



s. 


Yalue of coefficient, 1 — z, for centrifugal force. 


S. 


Hempen 
rope. 


Velocity of rope, in feel 


, per second. 


Wire rope. 


20 


40 


60 


80 


100 


Lb. 












Lb. 


400 


.987 


.948 


.882 


.791 


.674 


3600 


600 


.991 


.965 


.922 


.861 


.783 


5400 


800 


.993 


.974 


.941 


.896 


.837 


7200 


1000 


.995 


.980 


.953 


.916 


.870 


9000 


1200 


.996 


.982 


.961 


.930 


.892 


10800 


1400 


.996 


.985 


.966 


.940 


.907 


12600 



470 



Rope Driving. 



This table serves both for hemp and for wire rope by taking the nine- 
fold value of S in the right-hand column for wire rope. It should be 
observed that the velocities are in feet per second. It will be seen that for 
high speeds a high stress in the tension organ is necessary, in order to 
oppose the action of the centrifugal force. 

In order to simplify practical calculations we may substitute for the 
exponent, fa(l — z), in each case the form, /'a,— that is, instead of using 
the actual coefficient of friction,/, taking another one,/', which is equal 
to (1 — z)f. If it is a transmission system which is under consideration, 
the friction of the cord, belt, chain," etc., must at least equal the trans- 
mitted force, P; hence, also, must the stress be that of a cord friction >P, 
which gives, for a minimum value of T, 



ef' a 
ef ,a — 1 



whence 



p-V 



Both of these values are absolute numbers. 



The ratio, p , indicates the 



amount of stress which must be given to the tension organ, and hence 

T 
may be called the stress modulus, and is designated as t. The ratio, — , we 

may, in like manner, call the modulus of cord friction, this being under- 
stood to apply to any wrapping connector, and indicate as p. 
A series of values for p and t are given in the following table : 



Moduli for Cord Friction and Stress. 



/*. 


T 
T= P' 


T 
p = f 


fa. 


T 
r== -p' 


T 

p = Y • 


.1 


10.41 


1.11 


1.6 


1.25 


4.95 


.2 


5.52 


1.22 


1.7 


1.22 


5.47 


.3 


3.86 


1.35 


1.8 


1.20 


6.05 


.4 


3.03 


1.49 


1.9 


1.18 


6.69 


.5 


2.54 


1.65 


2.0 


1.16 


7.39 


.6 


2.22 


1.82 


2.2 


1.13 


9.03 


.7 


1.99 


2.01 


2.4 


1.10 


11.02 


.8 


1.86 


2.23 


2.6 


1.08 


13.46 


.9 


1.69 


2.46 


2.8 


1.07 


16.44 


1.0 


1.58 


2.72 


3.0 


1.05 


20.09 


1.1 


1.50 


3.00 


3.2 


1.04 


24.53 


1.2 


1.43 


3.32 


3.4 


1.03 


29.96 


1.3 


L.37 


3.67 


3.6 


1.03 


36.60 


1.4 


1.33 


4.06 


3.8 


1.02 


44.70 


1.5 


1.29 


4.48 


4.0 


1.02 


54.60 



The superficial pressure, p, of the tension organ upon the circumference 
of the pulley increases as the belt or cord passes from the slack to the 

tight side. It is equal to r%rr-, in which b' is the breadth of the surface 
b'Rda 



Pulleys. 



471 



of contact of the belt. Now, for any cross-section, q, the force Q 
hence, we have 

P = Q 

S b'R' 



qS; 



from which it will be seen that the pressure, p, can easily be kept within 
moderate limits. 

Within the limits of injurious action of centrifugal force it is desirable 
that the lineal speed of belts or cords be kept as high as practicable, since 
the power transmitted is directly proportioned to the speed. Belt trans- 
missions are therefore best designed with pulleys of large diameter, and 
small pulleys employed only when the rotative speeds are such as to make 
their use imperative. 




The general dimensions of belt pulleys may be taken as follows : 
Let A = the number of arms, and let the other dimensions be as in the 
figure, then 

E \ 



which gives, for 



5 +-T- 



10 11 12 



The width, h, of the arm, if prolonged to the middle of the hub, may 
be obtained from 

b 7? 

h = 0.25 inch + — + &—. 



The width, hi, of the arm at the rim is equal to 0.8ft, and the corresponding 
thicknesses are e = %h and e\ = y^n- 

Pulleys with two or three sets of arms may be considered as two or 
three separate pulleys combined in one, except that the proportions of the 
arms should be 0.8 or_0.7 times that of single-arm pulleys, or in the pro- 
portion of f% and f% 



The thickness of the rim may be made k = § to VA, this being frequently 
Lrned much thinner. The width of face should be " 
of the belt. 



turned much thinner. The width of face should be from § to f the width 



The thickness of metal in the hub may be made W=h to%h. The 
length of hub may = b for single-arm pulleys, and 2b for double-arm 
pulleys. 

In order to cause the belt to run in the middle of the pulley, the face 
should be made crowning or rounded, the rise being about ^ of the width 
of the face. Tight and loose pulleys should be made with rounded faces, 
and the wide face pulley from which they are driven made with straight 
face. 



472 Belting. 



Three causes of loss exist in belt transmissions,— viz., journal friction, 
belt stiffness, and belt creeping. For horizontal belting we have for the 
journal friction expressed at the circumference of the pulley a loss, E~, 
when T= 2.5P, t = 1.5P: 



F' T+t_ 



t J \2R ^ 2R 1 J ^U 1 " JjJ' 






in which d and dj are the journal diameters and / the coefficient of jour- 
nal friction. This loss is doubtless the greatest of the three. According 
to Eytelwein, the coefficient of stiffness, s, for force, S', which includes 
both'pulleys. is 

P 

4 
in which s = 0.009— = 0.012. 

IT 

The loss from creep is due to the fact that the greater stress on the 
driving pulley over that on the driven requires for a given volume of belt 
a longer arc of contact. For the expenditure of force, G', for creep on 
both pulleys, we have for a stress, Si, on the leading side of the belt, 

i-JL 

G* - T 0.4^ 



P s J^ E + Si' 

+ s 1 

In this E is the modulus of elasticity of the belt, which for leather is 
20,000 to 30,000 pounds. The losses from stiffness and creep are small. 

Example. Let d and d x = 4 inches, E= R x = 20 inches, 8 = 0.2, / = 0.08, 
S = 0.012, E = 28,440, Si = 425, we have 

F* = P 8 X °- 08 X 0.4 = 0.08P; 

also, S°- = P(0.048 X 2)-^- = 0.0048P, 

The total loss is, therefore, 0.08 + 0.0048 + 0.0059 = 9.1 per cent. 



Cone Pulleys. 

When a number of pulleys are placed side by side in order to enable 
varied speeds to be obtained with belt transmission, and are united together 
in one member, we obtain what is called a cone pulley, such pulley being 
used in pairs. This construction involves the problem of determining the 
proper radii for the various pulleys, so that the same belt shall serve for 
all the changes,— i.e., so that the* length of the belt shall be the same for 
each pair of pulleys in the set. The problem may be solved as follows : 

Crossed Belts (Fig. a).— The belt makes the angle, /3, with the centre 
line of the pulleys, R and R x ; and the half length of the belt, I = 

rI~-\-^\ +Pi(^- + /s] + a cos £, a being the distance from centre to 

centre of the pulley. We then have 



(P + PO^ + is) +a^/l- 



(R + Ri)* 



This value is constant when R + P t is constant,— that is, when the in- 
crease to the radius of one pulley is equal to the decrease in the radius of 



Cone Pulleys. 



473 



the other. Crossed belts are seldom used for this service, however, be- 
cause of the injurious friction between the rubbing parts of the belt. 




Open belts (Fig. b).— In this case we have 

I = (E + R^— + (E — EJp + a cos 8, 
and, also, a sin 8 = E — Ei, which gives 

E = (j8 sin 8 + cos 8) -f — sin ,8, 

7T 7T 2 



*1 



- —(6 sin 8 + cos 8) — — sin 8. 

7T Z 



This function is transcendental, but may be graphically represented in 
the following manner : in the rectangle, ABB' A' , with a radius, AB = a, 
strike the quadrant, BMC, about the centre, A. Within this arc will fall 
all the values of 8 which can occur. For any value of 8 = CAM, draw 




C Q 



MN perpendicular to MA and make MN= the arc, MC= ap. Drop the 
perpendicular, MP, to AC, and draw NO perpendicular to MP. NO will 
then = ap sin p. Through iVdraw QNK parallel to AB, and we have AQ = 
PQ -f ^4P = a(j8 sin 8 + cos 8). By taking successively all the values of 8 
between 0° and 90° in this manner, we can determine the path of the point, 
N, which will be the evolute of a circle, CND, BD being equal to the 

length of the arc, BMC-= -^-a. If we now draw DE parallel to BA, and 

take its middle point, F, we have DF = EF = — , and hence the propor- 
tion : 



DF:DB = ~: 



TK- 



= a : 77, and by similar triangles : 
-QA == —(8 sin 8 + cos 8). 



474 Cone Pulleys. 



This value is dependent upon — . If we prolong BF until it intersects 

7T 

AC prolonged, the resulting length, AA' = BB f , will bear to A'B' the 
ratio,—. By then working BG = I, and drawing GH parallel to A'B' , we T ^ 

have GH=—. This length being transferred to IK gives IT — — — 

7T IT 

(£ sin /3 + cos /3). We then have only to use ± -^ sin j8 to solve the 

IT 2 

problem. 

Make AR = — , and we have the perpendicular, RS = -^-sin /3. By lay- 
ing this length off above and below Ton QK we obtain the points, U and 
V, and this finally gives IU for the radius, R, of the larger cone pulley, 
and IV = R\, the radius of the corresponding smaller cone pulley. 

By solutions for successive values of /3 we obtain the curve, DXJXVE, 
which can be used for the determination of the radii of any desired pair 
of pulleys, each pair of ordinates measured from HI belonging to corre- 
sponding pulley on each cone. 

In practice it is usual to find one of the cone pulleys given and the 
dimensions of the other required. In this case VU may be taken as the 
difference, R — R u between the radii, were the steps uniform. By taking 
this difference, R — JR lf in the dividers, and finding the equivalent ordi- 
nate, UV, on the curve, and then adding VI = R u the axis, HI, is found. 

In order to use the curve conveniently, it may also be laid off left- 
handed, as shown in the dotted lines, D'XE'. 

The use of the diagram will be rendered still more convenient if we 
omit the unnecessary value, I. This enables us to distort the curve in the 
direction of the abscissas to any desired extent. This has been done in the 
proportional diagram on page 475, due to Professor Eeuleaux. 

The method of using the diagram is as follows : 

The sides, AB and BE, of the rectangle represent the distance, a, be- 
tween the centres of the pulleys ; all radii are given in proportional parts 
of a, for which reason AB is subdivided, the size of the diagram being 
selected so that A B = 18 to 20 inches. If, then, la and Va are two given 
radii for a pair of pulleys on a pair of cones, we take the vertical chord of 
the curve which = Va — la, prolong the chord downward until its length 
= la, and draw the axis, abed, parallel to AE. Then, for the other pairs 
of pulleys on the cones, we have 62 and 52', c3 and c3', etc., which can be 
taken directly from the diagram with the dividers. If the given pair of 
radii to which the cones are to be made are equal, the chord R — Ri = 0, 
and the axis will pass through Xat right angles to CX. 

If it is desired to construct a pair of cone pulleys to any given speed 
ratio, this can readily be done. If, for example, the given ratio is 1 : 1, we 
lay off toward Cthe corresponding radius, Xd, and prolong the axial line, 
dd r , to its intersection, d, with BE. Then lay off the given geometric ratio 
on CX, considering Xd as 1 (shown in the diagram by the small circles for 
the ratios %, %, %, f, § ), and draw rays from d' through the points of di- 
vision, and these rays will intersect the curve at the corresponding points 
for the pulley radii, R]. We then have for the radii, 

al and aP for the ratio 1 : 4 
b'2 and fr2' for the ratio 2 : 4 
c3 and c'S f for the ratio 3 : 4 
dX and dX for the ratio 4 : 4 
c5 and eb' for the ratio 5 : 4 
e6 and e6' for the ratio 6 : 4 

If the slowest and fastest speeds for any set of cone pulleys be given in 
revolutions per minute, as n and n x , x being the number of speed changes, J 
or steps of the cone, we have for a, the geometric ratio of the series. 



-Mt- 



Cone Pulleys. 



475 



B 


W 








E 


= \ 




ji 


, 4 

' if 




u' 




1' 

\2' 

\ c 

1 Y 


\ / 
\ / 

v 

/ 1 \ 

/j\l V 
PC N \ 

%v\ 

\ \ \\ 

\i ] \ \ \\ 

1 | ^<% 


^ i 
i 

i 
i 

i 
i 
i 

i 
i 

i 
i 
I 








i i 3 7 


i 
i 
i 
i 
i 
i 

i 

i 
i 

i 
i 

V i 








/ a b c 


def 


/ 




Nil 








. / 




i 

i 


2 1 






. / 




1 
1 
1 
1 
j 


j |U 




a b c d \ 


/ /^ 




y 


r 






Y \ 



Thus, for a cone of four steps, with an entire speed ratio of four to one, we 
have x = 4 and x — 1 = 3; hence, 



•Vf 



f 4 = 1.58. 



Then, if the first speed be 100 revolutions, the succeeding speeds will be 
100 X 1.58 = 158 ; 158 X 1.58 = 249.6 ; and 249.6 X 1.58 = 394, or say 400. 

When, as in many lathes, a back-gear system is introduced, it is desira- 
ble that the gear ratio should be so arranged that the speeds rmy proceed 



476 



Cone Pulleys. 



in a geometric ratio throughout all the changes. This is readily done 
according to the same principle. The introduction of the back gear 
simply doubles the number of speed changes ; in the above case it converts 
a lathe with a 4-step cone and four speed changes into one with eight 
changes. The speed ratio of the back gear, therefore, corresponds to the 
next term in the series, or at a 4 = 1.58 4 = 6.25. 

If, to take another example, we have a lathe with a 5-step cone, with 
back gear, the whole should give ten changes. If these are to range from 
100 to 600, we have 



9/ — 

V 6 = 1.22. 



The series will then be 



100 X 1.220 = ioo 
100 X 1.221 = 122 
100 X 1.222 = 149 
100 X 1 223 = i8i 
100 X 1.224 = 221 

for the cone acting direct. 

The back-gear ratio will then give the next term in the series, or 
1.22 5 = 2.70, which, starting with the first step in the cone again, gives 



100 X 1.225 = 270 
100 X 1.226 = 330 
100 X 1.227 _ 403 
100 X 1.228 = 492 
100 X 1.229 = 6 oo. 



When a lathe is not carefully proportioned in this manner it may have 
what is termed a "lump" in its speed, the change produced by throwing 
in the back gear not conforming to the regular geometric ratio of the steps 
of the cone. 

The simplest and most usual arrangements of belting are the plain open 
and the crossed belts. In these, as in all belt transmissions, the velocity 
ratio is inversely as the diameter of the pulleys. 




For these simple arrangements the belts are self-guiding, the only 
requirements being that the shafts shall be truly parallel to each other 
and one or both pulleys be made with crowning face. 

For inclined and intersecting axes self-guiding belts are not suitable, 



Angular Belting. 



477 



except in the case of inclined axes, in which the trace, SS, of the intersec- 
tion of the planes of the two pulleys passes through the points at which 
the belt leaves the pulleys. The leading line then falls in the middle 
plane of each pulley, but the following side of the belt does not ; hence, 
jt such systems can only be run in one direction. The leaving points in the 
figures are at a and fy. The arrangement gives an open belt when the 
angle, 0, between the planes of the pulleys = 0°, and a crossed belt when 
j8 — 180°. In the intermediate positions a partial crossing of the belt is 
produced. If |8 = 90°, the belt is half crossed (or, as commonly called, 
quarter twist) ; if £ = 45°, it is quarter crossed. 

The leading-orf angle may be made as much as 25°. which occurs when 
the distance between the axes is equal to twice the diameter of the largest 





pulley. Another rule for the minimum distance between shafts for quarter- 
twist belts is to make the distance never less than \/bD. 

In general, the rule to be observed for any such arrangement of belting 
is that each part of the belt must lie in the plane of the pulley toward 
which it is moving. 

It is evident that if such a system has its motion reversed the belt will 
leave the pulleys. Under such conditions guide pulleys are introduced, as 
shown in the illustration. 



478 



Rope Transmission. 



The introduction of electric driving of machinery is rendering quarter- 
twist belts and similar contrivances of minor importance in connection 
with the transmission of power, hut such belts will probably continue to 
be used in connection with machines themselves, and hence care must be 
taken in their application. 

In arranging belt transmissions the direction of motion should be made," 
when possible, so as to bring the slack side of the belt on the upper part 
for belts in the horizontal or inclined positions. This brings the sag of the 
belt in such a position as to increase the arc of contact about the pulleys 
and diminishes the probabilitv of slipping. When practicable, machines 
should be so placed with regard to the line shaft that belts on adjacent 




pulleys pull in opposing directions, as in that manner much of the pressure 
due to belt pull may be taken off of the bearings of the shaft, the pulls of 
the belts neutralizing each other. If possible, one pulley should never 
be placed vertically over another, since the weight of the belt acts to 
diminish the contact with the lower pulley. When such an arrangement 
must be employed a tightening pulley may be found necessary, placed 
upon the slack side of the belt. 

Belts are usually joined by lacings, but whenever possible they should 
be scarfed and cemented, this making a much neater joint and rendering 
the joint as effective as any other portion of the belt 

Rope Transmission. 

The transmission of power over longer distances than are practicable 
for belting may be accomplished by use of rope running at high velocities, 
and hence requiring but small diameters. This form of transmission was 
at one time thought to offer great possibilities for long-distance transmis- 
sion, but the development of electrical transmission has caused it to be 
superseded. For many purjx)ses, however, for spans of not less than 70 or 
more than 400 feet, wire-rope transmission may be used with success. The 
complete computations for wire-rope transmission are to be found in 
Reuleaux's "Constructor," but for general purposes the practical rules of 
Messrs. John A. Roebling's Sons Company may be employed. 

The rope used for transmission purposes may be either 6-strand, of 
seven wires each, or with nineteen wires to the'strand. For the 7-wire 
rope the diameter of the sheaves should be not less than 100 times the 
diameter of the rope, and for a 19-wire rope the minimum diameter of 
sheave is 60 times the rope diameter. The wheels are made with a deep 
V-groove, the bottom of the groove on which the rope runs being provided 
with a filling composed of alternate blocks of leather and rubber. 



Rope Transmission. 479 

The tension upon the rope in a transmission is that due to the weight 
of the rope itself, and since this is the measure of the power transmitted 
for a given speed it is entirely practicable to provide such a sag or deflec- 
tion to the rope as will give the tension desired in practice. According to 
Messrs. Roebling, the sag of both parts of a horizontal transmission should 
be sV # part of the span when the rope is stationary. When the rope is 
running the deflection of the upper part will become about £g °* the span, 
and that of the lower part about ■£$ of the span. Under such conditions 
the difference of tension, J, or pull on the tight side of the rope, will be 
three times the weight of a single portion of rope between the sheaves. If 
V is the velocity of the rope, in feet, per minute, the horse-power trans- 
mitted will be 

TV 
IP-- 



33000 ' 



The rope diameters used range from % inch to 1 inch, and the weights 
will be found in the tables on pages 342-344. 

For Manila-rope driving the formulas of Mr. C. W. Hunt may be used 
to advantage. He recommends ropes of 1 to 2 inches in diameter, and 
estimates the strength of good Manila ropes for driving as about 7000- 
pounds per square inch. The working stress, however, should be only 
about 200 pounds per square inch, this making allowance for wear and for 
the reduction in strength at the splice. 

The power transmitted by ropes depends upon the tension and the 
speed, the power increasing with the speed until the influence of centrifu- 
gal force begins to preponderate. 

Let 

T = tension on driving side of rope ; 
t = tension on slack side of rope ; 
F= tension due to centrifugal force ; 
v = velocity of rope, in feet, per minute ; 
W= weight of rope, in pounds, per foot ; 
g = acceleration of gravity. 

The value of W, the weight per foot for a rope of diameter, D, or cir- 
cumference, C, is 

TF=0.3Z>2 = 0.032 C 2 . 

We have for the tension due to centrifugal force 

9 

Assuming that the tension on the slack side necessary for giving adhe- 
| sion is equal to one-half the force doing useful work on* the driving side 
I and calling this available tension for useful work R, we have 

R = %{T-F). 

, The tension on the slack side to give the required adhesion will, therefore, 
1 be equal to %( T— F), whence we have 

t = V 3 (T-F) +F. 

Since F increases with the square of the velocity, there are, with increas- 
ing speeds, a decreasing useful force and an increasing tension, t, on the 
; slack side. The horse-power may, therefore, be obtained from the follow- 
ing formula : 

= 2v{T-F) 
3 X 33000 ' 

The following table has been computed from this formula- 



480 



Rope Transmission. 



Horse=power of Manila=rope Transmission. 

C. W. Hunt. 



(4-1 



S 


Speed of the rope, in feet, per minute. 


50 


II. 

cJ 


1500 


2000 


2500 


| | 
3000 3500 4000 


j 
4500 5000 


6000 


7000 


8000 


Small e 
diam 
of pu 


Inch. 


H.-P. 


H.-P. 


H.-P. 


H.-P. H.-P.'h.-P. 


H.-P. H.-P. 


H.-P. 


H.-P. 


H.-P. 


Inch. 


% 


1.45 


1.9 


2.3 


2.7 


3.0 


3.2 


3.4 


3.4 


3.1 


2.2 





20 


% 


2.3 


3.2 


3.6 


4.2 


4.6 


5.0 


5.3 


5.3 


4.9 


3.4 





24 


% 


3.3 


4.3 


5.2 


■ 5.8 


6.7 


7.2 


7.7 


7.7 


7.1 


4.9 





30 


% 


4.5 


5.9 


7.0 


8.2 


9.1 


9.8 


10.8 


10.8 


9.3 


6.9 





36 


1 


5.8 


7.7 


9.2 


10.7 


11.9 


12.8 


13.6 


13.7 


12.5 


8.8 


* 


42 


i x 4 


9.2 


12.1 


14.3 


16.8 


18.6 


20.0 


21.2 


21.4 


19.5 


13.8 





54 


VA 


13.1 


17.4 


20.7 


23.1 


26.8 


28.8 


30.6 


30.8 


28.2 


19.8 





60 


1% 


18.0 


23.7 


28.2 


32.8 


36.4 


39.2 


41.5 


41.8 


37.4 


27.6 





72 


2 


23.2 


30.8 


36.8 


42.8 


47.6 


51.2 


54.4 


54.8 


50.0 


35.2 





84 



Where large amounts of power are to be transmitted a number of ropes 
are used. In English practice separate ropes are generally employed, but 
in the United States the rope is made endless, passing around the grooves 
in the pulleys as many times as may be necessary, and finally over an idler 
guide pulley supported in a tension carriage, the required initial tension 
being secured by weighting. In the American system ropes of small diam- 
eter are generally employed. 

The form of grooves employed for rope driving, according to Unwin, 
are given in the illustration. The unit for the proportional figures is 7, the 




girth of the rope. If the pulley is a guide pulley merely, the rope should 
rest on the bottom of the groove. The sides of the groove are usually in- 
clined at 45°. 

Mr. Spencer Miller has proposed that the angle of the sides of the 
grooves should be varied to suit the difference in the drameters of the 
pulleys, the angles being equal only when both pulleys are of the same 
diameter. This may well be done when the pulleys are made to order, 
but it is impracticable if pulleys are to be carried in stock. 

The following table gives the transmitting power of cotton driving 
ropes, according to good British practice. 



Rope Transmission. 



481 



Horse=power of Cotton=rope Transmission. 



Speed, in 






Diameter of ropes 


, in inches. 






feet, per 






































minute. 


1 


IX 


m 


iVs 


IK 


i% 


1% 


1% 


2 




H.-P. 


H.-P. 


H.-P. 


H.-P. 


H.-P. 


H.-P. 


H.-P. 


H.-P. 


H.-P. 


2500 


10.8 


13.4 


16.7 


20.5 


24.3 


28.5 


33.2 


38.1 


43.4 


2600 


11.1 


13.9 


17.2 


20.8 


25.0 


29.4 


34.1 


39.4 


44.7 


2700 


11.4 


14.3 


17.7 


21.7 


25.7 


30.2 


35.3 


40.6 


46.0 


2800 


11.8 


14.7 


18.2 


22.3 


26.4 


31.0 


36.2 


41.7 


47.3 


2900 


12.1 


15.1 


18.7 


22.9 


27.1 


31.9 


37.2 


42.8 


48.6 


3000 


12.3 


15.4 


19.1 


23.4 


27.8 


32.6 


38.1 


43.8 


49.5 


3100 


12.5 


15.7 


19.5 


24.0 


28.4 


33.4 


39.0 


44.8 


50.6 


3200 


12.9 


16.1 


19.9 


24.5 


29.0 


34.0 


39.9 


45.8 


52.0 


3300 


13.2 


16.5 


20.3 


25.0 


29.6 


34.8 


40.8 


46.8 


53.2 


3400 


13.4 


16.7 


20.6 


25.5 


30.1 


35.4 


41.6 


47.7 


54.3 


3500 


13.6 


16.9 


20.9 


26.0 


30.6 


36.2 


42.3 


48.6 


55.2 


3600 


13.9 


17.1 


21.2 


26.4 


31.1 


36.5 


43.0 


49.5 


56.0 


3700 


14.1 


17.3 


21.5 


26.8 


31.5 


37.1 


43.6 


50.2 


56.8 


3800 


14.2 


17.5 


21.7 


27.0 


31.9 


37.5 


44.2 


50.8 


57.6 


3900 


14.4 


17.7 


21.9 


27.3 


32.2 


37.9 


44.8 


51.4 


58.2 


4000 


14.5 


17.8 


22.1 


27.5 


32.6 


38.4 


45.3 


51.9 


58.9 


4100 


14.6 


17.9 


22.3 


27.8 


32.9 


38.7 


45.8 


52.4 


59.6 


4200 


14.7 


18.0 


22.5 


28.0 


33.1 


39.0 


46.3 


52.8 


60.3 


4300 


14.8 


18.0 


22.6 


28.1 


33.3 


39.3 


46.6 


53.2 


60.6 


4400 


14.9 


18.1 


22.7 


28.2 


33.4 


39.6 


46.8 


53.5 


60.9 


4500 


15.0 


18.1 


22.7 


28.3 


33.5 


39.7 


47.0 


53.8 


61.2 


4600 


15.1 


18.1 


22.7 


28.4 


33.6 


39.7 


47.2 


54.0 


61.4 


4700 


15.1 


18.1 


22.6 


28.4 


33.7 


39.8 


47.4 


54.2 


61.5 


4800 


15.1 


18.0 


22.6 


28.5 


33.7 


39.8 


' 47.5 


54.2 


61.5 


4900 


15.0 


18.0 


22.5 


28.5 


33.7 


39.9 


47.6 


54.3 


61.6 


5000 


15.0 


17.9 


22.4 


28.4 


33.6 


39.8 


47.5 


54.3 


61.5 


5100 


14.9 


17.8 


22.3 


28.3 


33.4 


39.6 


47.4 


54.0 


61.3 


5200 


14.8 


17.6 


22.0 


28.2 


33.2 


39.3 


47.2 


53.8 


61.1 


5300 


14.7 


17.4 


21.8 


28.0 


33.0 


39.0 


47.0 


53.6 


60.9 


5400 


14.6 


17.2 


21.6 


27.7 


32.7 


38.6 


46.8 


53.3 


60.4 


.5500 


14.5 


17.0 


21.3 


27.3 


32.3 


38.2 


46.1 


52.8 


59.8 



HEAT. 

Heat is defined as a form of molecular energy which is manifested by 
the changes which it produces in the form or state of the bodies upon 
which it acts. The most readily observed effect of heat is that of the 
expansion of the bodies to which it is applied ; and this effect is used both 
for the measurement of quantities of heat and for its useful application 
by conversion into mechanical work. 

Heat can be transferred from one body to another, the hotter body 
parting with heat to the body which is less hot. The scale of quantities 
upon which S'icn transfers are compared is called the scale of temperatures. 
When there is no tendency for the transfer of heat from one body to 
another the two bodies are said to be at the same temperature. 

There are, in nature, certain temperatures which can be identified by 

31 



482 Heat. 

positive phenomena which occur with them. Among them are the melting- 
point of ice and the boiling-point of water, these being considered as 
occurring at the average atmospheric pressure of 14.7 pounds to the square 
inch, corresponding to 29.922 inches, or 760 millimetres of mercury on the 
barometer. Having these, or certain other standards of temperature, it is \i 
practicable to make scales by which other temperatures may be compared. 

The practical method of making instruments for the measurement of 
temperatures is to use the expansive effect of heat upon certain liquids or 
upon a gas. For temperatures within the range of its freezing and boiling 
points mercury is generally used. 

There are three forms of mercurial thermometers, or temperature- 
indicating instruments, in use. These all consist of sealed tubes of fine 
bore, there being a bulb at one end containing mercury. The expansion 
or contraction of the mercury in the bulb causes the portion in the tube to 
move, the extent of this movement indicating the changes in temperature. 
The three thermometers differ from each other only in the graduation and 
numbering of the scales upon the tube. 

In the Centigrade thermometer the position of the mercury at the melt- 
ing-point of ice is taken as the zero of the scale, while the boiling-point of 
water is called 100, the space between being divided into 100 equal parts, 
called degrees. 

The Fahrenheit thermometer was originally designed to range between 
two al together different standard points, one of these being the tempera- 
ture of pounded ice and salt, the other the normal temperature ot the 
human body. The space between these was divided duodecimally, and 
these large divisions subdivided by repeated bisection into halves, quarters, 
and eighths, thus making 96 divisions. Owing to the erroneous measure- 
ments made by Fahrenheit in constructing his early instruments the 
temperature of the human body was taken too low, and it is really equal 
to 98 degrees above the Fahrenheit zero. This scale, if prolonged upward 
to the boiling-point of water, reaches that temperature at 212 degrees, and 
it is often erroneously stated that Fahrenheit's scale was originally derived 
between those points. 

The remarkable uniformity of the early thermometers made by Fahren- 
heit caused his instruments to be used for work involving scientific accu- 
racy, and it is still the scale most extensively used in steam engineering in 
English-speaking countries. 

The Reaumur scale has its zero at the melting-point of ice, as in the 
centigrade scale, and the graduations were intended to correspond to the 
expansion of the mercury in the bulb by T ^ of its original volume for 
each degree. Upon this scale the boiling-point of water is reached at 80 
degrees above zero, and the scale is generally so defined. The Reaumur 
scale is now rarely used ; but many old measurements of importance are 
recorded in it, and hence it is valuable for purposes of comparison. 

In converting the several scales from one to the other the following 
formulas are used : 

Zero Fahr. = —17.77° Cent. = —14.22° Reau. 

Melting-point of Ice. 

Zero Cent. = 32 Fahr. = zero Reau. 

Botling=point of Water. 

212° Fahr. = 100° Cent. = 80° Reau. 
9° Fahr. = 5° Cent. = 4° Reau. 

Formulas. 

Cent. = | (Fahr. t 32) = f Reau. 
Fahr. = § Cent. ± 32 = £ Reau. ± 32. 
Reau. = I Cent. = |(Fahr. q= 32). 

In the accompanying tables the corresponding values of Fahrenheit 
and Centigrade degrees are given for the temperatures generally used in 
engineering. The main tables give the values for even degrees, and by 
means of the supplementary tables the values for tenths of a degree may 
be taken out. 



Thekmometers. 



483 



Fahrenheit to Centigrade. 



Fahr. 


Cent. 


Fahr. 


Cent. 


Fahr. 


Cent. 


Fahr. 


Cent. 


Fahr. 


Cent. 


—5 


—20.55 


57 


13.88 


119 


48.33 


181 


82.77 


243 


117.22 


-4 


—20.00 


58 


14.44 


120 


48.88 


182 


83.33 


244 


117.77 


—3 


—19.44 


59 


15.00 


121 


49.44 


183 


83.88 


245 


118.33 


2 


—18.88 


60 


15.55 


122 


50.00 


184 


84.44 


246 


118.88 


— 1 


—18.33 


61 


16.11 


123 


50.55 


185 


85.00 


247 


119.44 


Zero. 


—17.77 


62 


16.66 


124 


51.11 


186 


85.55 


248 


120.00 


+1 


—17.22 


63 


17.22 


125 


51.66 


187 


86.11 


249 


120.55 


2 


—16.66 


64 


17.77 


126 


52.22 


188 


86.66 


250 


121.11 


3 


—16.11 


65 


18.33 


127 


52.77 


189 


87.22 


251 


121.66 


4 


—15.55 


66 


18.88 


128 


53.33 


190 


87.77 


252 


122.22 


5 


—15.00 


67 


19.44 


129 


53.88 


191 


88.33 


253 


122.77 


6 


—14.44 


68 


20.00 


130 


54.44 


192 


88.88 


254 


123.33 


7 


—13.88 


69 


20.55 


131 


55.00 


193 


89.44 


255 


123.88 


8 


—13.33 


70 


21.11 


132 


55.55 


194 


90.00 


256 


124.44 


9 


—12.77 


71 


21.66 


133 


56.11 


195 


90.55 


257 


125.00 


10 


—12.22 


72 


22.22 


134 


56.66 


196 


91.11 


258 


125.55 


11 


—11.66 


73 


22.77 


135 


57.22 


197 


91.66 


259 


126.11 


12 


—11.11 


74 


23.33 


136 


57.77 


198 


92.22 


260 


126.66 


13 


—10.55 


75 


23.88 


137 


58.33 


199 


92.77 


261 


127.22 


14 


—10.00 


76 


24.44 


138 


58.88 


200 


93.33 


262 


127.77 


15 


— 9.44 


77 


25.00 


139 


59.44 


201 


93.88 


263 


128.33 


16 


— 8.88 


78 


25.55 


140 


60.00 


202 


94.44 


264 


128.88 


17 


— 8.33 


79 


26.11 


141 


60.55 


203 


95.00 


265 


129.44 


18 


— 7.77 


80 


26.66 


142 


61.11 


204 


95.55 


266 


130.00 


19 


— 7.22 


81 


27.22 


143 


61.66 


205 


96.11 


267 


130.55 


20 


— 6.66 


82 


27.77 


144 


62.22 


206 


96.66 


268 


131.11 


21 


- 6.11 


83 


28.33 


145 


62.77 


207 


97.22 


269 


131,66 


22 


— 5.55 


84 


28.88 


146 


63.33 


208 


97.77 


270 


132.22 


23 


— 5.00 


85 


29.44 


147 


63.88 


209 


98.33 


271 


132.77 


24 


— 4.44 


86 


30.00 


148 


64.44 


210 


98.88 


272 


133.33 


25 


— 3.88 


87 


30.55 


149 


65.00 


211 


99.44 


273 


133.88 


26 


— 3.33 


88 


31.11 


150 


65.55 


212 


100.00 


274 


134.44 


27 


— 2.77 


89 


31.66 


151 


66.11 


213 


100.55 


275 


135.00 


28 


— 2.22 


90 


32.22 


152 


66.66 


214 


101.11 


276 


135.55 


29 


— 1.66 


91 


32.77 


153 


67.22 


215 


101.66 


277 


136.11 


30 


— 1.11 


92 


33.33 


154 


67.77 


216 


102.22 


278 


136.66 


31 


— .55 


93 


33.88 


155 


68.33 


217 


102.77 


279 


137.22 


32 


Zero. 


94 


34.44 


156 


68.88 


218 


103.33 


280 


137.77 


33 


+ .55 


95 


35.00 


157 


69.44 


219 


103.88 


281 


138.33 


34 


1.11 


96 


35.55 


158 


70.00 


220 


104.44 


282 


138.88 


35 


1.66 


97 


36.11 


159 


70.55 


221 


105.00 


283 


139.44 


36 


2.22 


98 


36.66 


160 


71.11 


222 


105.55 


284 


140.00 


37 


2.77 


99 


37.22 


161 


71.66 


223 


106.11 


285 


140.55 


38 


3.33 


100 


37.77 


162 


72.22 


224 


106.66 


286 


141.11 


39 


3.88 


101 


38.33 


163 


72.77 


225 


107.22 


287 


141.66 


40 


4.44 


102 


38.88 


164 


73.33 


226 


107.77 


288 


142.22 


41 


5.00 


103 


39.44 


165 


73.88 


227 


108.33 


289 


142.77 


42 


5.55 


104 


40.00 


166 


74.44 


228 


108.88 


290 


143.33 


43 


6.11 


105 


40.55 


167 


75.00 


229 


109.44 


291 


143.88 


44 


6.66 


106 


41.11 


168 


75.55 


230 


110.00 


292 


144.44 


45 


7.22 


107 


41.66 


169 


76.11 


231 


110.55 


293 


145.00 


46 


7.77 


108 


42.22 


170 


76.66 


232 


111.11 


294 


145.55 


47 


8.33 


109 


42.77 


171 


77.22 


233 


111.66 


295 


146.11 


48 


8.88 


110 


43.33 


172 


77.77 


234 


112.22 


296 


146.66 


49 


9.44 


111 


43.88 | 


173 


78.33 


235 


112.77 


297 


147.22 


50 


10.00 


112 


44.44 | 


174 


78.88 


236 


113.33 


298 


147.77 


51 


10.55 


113 


45.00 ; 


175 


79.44 


237 


113.88 


299 


148.33 


52 


11.11 


114 


45.55 i 


176 


80.00 


238 


114.44 


300 


148.88 


53 


11.66 


115 


46.11 


177 


80.55 


239 


115.00 


400 


204.44 


54 


12.22 


116 


46.66 


178 


81.11 


240 


115.55 


600 


315.55 


55 


12.77 


117 


47.22 ; 


179 


81.66 


241 


116.11 


800 


433.33 


56 


13.33 


118 


47.77 I 


180 


82.22 


242 


116.66 


1000 


537.77 



For Supplementary Tables, see page 485. 



484 



Thermometers. 







Centigrade to 


Fahrenheit. 






Cent. 


Fahr. 


Cent. 


Fahr. 


Cent. 


Fahr. 


Cent. 


Fahr. 


Cent. 


Fahr. 


—273.00 


—460.7 


16 


60.8 


330 


626 


950 


1742 


1570 


2858 


—260.00 —436.0 


17 


62.6 


340 


644 


960 


1760 


! 1580 


2876 


—250.00; —418.0 


18 


64.4 


1 350 


662 


970 


1778 


1590 


2894 


—240. 00 : —400.0 


19 


66.2 


360 


680 


980 


1796 


; 1600 


2912 


—230.00! —382.0 


20 


68.0 


370 


698 


990 


1814 


1610 


2930 


—220.00 —364.0 


21 


69.8 


380 


716 


1000 


1832 


1620 


2948 


—210.00 —346.0 


22 


71.6 


390 


734 


1010 


18-50 


1630 


2966 


—200.00 —328.0 


23 


73.4 


400 


752 


1020 


1868 


1640 


2984 


—190.00 —310.0 


24 


75.2 


; 410 


770 


1030 


1886 


16-50 


3002 


—180.00 —292.0 


25 


77.0 


420 


788 


1040 


1904 


1660 


i 3020 


—170.00 -274.0 


26 


78.8 


430 


806 


1050 


1922 


1670 


3038 


—160.00 —256.0 


27 


80.6 


440 


824 


1060 


1940 


1680 


30.56 


— 150.00! —238.0 


28 


82.4 


450 


842 


1070 


1958 


1690 


3074 


—140.00 —22*0.0 


29 


84.2 


460 


860 


1080 


1976 


1700 


3092 


—130.00 —202.0 


30 


86.0 


470 


878 


1090 


1994 


1710 


3110 


—120.00 —184.0 


31 


87.8 


i 480 


896 


1100 


2012 


1720 


3128 


—110.00 —166.0 


32 


89.6 


490 


914 


1110 


2030 


1730 


3146 


—100.00 —148.0 


33 


91.4 


500 


932 


1120 


2048 


1740 


3164 


— 90.00 —130.0 


34 


93.2 


510 


950 


1130 


2066 


17.50 


3182 


— 80.00 —112.0 


35 


95.0 


520 


968 


1140 


2084 


1760 


3200 


— 70.00 — 94.0 


36 


96.8 


530 


986 


11.50 


2102 


1770 


3218 


— 60.00— 76.0 


37 


98.6 


540 


1004 


1160 


2120 


1780 


3236 


— 50.00 : — 58.0 


38 


100.4 


550 


1022 


1170 


2138 


1790 


3254 


— 40.00 — 40.0 


39 


102.2 


560 


1040 


1180 


2156 


1800 


3272 


— 30.00 — 22.0 


40 


104.0 


570 


1058 


1190 


2174 


1810 


3290 


— 20.00 — 4.0 


41 


105.8 


i 580 


1076 


1200 


2192 


1820 


3308 


— 19.00 — 2.2 


42 


107.6 


590 


1094 


1210 


2210 


1830 


3326 


— 18.00 ; — 0.4 


43 


109.4 


600 


1112 


1220 


2228 1 


1840 


3344 


— 17.77 Zero. 


44 


111.2 


610 


1130 


1230 


2246 , 


18.50 


3362 


— 17.00 + 1.4 


45 


113.0 


620 


1148 


1240 


2264 


1860 


3380 


— 16.00 + 3.2 


46 


114.8 


630 


1166 


1250 


2282 


: 1870 


3398 


— 15.00: + 5.0 


47 


116.6 


640 


1184 


1260 


2300 ; 


i 1880 


3416 


— 14.00! + 6.8 


48 


118.4 


> 650 


1202 


1270 


2318 


1890 


3434 


— 13.00 + 8.6 


49 


120.2 


660 


1220 


1280 


2336 ; 


1900 


3452 


— 12.00| + 10.4 


50 


122.0 


670 


1238 ; 


1290 


2354 


1910 


3470 


— 11.00 + 12.2 


60 


140.0 


i 680 


1256 


1300 


2372 1 


1920 


3488 


— 10.00 + 14.0 


70 


158.0 


690 


1274 


1310 


2390 


1930 


3.506 


— 9.00 4- 15.8 


80 


176.0 


700 


1292 ! 


1320 


2408 


1940 


3524 


— 8.00 + 17.6 


90 


194.0 


710 


1310 


1330 


2426 


1950 


3542 


— 7.00 + 19.4 


100 


212.0 


720 


13-28 


1340 


2444 


1960 


3560 


— 6.00 - 21.2 i 


■ 110 


230.0 


730 


1346 


1350 


2462 i 


1970 


3578 


— 5.00 - 23.0 


120 


248.0 


740 


1364 


1360 


2480 


1980 


3596 


— 4.00 4- 24.8 


130 


266.0 


750 


1382 


1370 


2498 : 


1990 


3614 


— 3.00 4- 26.6 


140 


2S4.0 


760 


1400 


1380 


2516 ; 


2000 


3632 


— 2.00 4- 28.4 


150 


302.0 


770 


1418 


1390 


2534 


2010 


3650 


— 1.00 4- 30.2 


, 160 


320.0 


780 


1436 


1400 


2552 


2020 


3668 


Zero. + 32.0 


170 


338.0 


790 


1454 


1410 


2570 


2030 


3686 


+ 1 


4- 33.8 


1 180 


356.0 


800 


1472 ! 


1420 


2588 


2040 


3704 


2 


35.6 


190 


374.0 


810 


1490 ! 


1430 


2606 


20.50 


3722 


3 


37.4 


200 


392.0 


820 


1508 


1440 


2624 


2060 


3740 


4 


39.2 


210 


410.0 


830 


1526 ! 


1450 


2642 


2070 


3758 


5 


41.0 


220 


428.0 


840 


1544 


1460 


2660 


2080 


3776 


6 


42.8 


230 


446.0 


850 


1562 


1470 


2678 


2090 


3794 


7 


44.6 


240 


464.0 


860 


1580 


1480 


2696 


2100 


3812 


8 


46.4 


250 


482.0 


870 


1598 ! 


1490 


2714 


2110 


3830 


9 


48.2 


260 


500.0 


880 


1616 


1500 


2732 


2120 


3848 


10 


50.0 


270 


518.0 


890 


1634 ! 


1510 


2750 


2130 


3866 


11 


51.8 


280 


536.0 


900 


1652 


1520 


2768 


2140 


3884 


12 


53.6 ! 


290 


554.0 


910 


1670 


1530 


2786 i 


2150 


3902 


13 


55.4 


300 


572.0 


920 


1688 1 


1540 


2804 


2160 


3920 


14 


57.2 


310 


590.0 


930 


1706 . 
1724 ! 


1550 


2822 


2180 


3956 


15 


69.0 


320 


608.0 


940 | 


1560 


2840 j 


2200 


3992 



For Supplementary Tables, see page 485. 



Thermometees. 



485 



SUPPLEMENTARY TABLES. 

El Number of Degrees Cent. = Number of Degrees Fahr. 



■ 
o • 


Tenths of a degree — Centigrade scale. 


£ s 
- 


.0 


.1 


.2 


.3 .4 


.5 .6 


.7 


.8 


.9 




Fahr. 


Fahr. 


Fahr. 


Fahr. Fahr. 


Fahr. 


Fahr. 


Fahr. 


Fahr. 


Fahr. 





.00 


.18 


.36 


..54 .72 


.90 


1.08 


1.26 


1.44 


1.62 


1 


1.80 


1.98 


2.16 


2.34 


2.52 


2.70 


2.88 


3.06 


3.24 


3.42 


2 


3.60 


3.78 


3.96 


4.14 


4.32 


4.50 


4.68 


4.86 


5.04 


5.22 


3 


5.40 


5.58 


5.76 


5.94 


6.12 


6.30 


6.48 


6.66 


6.84 


7.02 


4 


7.20 


7.38 


7.56 


7.74 


7.92 


8.10 


8.28 


8.46 


8.64 


8.82 


5 


9.00 


9.18 


9.36 


9.54 


9.72 


9.90 


10.08 


10.26 


10.44 


10.62 


6 


10.80 


10.98 


11.16 


11.34 


11.52 


11.70 


11.88 


12.06 


12.24 


12.42 


7 


12.60 


12.78 


12.96 


13.14 13.32 


13.50 


13.68 


13.86 


14.04 


14.22 


8 


14.40 


14.58 


14.76 


11.94 


15.12 


15.30 


15.48 


15.66 


15.84 


16.02 


9 


16.20 


16.38 


16.56 


16.74 


16.92 


17.10 


17.28 


17.16 


17.64 


17.82 



Number of Degrees Fahr. = Number of Degrees Cent. 



■ 






Tenths of a 


degree — Fahrenheit scale. 






<s — 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 




Cent. 


Cent. 


Cent. 


Cent. 


Cent. 


Cent. 


Cent. 


Cent. 


Cent. 


Cent. 





.00 


.06 


.11 


.17 


.22 


.28 


.33 


.39 


.44 


.50 


1 


.56 


.61 


.67 


.72 


.78 


.83 


.89 


.94 


1.00 


1.06 


2 


1.11 


1.17 


1.22 


1.28 


1.33 


1.39 


1.44 


1.50 


1.56 


1.61 


3 


1.67 


1.72 


1.78 


1.83 


1.89 


1.94 


2.00 


2.06 


2.11 


2.17 


4 


2.22 


2.28 


2.33 


2.39 


2.44 


2.50 


2.56 


2.61 


2.67 


2.72 


5 


2.78 


2.83 


2.89 


2.94 


3.00 


3.06 


3.11 


3.17 


3.22 


3.28 


6 


3.33 


3.39 


3.44 


3.50 


3.56 


3.61 


3.67 


3.72 


3.78 


3.83 


7 


3.89 


3.94 


4.00 


4.06 


4.11 


4.17 


4.22 


4.28 


4.33 


4.39 


8 


4.44 


4.50 


4.56 


4.61 


4.67 


4.72 


4.78 


4.83 


4.89 


4.94 


9 


5.00 


5.06 


5.11 


5.17 


5.22 


5.28 


5.33 


5.39 


5.44 


5.50 



By the use of the above tables any value may be obtained in connection 
with the preceding tables. Thus, to convert 1375.4° C. to Fahrenheit we 
have 

1370.0° C. = 2498.00° F. 
5.0° C. = 9.00° F. 
0.4° C. = 0.72° F. 

1375.4° C. = 2-507.72° F. 



486 



Expansion. 



Coefficients of Expansion. 

Per Degree of Fahrenheit Scale. 



Solids. 



VGlass 

> Wrought-iron 

Soft iron 

Cast-iron 

Cast-steel 

Hardened steel 

| Copper 

Lead 

Gold, pure 

Gold, hammered 

Silver, pure 

Silver, hammered 

Brass, common cast. . . . 
Brass, wire or sheet — 

j- Platinum, pure 

Platinum, hammered . . 

Palladium 

Roman cement 

Zinc, pure or cast 

Zinc, hammered 

Tin, cast 

Tin, hammered 

Fire-brick 

Good red brick 

Marble 

Granite 

Bismuth 

Antimony 

^Mercury 

I Water 

Salt, dissolved 

Sulphuric acid 

Turpentine and ether. . 

Oil, common 

Alcohol and nitric acid 
All permanent gases . . . 



Linear. 



.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 



00478 
00546 
00660 
00656 
00895 
00680 
00618 
00600 
00689 
00955 
01092 
01580 
00815 
00830 
01060 
01116 
01043 
01075 
00491 
00520 
00530 
00555 
00797 
01633 
01722 
01207 
01500 
00235 
00305 
00613 
00438 
00773 
00602 
03333 
03416 
03500 
08806 
17066 
18904 
09250 
11111 
12966 
14814 
15151 
69416 



Surface. 



.000 00956 
.000 01093 
.000 01320 
.000 01312 
.000 01790 
.000 01360 
.000 01236 
.000 01200 
.000 01378 
.000 01910 
.000 02184 
.000 03160 
.000 01630 
.000 01660 
.000 02120 
.000 02232 
.000 02086 
.000 02150 
.000 00982 
.000 01040 
.000 01060 
.000 OHIO 
.000 01594 
.000 03266 
.000 03444 
.000 02414 
.000 03000 
.000 00470 
.000 00610 
.000 01226 
.000 00876 
.000 01546 
.000 01204 
.000 06666 
.000 06833 
.000 07000 
.000 17612 
.000 34133 
.000 37808 
.000 18500 
.000 22222 
.000 25933 
.000 29629 
.000 30302 
.001 38832 



Volume. 



.000 01434 
.000 01639 
.000 01980 
.000 01968 
.000 02686 
.000 02040 
.000 01854 
.000 01800 
.000 02067 
.000 02865 
.000 03276 
.000 04740 
.000 02445 
.000 02490 
.000 03180 
.000 03348 
.000 03129 
.000 03225 
.000 01473 
.000 01560 
.000 01590 
.000 01665 
.000 02391 
.000 04899 
.000 05166 
.000 03621 
.000 04500 
.000 00705 
.000 00915 
.000 01839 
.000 01314 
.000 02319 
.000 01806 
.000 10000 
.000 10250 
.000 10500 
.000 26420 
.00051198 
.000 56713 
.000 27750 
.000 33333 
.000 38900 
.000 44443 
.000 45453 
.002 08250 



According to the investigations of M. Guillaume, an alloy of nickel- 
steel, containing 36 per cent, of nickel, has a coefficient of expansion only « 
A that of platinum, or about 0.0000003 for 1° F. Wires made of this alloy 
nave been used for the measurement of geodetic base lines, without 
requiring any temperature correction. 



Expansion. 



487 



Coefficients of Expansion. 

Per Degree of the Centigrade Scale. 



Substance. 



Aluminum 

Brass, cast 

Brass wire 

Bronze 

Carbon, gas 

Carbon, graphite 

Copper 

German silver . . . 

Gold... 

Glass, crown 

Glass, flint 

Iron, cast 

Iron, wrought. .. 

Steel, hard 

Steel, soft 

Lead 

Nickel 

Platinum 

Silver 

Tin 

Zinc 



Linear. 



.000 0231 
.000 0187 
.000 0193 
.000 0184 
.000 0054 
.000 0077 
.000 0168 
.000 0184 
.000 0144 
.000 0090 
.000 0079 
.000 0106 
.000 0114 
.000 0132 
.000 0109 
.000 0292 
.000 0128 
.000 0090 
.000 0192 
.000 0223 
.000 0292 



Surface. 



.000 0462 
.000 0374 
.000 0386 
.000 0368 
.000 0108 
.000 0154 
.000 0336 
.000 0368 
.000 0288 
.000 0180 
.000 0158 
.000 0212 
.000 0228 
.000 0264 
.000 0218 
.000 0584 
.000 0256 
.000 0180 
.000 0384 
.000 0446 
.000 0584 



Volume. 



.000 0693 
.000 0561 
.000 0579 
.000 0552 
.000 0162 
.000 0231 
.000 0504 
.000 0552 
.000 0432 
.000 0270 
.000 0237 
.000 0318 
.000 0342 
.000 0396 
.000 0327 
.000 0876 
.000 0384 
.000 0270 
.000 0576 
.000 0669 
.000 0876 



488 



Expansion. 



Linear Expansion or Contraction, in Inches, of Cast=iron. 

Lengths in Feet. 





Difference in temperature. — Fahrenheit. 


Leni 


100° 


150° 


200° 


250° 


300° 


400° 


500° 


600° 


800° 


Feet. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


1 


.0072 


.0110 


.0150 


.0192 


.0237 


.0336 


.0444 


.0561 


.0787 


2 


.0144 


.0220 


.0300 


.0384 


.0474 


.0632 


.0885 


.1123 


.1574 


3 


.0216 


.0330 


.0450 


.0576 


.0711 


.1008 


.1332 


.1684 


.2361 


4 


.0288 


.0440 


.0600 


.0768 


.0948 


.1344 


.1776 


.2246 


.3148 


5 


.0360 


.0550 


.0750 


.0960 


.1185 


.1680 


.2220 


.2805 


.3935 


6 


.0432 


.0660 


.0900 


.1152 


.1422 


.2016 


.2664 


.3368 


.4722 


7 


.0504 


.0770 


.1050 


.1344 


.1659 


.2352 


.3108 


.3929 


.5509 


8 


.0576 


.0880 


.1200 


.1536 


.1896 


.2688 


.3552 


.4496 


.6396 


9 


.0648 


.0990 


.1350 


.1728 


.2133 


.3024 


.3996 


.5052 


.7083 


10 


.0720 


.1102 


.1502 


.1926 


.2376 


.3360 


.4440 


.5616 


.7872 


11 


.0792 


.1214 


.1652 


.2125 


.2615 


.3696 


.4884 


.6177 


.8659 


12 


.0864 


.1316 


.1802 


.2318 


.2853 


.4032 


.5328 


.6739 


.9446 


13 


.0936 


.1417 


.1952 


.2510 


.3090 


.4368 


.5772 


.7300 


1.0233 


14 


.1008 


.1519 


.2102 


.2703 


.3328 


.4704 


.6216 


.7862 


1.1020 


15 


.1080 


.1620 


.2253 


.2895 


.3565 


.5040 


.6660 


.8423 


1.1808 


16 


.1152 


.1722 


.2403 


.3088 


.3803 


.5376 


.7104 


.8985 


1.2595 


17 


.1224 


.1823 


.2553 


.3280 


.4040 


.5712 


.7548 


.9546 


1.3382 


18 


.1296 


.1925 


.2703 


.3472 


.4278 


.6048 


.7992 


1.0108 


1.4169 


19 


.1368 


.2026 


.2853 


.3665 


.4515 


.6384 


.8436 


1.0669 


1.4956 


20 


.1440 


.2203 


.3005 


.3852 


.4752 


.6720 


.8880 


1.1232 


1.5744 


21 


.1512 


.2305 


.3155 


.4045 


.4995 


.7056 


.9324 


1.1793 


1.6531 


22 


.1584 


.2407 


.3305 


.4238 


.5228 


.7392 


.9768 


1.2394 


1.7318 


23 


.1656 


.2508 


.3455 


.4430 


.5465 


.7728 


1.0212 


1.2915 


1.8105 


24 


.1728 


.2610 


.3606 


.4623 


.5703 


.8064 


1.0656 


1.3477 


1.8892 


25 


.1800 


.2711 


.3756 


.4815 


.5940 


.8400 


1.1100 


1.4038 


1.9679 


26 


.1872 


.2813 


.3906 


.5008 


.6179 


.8736 


1.1544 


1.4600 


2.0467 


27 


.1944 


.2914 


.4056 


.5200 


.6415 


.9072 


1.1988 


1.5161 


2.1254 


28 


.2016 


.3016 


.4206 


.5393 


.6553 


.9408 


1.2432 


1.5723 


2.2041 


29 


.2088 


.3117 


.4356 


.5585 


.6890 


.9744 


1.2876 


1.6284 


2.2829 


30 


.2160 


.3304 


.4507 


.5778 


.7128 


1.0080 


1.3320 


1.6848 


2.3616 


31 


.2232 


.3405 


.4657 


.5970 


.7365 


1.0416 


1.3764 


1.7409 


2.4403 


32 


.2304 


.3507 


.4807 


.6163 


.7603 


1.0752 


1.4208 


1.7971 


2.5190 


33 


.2376 


.3608 


.4957 


.6355 


.7841 


1.1088 


1.4652 


1.8533 


2.5977 


34 


.2448 


.3710 


.5107 


.6548 


.8078 


1.1424 


1.5096 


1.9094 


2.6764 


35 


.2520 


.3811 


.5258 


.6740 


.8316 


1.1760 


1.5540 


1.9656 


2.7552 


36 


.2592 


.3913 


.5408 


.6933 


.8553 


1.2096 


1.5984 


2.0217 


2.8339 


37 


.2664 


.4014 


.5558 


.7125 


.8791 


1.2432 


1.6428 


2.0779 


2.9126 


38 


.2736 


.4116 


.5708 


.7298 


.9028 


1.2768 


1.6872 


2.1340 


2.9913 


39 


.2808 


.4217 


.5858 


.7490 


.9266 


1.3104 


1.7316 


2.1902 


3.0701 


40 


.288 


.4406 


.6009 


.7704 


.9501 


1.344 


1.776 


2.2464 


3.1488 


45 


.324 


.4957 


.6760 


.8667 


1.0692 


1.512 


1.998 


2.5272 


3.5424 


50 


.360 


.5508 


.7512 


.9630 


L.1880 


1.680 


2.220 


2.8080 


3.9360 


55 


.396 


.(5059 


.8263 


1.0593 


1.3068 


1.848 


2.442 


3.0888 


4.3296 


60 


.432 


.6610 


.9014 


1.1556 


1.4256 


2.016 


2.664 


3.3696 


4.7132 


65 


.468 


.7150 


.9765 


1.2519 


1.5444 


2.184 


2.886 


3.6540 


5.1068 


70 


.504 


.7711 


1.0517 


1.3482 


1.6632 


2.352 


3.108 


3.9312 


5.5104 


75 


.540 


.8262 


1.1268 


1.4445 


1.7S20 


2.520 


3.330 


4.2120 


5.9040 


80 


.576 


.8813 


1.2019 


■1.5408 


1.9008 


2.688 


3.552 


4.4948 


6.2976 


85 


.612 


.9364 


1.2770 


1.6371 


2.0196 


2.856 


3.774 


4.7756 


6.6912 


90 


.648 


.9914 


1.3521 


1.7334 


2.1384 


3.024 


3.996 


5.0544 


7.0848 


95 


.684 


1.0465 


1.4272 


1.8297 


2.2572 


3.192 


4.218 


5.3352 


7.4784 


100 


.720 


L.1016 


1.5024 


1 .9200 


2.3760 


3.360 


4.440 


5.6160 


7.8720 


.000 006 


612 


626 


642 


660 


700 


740 


780 


820 



Expansion per degree. — Fahrenheit. 



Multiply by 1.1, for wrought-irou ; 1.5, for copper ; 1.6, for brass ; 2.6, for zinc. 



FUSING-POINTS. 



489 



For the determination of temperatures above the boiling-point of mer- 
3ury various forms of pyrometers are used. The most reliable for practical 
iise is the thermo-electric pyrometer of Le Chatelier, composed of a thermo- 
electric couple, one member of which is of pure platinum and the other of 
£, platinum alloyed with 10 per cent, of rhodium. For a full description of 
this and other methods of measurement of high temperatures reference 
may be made to the work entitled "High Temperature Measurements," by 
Le Chatelier and Boudouard, and translated by George K. Burgess.* 

The following table gives the melting-points of the elements used in 
engineering. 

Fusing=points. 



Substance. 



Aluminum 

Antimony 

Copper 

Gold 

Iron, pure 

Iron, white pig 
Iron, gray pig . 

Steel 

Lead 

Manganese — 

Nickel 

Platinum 

Silver 

Tin 

Zinc 

Brass 

Bronze.. 



Fahr. 



Cent. 



Degrees. 
1213 

815 
1949 
1947 
2975 
1967 
2192. 
2507 

621 
3452 
2732 
3452 
1751 

446 

787 
1859 
1652 



Degrees. 

657 

435 

1065 

1064 

1635 

1075 

1200 

1375 

327 

1900 

1500 

1900 

955 

230 

419 

1015 

900 



Substance. 



Degrees. Degrees. 



Glass : 

Glass, lead free... 
Delta metal 

Fusible Metals. 

3 Tin 

5 Lead 

8 Bismuth 

4 Tin 

4 Lead 

1 Bismuth 

3 Tin | 

2 Lead J 

1 Tin ■> 

1 Lead ) 

1 Tin | 

2 Lead J 



Fahr. 



Cent. 



1832 
2192 
1742 



212 

263 

275 
304 
361 



1000 

1200 

950 



100 

128 

135 
151 

183 



Expansion of Gases. 

All perfect gases, so called, expand and contract alike under the action 
of heat. That is to say, every substance, when in the gaseous state, and 
not near its point of liquefaction, has the same coefficient of expansion, 

this coefficient being — — of its volume, or 0.003665 for each degree Centi- 
grade, or -r^rj- part = 0.002035 for each degree Fahrenheit. 

Since a gas contracts -^— part of its volume when its temperature is 

lowered 1° C, such a rate of contraction would theoretically reduce its 
volume to zero at a temperature of —273° C. = — 459.4° F. Since all gases 
reach their liquefying point before this low temperature is attained, how- 
ever, no such contraction exists. At the same time, it may be said that if 
heat is considered as a motion of the molecules of a substance, that motion 
is to be considered as having ceased when the temperature has reached 
—273° C. 

This temperature of —273° C. = — 459.4° F. is, therefore, called the abso- 
lute zero, and from it all temperatures should properly be reckoned. When- 



* New York, John Wiley & Sons. 



490 Heat Units. 



ever a temperature is mentioned as being in degrees absolute, either in the 
Centigrade or the Fahrenheit scale, it is understood to be counted from the 
absolute zero, and therefore is equal to the observed temperature plus 273 
or 459.4, as the case may be. 

The lowest temperature which has thus far been attained is that pro-* 
duced by the evaporation of liquid hydrogen by Dewar, = —252° C. 



Heat Units. 

In expressing quantities of heat the temperature alone is not sufficient, 
since the substance in which the change of temperature is produced must 
be considered. The substance chosen as a standard is pure water, at or 
near its point of greatest density. 

Two heat units are in general use. 

The British Thermal Unit, abbreviated B. T. U., is the quantity of 
heat required to raise the temperature of 1 pound of water 1° F., at or near 
the temperature of 39.1°. 

The limitation of the part of the scale, at or near which the measure- 
ment should be made, need be considered only for very precise physical 
work, since the variation in the quantity of heat corresponding to an in- 
terval of one degree in a given weight of water varies but slightly for 
different parts of the scale. 

In the metric system the kilogramme of water is taken, and the degree 
on the Centigrade scale. The unit is the Calorie, being the quantity of 
heat required to raise 1 kilogramme of water 1° C., at or near the tem- 
perature of 4° C. In French the calorie is sometimes abbreviated Cal., 
and in German it is written W. E. (Warme Einheit). 

1 B. T. U = 0.252 calorie. 
1 calorie = 3.968 B. T. U. 

When the effect of the application of a given number of British thermal 
units or calories upon a given weight of any substance is under considera- 
tion, care must be taken to take into account the relation of the weights 
in making the conversion, or errors may be made. Thus, 1 calorie per kilo- 
gramme is only 1.8 times greater than 1 British thermal unit per pound, 
since the calorie is considered in connection with a weight equal to 2.2 
pounds, and, conversely, 1 British thermal unit per pound is equal to 0.555 
calories per kilogramme. 

The following tables will be found convenient for transforming quanti- 
ties in one kind of heat units to another. 



Heat Units. 



491 



Conversion of British Thermal Units into Calories. 



B. T. TJ. 


Calories. 


B. T. TJ. 


Calories. 


B. T. U. 


Calories. 


1 


.252 


34 


8.568 


67 


16.884 


2 


.504 


35 


8.820 


68 


17.136 


3 


.756 


36 


9.072 


69 


17.388 


4 


1.008 


37 


9.324 


70 


17.640 


5 


• 1.260 


38 


9.576 


71 


17.892 


6 


1.512 


39 


9.828 


72 


18.144 


7 


1.764 


40 


10.080 


73 


18.396 


8 


2.016 


41 


10.332 


74 


18.648 


9 


2.268 


42 


10.584 


75 


18.900 


10 


2.520 


43 


10.836 


76 


19.152 


11 


2.772 


44 


11.088 


77 


19.404 


12 


3.024 


45 


11.340 


78 


19.656 


13 


3.276 


46 


11.592 


79 


19.908 


11 


3.528 


47 


11.844 


80 


20.160 


15 


3.780 


48 


12.096 


81 


20.412 


16 


4.032 


49 


12.348 


82 


20.664 


17 


4.284 


50 


12.600 


83 


20.916 


18 


4.536 


51 


12.852 


84 


21.168 


19 


4.788 


52 


13.104 


85 


21.420 


20 


5.040 


53 


13.356 


86 


21.672 


21 


5.292 


54 


13.608 


87 


21.924 


22 


5.544 


55 


13.860 


88 


22.176 


23 


5.796 


56 


14.112 


89 


22.428 


24 


6.048 


57 


14.364 


90 


22.680 


25 


6.300 


58 


14.616 


91 


22.932 


26 


6.552 


59 


14.868 


92 


23.184 


27 


6.804 


60 


15.120 


93 


23.436 


28 


7.056 


61 


15.372 


94 


23.688 


29 


7.308 


62 


15.624 


95 


23.940 


30 


7.560 


63 


15.876 


96 


24.192 


31 


7.812 


64 


16.128 


97 


24.444 


32 


8.064 


65 


16.380 


98 


24.696 


33 


8.316 


66 


16.632 


99 


24.948 




. 











492 



Heat Units. 



Conversion of Calories into British Thermal Units. 



Calories. 


B. T. U. 


Calories. 


B. T. U. 


Calories. 


B. T.U. 


1 


3.97 


34 


134.92 


67 


265.88 


2 


7.94 


35 


138.89 


68 


269.85 


3 


11.90 


36 


142.86 


69 


273.81 


4 


15.87 


37 


146.83 


70 


277.78 


5 


19.84 


38 


150.80 


71 ■ 


281.75 


6 


23.81 


39 


154.76 


72 


285.72 


7 


27.78 


40 


158.73 


73 


289.69 


8 


31.75 


41 


162.70 


74 


293.66 


9 


35.71 


42 


166.67 


75 


297.62 


10 


39.68 


43 


170.64 


76 


301.59 


11 


43.65 


44 


174.61 


77 


305.56 


12 


47.62 


45 


178.57 


78 


309.53 


13 


51.59 


46 


182.54 


79 


313.50 


14 


55.56 


47 


186.51 


80 


317.47 


15 


59.52 


48 


190.48 


81 


321.43 


16 


63.49 


49 


194.45 


82 


325.40 


17 


67.46 


50 


198.42 


83 


329.37 


18 


71.43 


51 


202.38 


84 


333.34 


19 


75.40 


52 


206.35 


85 


337.31 


20 


79.37 


53 


210.32 


86 


341.28 


21 


83.33 


54 


214.29 


87 


345.24 


22 


87.30 


55 


218.26 


88 


349.21 


23 


91.27 


56 


222.23 


89 


353.18 


24 


95.24 


57 


226.19 


90 


357.15 


25 


99.21 


58 


230.16 


91 


361.12 


26 


103.18 


59 


234.13 


92 


365.09 


27 


107.14 


60 


238.10 


93 


369.05 


28 


111.11 


61 


242.07 


94 


373.02 


29 


115.08 


62 


246.04 


95 


376.99 


30 


119.05 


63 


250.00 


96 


380.96* 


31 


123.02 


64 


253.97 


97 


384.93 


32 


126.99 


65 


257.94 


98 


388.90 


33 


130.95 


66 


261.91 


99 


392.86 



Heat Units. 



493 



Conversion of Calories into Foot=pounds. 



Calories. 


Foot-pounds. 


Calories. 


Foot-pounds. 


Calories. 


Foot-pounds. 


1 


3 091 


34 


105 106 


67 


207 121 


2 


6183 


35 


108 198 


68 


210 212 


3 


9 274 


36 


111 289 


69 


213 304 


4 


12 365 


37 


114 380 


70 


216 395 


5 


15 457 


38 


117 472 


71 


219 487 


6 


18 548 


39 


120 563 


72 


222 578 


7 


21640 


40 


123 654 


73 


225 669 


8 


24 731 


41 


126 746 


74 


228 761 


9 


27 822 


42 


129 837 


75 


231 852 


10 


30 914 


43 


132 928 


76 


234 943 


11 


34 005 


44 


136 020 


77 


238 035 


12 


37 096 


45 


139 111 


78 


241 126 


13 


40188 


46 


142 203 


79 


244 217 


14 


43 279 


47 


145 294 


80 


247 309 


15 


46 370 


48 


148 387 


81 


250 400 


16 


49 462 


49 


151 477 


82 


253 492 


17 


52 553 


50 


154 568 


83 


256 583 


18 


55 644 


51 


157 659 


84 


259 674 


19 


58 736 


52 


160 751 


85 


262 766 


20 


61827 


53 


163 824 


86 


265 857 


21 


64 919 


54 


166 933 


87 


268 948 


22 


68 010 


55 


170 025 


88 


272 040 


23 


71101 


56 


173 116 


89 


275 131 


24 


74 193 


57 


176 208 


90 


278 222 


25 


77 284 


58 


179 299 


91 


281 314 


26 


80 375 


59 


182 390 


92 


284 405 


27 


83 467 


60 


185 482 


93 


287 496 


28 


86 558 


61 


188 573 


94 


290 588 


29 


89 649 


62 


191 664 


95 


293 679 


30 


92 741 


63 


194 756 


96 


296 771 


31 


95 835 


64 


197 847 


97 


299 862 


32 


98 924 


65 


200 938 


98 


302 953 


33 


102 015 


66 


204 030 


99 


306 045 















494 



Heat Units. 



Conversion of Foot=pounds into Calories. 



Foot- 
pounds. 


Calories. 


Foot- 
pounds. 


Calories. 


Foot- 
pounds. 


Calories. 


1 


.000 323 


34 


.010 998 


67 


.021 676 


2 


.000 647 


35 


.011 322 


68 


.021 997 


3 


.000 970 


36 


.011 645 


69 


.022 320 


4 


.001 294 


37 


.011 969 


70 


.022 644 


5 


.001 617 


38 


.012 292 


71 


.022 967 


6 


.001 941 


39 


.012 616 


72 


.023 291 


7 


.002 264 


40 


.012 939 


73 


.023 614 


8 


.002 588 


41 


.013 263 


74 


.023 938 


9 


.002 911 


42 


.013 586 


75 


.024 261 


10 


.003 235 


43 


.013 910 


76 


.024 584 


11 


.003 558 


44 


.014 233 


77 


.024 908 


12 


.003 882 


45 


.014 557 


78 


.025 231 


13 


.004 205 


46 


.014 880 


79 


.025 555 


14 


.004 529 


47 


.015 204 


80 


.025 878 


15 


.004 852 


48 


.015 527 


81 


.026 202 


16 


.005 176 


49 


.015 851 


82 


.026 525 


17 


.005 499 


50 


.016 174 


83 


.026 849 


18 


.005 823 


51 


.016 497 


84 


.027 172 


19 


.006 146 


52 


.016 821 


85 


.027 496 


20 


.006 470 


53 


.017 144 


86 


.027 819 


21 


.006 793 


54 


.017 468 


87 


.028 143 


22 


.007 117 


55 


.017 791 


88 


.028 466 


23 


.007 440 


56 


.018 115 


89 


.028 790 


24 


.007 764 


57 


.018 438 


90 


.029 113 


25 


.008 087 


58 


.018 762 


91 


.029 437 


26 


.008 410 


59 


.019 085 


92 


.029 760 


27 


.008 734 


60 


.019 409 


93 


.030 084 


28 


.009 057 


61 


.019 732 


94 


.030 407 


29 


.009 381 


62 


.020 056 


95 


.030 731 


30 


.009 704 


63 


.020 379 


96 


.031 054 


31 


.010 028 


64 


.020 703 


97 


.031 378 


32 


.010 351 


65 


.021 026 


98 


.031 701 
.032 025 * 


33 


.010 675 


66 


.021 350 


99 















Specific Heat. 



495 



Mechanical Equivalent of Heat. 

In the conversion of heat into mechanical energy there is always a defi- 
nite amount of work produced for a definite quantity of heat. One British 
•"thermal unit is equal to 778 foot-pounds, and one calorie is equal to 428 
kilogrammetres. The maximum amount of energy which can be obtained 
for any given number of heat units is, therefore, found by multiplying by 
778. Conversely, 1 foot-pound = 7 £ F = 0.001285 heat unit. 

Specific Heat. 

We have seen that heat requires for its determination the production of 
a determinate change in temperature of a definite weight of a given sub- 
stance. For the purpose of establishing a unit, water has been chosen as 
the standard substance. The quantity of heat required to raise the tem- 
perature of other substances is different from that required for water. 
The ratio of the quantity of heat required for any substance by that re- 
quired for water is called the Specific Heat of the substance. Thus, it is 
found that it takes only about one-ninth as much heat to raise the tem- 
perature of a pound of iron one degree that it does to raise a pound of 
water; hence, the specific heat of iron is one-ninth, or, more precisely, 
= 0.1138. 

The methods of measuring specific heats vary according to the charac- 
ter of the substances. For metals the most convenient is the method of 
mixtures, in which a known weight of the metal is raised to a definite 
temperature and then plunged into a given weight of water at a known 
temperature. The rise in temperature of the water gives the number of 
heat units which have been imparted to it, and these have obviously been 
derived from the metal which has been cooled. We then have, if 

x "= specific heat of metal required ; 

T= fall in temperature of metal; 

t = rise in temperature of water ; 
W = weight of metal ; 
w = weight of water ; 

_ wt 
X ~~Wt' 

The specific heats of various substances are not constant, but gradually 
increase with the temperature. The following table gives the mean be- 
tween 10° C. and 100° C, the usual working temperatures. Fuller tables 
for various ranges of temperatures are to be found in the Smithsonian 
Physical Tables. 



Table of Specific Heats. 

Solids (mean specific heat between 10° C. and 100° C). 



Copper 0951 

Silver 0570 

Iron 1138 

Zinc 0955 

Tin 0562 

Lead 0314 

Gold 0324 

Platinum 0324 

Bismuth 0308 



Antimony 0508 

Brass 0939 

Magnesium 2499 

Aluminum 2143 

Glass 1877 

Ice 5040 

Sulphur 1777 

Graphite 2008 

Diamond 1469 



Liquids. 

Mercury 0333 I Alcohol 

Sulphuric acid 3430 Oil of turpentine . 

Ether 5030 Acetic acid 



.615 
.462 
.659 



496 



Latent Heat. 



Gases (at constant pressure). 



Air 2374 

Hydrogen 3.4090 

Oxygen 2175 

Chlorine 1210 



Nitrogen 2438 

Carbonic anhydride 2163 

Carbonic oxide 2450 a 

Steam 4805 



The above specific heats represent the quantity of heat, in British ther- 
mal units, required to raise the temperature of 1 pound of the substance 
1° F., or, in calories, required to raise the temperature of 1 kilogramme of 
the substance 1° C. 



Latent Heat. 

The phenomena of expansion follow the simple rule of direct relation 
to the temperature only when the substance does not suffer any change of 
state. Thus, there are determinate coefficients for solids, for liquids, and 
for gases. When, however, a substance under observation passes from the 
solid to the liquid state, and from the liquid to the gaseous state, certain 
amounts of heat are absorbed which do not raise the temperature, this 
heat being expended in molecular work, separating the molecules of the 
substance. The heat thus absorbed is said to be rendered latent— every sub- 
stance having a latent heat of fusion, required to convert it from a solid to a 
liquid, and another latent heat of vaporization. 

Thus, a pound of ice may be heated and its temperature will rise until 
the melting-point, 32° F. or 0° C, has been reached, when further applica- 
tion of heat, however intense, will cause no further rise in temperature 
until the ice has been entirely melted. Experiments have shown that 
142.6 British thermal units are required to convert a pound of ice at 32° 
into a pound of water at 32°, and hence the latent heat of fusion of water 
is said to be 142.6°. Further application of heat causes a rise in tempera- 
ture directly proportional to the quantity of heat supplied, 180 thermal 
units raising it to the boiling-point, 212° F. Here, again, the rise in tem- 
perature ceases until all the pound of water at 212° has been converted 
into steam at 212°. This operation requires 966 heat units, so that the 
latent heat of vaporization of water is 966°. 

The following table gives the latent heats of various substances. 



Substance. 



B. T. U. per pound. 



Fusion. 



Vaporiza- 
tion. 



Calories per kilogramme. 



Vaporiza- 
Fusion. tion 



Water 

Alcohol, ethyl 

Alcohol, methyl 

Ammonia 

Bisulphide of carbon. 

Sulphur dioxide 

Turpentine 

Iron, gray 

Iron, white 

Lead 

Mercury 

Silver ' 

Zinc 



142.60 



41.40 
59.40 
10.55 
5.08 
37.92 
50.63 



966.6 
371.0 
481.0 
529.0 
162.0 
164.0 
133.0 



79.24 



23.00 
33.00 
5.86 
2.82 
21.07 
28.13 



537 
205 
267 
294 
90 
91 
74 



Radiation. 



497 



Coefficients for Heat Transmission. 



Substance. 



Aluminum . . . 

Antimony 

Brass, yellow, 

Brass, red 

Copper 

German silver 
Iron 



Metric. 


British. 


.00036 


.00203 


.00004 


.00022 


.00025 


.00142 


.00028 


.00157 


.00072 


.00404 


.00009 


.00050 


.00016 


.00089 



Substance. 



Lead 

Mercury . . , 
Steel, hard 
Steel, soft . 

Silver 

Tin 

Zinc 



Metric. 



.00008 
.00002 
.00006 
.00011 
.00109 
.00015 
.00030 



British. 



.00045 
.00011 
.00034 
.00062 
.00610 
.00084 
.00170 



In the above table the metric coefficients give the quantity of heat, in 
calories, transmitted per second through a plate 1 centimetre thick, per 
square centimetre of surface, for a difference of 1° C, at a temperature of 
100° C. 

The British coefficients give the quantity of heat transmitted, in British 
thermal units, per second through a plate I inch thick, per square inch of 
surface, for a difference of 1° F., at a temperature of 212° F. 

The coefficients vary somewhat with the temperature, but the above 
will serve in practice. 

Radiation. 

For moderate differences in temperature the loss of heat by radiation 
may be taken as dependent upon the character of the surface, the area, 
and the difference in temperature. 

The coefficients of radiation, as determined by Peclet, give the number 
of heat units emitted per hour, per square foot of surface, for 1° F., or the 
number of calories emitted per hour, per square metre, per 1° C, as below : 



Coefficients of Radiation. 



Surface. 


B.T. U.,perl°F.,per 
square foot, per hour. 


Calories, per 1° C, per 
square metre, per hour. 


Silver, polished 


.02657 
.03270 
.04395 
.08585 
.0920 
.5662 
.5948 
.6480 
.6868 
.7215 
.7400 
1.0853 
1.4800 


.13 


Copper, polished 


.16 


Tin, polished 


.22 


Tinned iron, polished 

Iron, sheet-, polished 


.42 
.45 


Iron, ordinary « 


2.77 


Glass 


2.91 


Cast-iron, new 


3.17 


Cast-iron, rusted 


3.36 


Sawdust .' 


3.53 


Sand, fine 


3 62 


Water 


5.31 


Oil 


7.24 







The number of heat units radiated from any surface per hour may, 
therefore, be computed by multiplying the area by the difference in tem- 
perature between the hot surface and the surrounding air, and by the co- 
efficient corresponding to the character of the surface. It will be seen 
from the table that ordinary cast-iron is about six times as good a radiating 

32 



498 



Radiation. 



surface as polished sheet-iron, and about twenty-five times as effective as 
polished silver. 

The coefficients in the preceding table are sufficiently correct for use 
when the difference in temperature is not great. When, however, there is 
a considerable difference in the temperature of the heated body and the ' 
surrounding air, the i ate of cooling becomes more rapid. The following 
tables give the ratio of increase in the rate of cooling for larger differences 
in temperature. 

Ratio of Increase in Radiation for Temperatures from 
10° F. to 450° F. 

Temperatures of Air = 70° F. 



Difference in 




Difference in 




Difference in 




temperature, 
Fahr. 


Ratio. 


temperature, 
Fahr. 


Ratio. 


temperature, 
Fahr. 


Ratio. 


Degrees. 




Degrees. 




Degrees. 




10 


1.15 


160 


1.61 


310 


2.34 


20 


1.18 


170 


1.65 


320 


2.40 


30 


1.20 


180 


1.68 


330 


2.47 


40 


1.23 


190 


1.73 


340 


2.54 


50 


1.25 


200 


1.78 


350 


2.60 


60 


1.27 


210 


1.82 


360 


2.68 


70 


1.32 


220 


1.86 


370 


2.77 


80 


1.35 


230 


1.90 


380 


2.84 


90 


1.38 


240 


1.95 


390 


2.93 


100 


1.40 


250 


2.00 


400 


3.02 


110 


1.44 


260 


2.05 


410 


3.10 


120 


1.47 


270 


2.10 


420 


3.20 


130 


1.50 


280 


2.16 


430 


3.30 


140 


1.54 


290 


2.21 


440 


3.40 


150 


1.57 


300 


2.27 


450 


3.50 



Ratio of Increase in Radiation for Temperatures from 
10° C. to 240° C. 

Temperatures of Air = 20° C. 



Difference in 




Difference in 




Difference in 




temperature, 
Cent. 


Ratio. 


temperature, 
Cent. 


Ratio. 


temperature, 
Cent. 


Ratio. 


Degrees. 




Degrees. 




Degrees. 




10 


1.16 


90 


1.60 


170 


2.31 


20 


1.21 


100 


1.68 


180 


2.42 


30 


1.25 


110 


1.75 


190 


2.54 


40 


1.30 


120 


1.83 


200 


2.66 


50 


1.36 


130 


1.90 


210 


2.79 


60 


1.42 


140 


2.00 


220 


2.93 


70 


1.48 


150 


2.09 


230 


3.07 


80 


1.54 


160 


2.20 


240 


3.23 



In computing the number of heat units radiated from a given area and 
material the result should first be calculated by the coefficients of radiation, 
given on page 497, and the value thus obtained, multiplied by the ratio, 



Heat Emission. 



499 



corresponding to the difference in temperature, as given in the preceding 
tables. 

Heating Pipes (Iron). 





Units of heat (B. T. U 


. ) emitted, per square foot, per hour. 


Mean 




Temperature of air = 


. 70° F. 




temperature 

of pipes, 

Fahr. 


By convection. 


By radiation 
alone. 


By convection and radia- 
tion, combined. 














Air, still. 


Air, moving. 




Air, still. 


Air, moviug. 


Degrees. 












80 


5.04 


8.40 


7.43 


12.47 


15.83 


90 


11.84 


19.73 


15.31 


27.15 


35.04 


100 


19.53 


32.55 


23.47 


43.00 


56.02 


110 


27.86 


46.43 


31.93 


57.79 


78.36 


120 


36.66 


61.10 


40.82 


77.48 


101.92 


130 


45.90 


76.50 


50.00 


95.90 


126.50 


140 


55.51 


92.52 


59.63 


115.14 


152.15 


150 


65.45 


109.18 


69.69 


135.14 


178.87 


160 


75.68 


126.13 


80.19 


155.87 


206.32 


170 


86.18 


143.30 


91.12 


177.30 


234.42 


180 


96.93 


161.55 


102.50 


199.43 


264.05 


190 


107.90 


179.83 


114.45 


222.35 


294.28 


200 


119.13 


198.55 


127.00 


246.13 


325.55 


210 


130 49 


217.48 


139.96 


270.49 


357.48 


220 


142.20 


237.00 


155,27 


297.47 


392.27 


230 


153.95 


256.58 


169.56 


323.51 


426.14 


240 


165.90 


279.83 


184.58 


350.48 


464.41 


250 


178.00 


296.66 


200.18 


378.18 


496.84 


260 


189.90 


316.50 


214.36 


404.26 


530.86 


270 


202.70 


337.83 


233.42 


436.12 


571.25 


280 


215.30 


358.85 


251.21- 


466.51 


610.06 


290 


228.55 


380.91 


267.73 


496.28 


648.64 


300 


240.85 


401.41 


279.12 


519.97 


680.53 



Loss of Heat Through Walls. 

Loss, in British Thermal Units, per Square Foot, per Hour, for 1° F. 
Difference. 



Thickness, 
in inches. 


Brick. 


Stone. 


Thickness, 
in inches. 


Brick. 


Stone. 


4 

8 

12 

16 

20 


.273 
.223 
.188 
.163 
.144 


.330 
.312 
.295 
.280 
.267 


24 

28 
32 
36 
40 


.129 
.116 
.106 
.097 
.090 


.255 
.244 
.234 
.224 
.216 



500 



Air. 



AIR. 

Air is composed of a mixture of oxygen and nitrogen, in the propor- 
tion of 21 of oxygen to 79 of nitrogen, by volume; or 23 of oxygen to 77 
of nitrogen, by weight, with an average of 0.04 per cent, of carbonic acid. T 

A cubic metre of dry air, at 0° C. and a pressure of 760 millimetres of 
mercury, weighs 1.29305 kilogrammes. A cubic foot of dry air, at 32° F. 
and a pressure of 29.92 inches of mercury, weighs 0.08072 pound. Above 
its critical temperature of — 140° C. air may be considered as a permanent 
gas, expanding ^3 of its volume for each degree Centigrade increase in 
temperature, and ? £ T of its volume for an increase of 1° Fahrenheit. 

Taking the volume at freezing-point as unity, the weights arid volumes 
at other temperatures are given in the following table. 

Volume and Weight of Dry Air at Different 
Temperatures. 

Under a Constant Atmospheric Pressure of 29.92 Inches of Mercury, the 
Volume at 32° F. being 1. 



Tempera- 
ture, Fahr. 


Volume. 


Weight of a 
cubic foot. 


Tempera- 
ture, Fahr. 


Volume. 


Weight of a 
cubic foot. 


Degrees. 




Lb. 


Degrees. 




Lb. 





.935 


.0864 


500 


1.954 


.0413 


12 


.960 


.0842 


552 


2.056 


.0385 


22 


.980 


.0824 


600 


2.150 


.0376 


32 


1.000 


.0807 


650 


2.260 


.0357 


42 


1.020 


.0791 


700 


2.362 


.0338 


52 


1.041 


.0776 


750 


2.465 


.0328 


62 


1.061 


.0761 


800 


2.566 


.0315 


72 


1.082 


.0747 


850 


2.668 


.0303 


82 


1.102 


.0733 


900 


2.770 


• .0292 


92 


1.122 


.0720 


950 


2.871 


.0281 


102 


1.143 


.0707 


1000 


2.974 


.0268 


112 


1.163 


.0694 


1100 


3.177 


.0254 


122 


1.184 


.0682 


• 1200 


3.381 


.0239 


132 


1.204 


.0671 


1300 


3.584 


.0225 


142 


1.224 


.0659 


1400 


3.788 


.0213 


152 


1.245 


.0649 


1500 


3.993 


.0202 


162 


1.265 


.0638 


1600 


4.196 


.0192 


172 


1.285 


.0628 


1700 


4.402 


.0183 


182 


1.306 


.0618 


1800 


4.605 


.0175 


192 


1.326 


.0609 


1900 


4.808 


.0168 


202 


1.347 


.0600 


2000 


5.012 


.0161 


212 


1.367 


.0591 


2100 


5.217 


.0155 


230 


1.404 


.0575 


2200 


5.420 


.0149 


250 


1.444 


.0559 


2300 


5.625 


.0142 


275 


1.495 


.0540 


2400 


5.827 


.0138 


300 


1.546 


.0522 


2500 


6.032 


.0133 


325 


1.597 


.0506 


2600 


6.236 


.0130 


350 


1.648 


.0490 


2700 


6.440 


.0125 


375 


1.689 


.0477 


2800 


6.644 


.0121 


400 


1.750 


.0461 


2900 


6.847 


.0118 


450 


1.852 


.0436 


3000 


7.051 


.0114 



<2\2.45 



Air. 501 

On the Compression and Expansion of a Definite 
Weight of Air Enclosed in a Vessel. 

In this treatment no heat must be lost or gained by radiation from the 
sides of the vessel in which the air is enclosed. Let D and d represent the 
degrees of absolute temperatures of volumes, v and V, of the air to be 
experimented upon. 

The absolute zero is 461° below Fahr. zero and 273° Cent, below the 
freezing-point of water. D = 461 + T, d = 461 + t, and D — d =- T— t, 
Fahr. scale. 



Volume and Temperature, 

V /P\ 2 - 45 , V Id \2.46 

¥=U) ■ and v=\-d) ' 

Expansion, V = v I —=- ) ; compression, v = ( — J 

2 - 45 /7 2 - 45 /7 

Compression, D = d-J — ; expansion, d =-- D -\ — . 

Example. To what fraction must air of t = 65° be compressed, in order 
to fire tinder at a temperature of T= 550°, d = 461 + 65 = 526°, Z> = 550 + 
461 = 1011°? 

v [ 526 \ 2 - 45 
Formula. -= = ( — — J =0.20, the answer. 

Example. How much must air of T= 80° be expanded to reduce the 
temperature to t = 32°, or freezing-point of water ? 

V /541\ 2 - 45 
Formula. — = -— ) = 1.3308 times, the answer. 
v \493/ 

Example, v = 360 cubic inches of air, of temperature, T = 380° or 
D = 841°, is to be expanded until the temperature becomes t = 80° or d = 
541°. Required the volume, V, corresponding to that temperature ? 

/821\ 2 - 45 
Formula. V = 360 ( — J = 1025.9 cubic feet. 

Example. V = 20 cubic feet of air, of t = 32° or d = 493°, is to be com- 
pressed to v = 12 cubic feet. Required the temperature, t, of compression? 

2 - 45 /20 
Formula. D = 493 -i/— = 607.29° or T 7 = 146.29°. 



Pressure and Temperature. 

P /Z>\3.42 p / £>\3.42 

?=(t) ' and F=U) ' 

/ J) \ 3.42 / d n 3.41 

Compression, P = p ( — J ; expansion, p = Pi — \ 



502 Am. 



3.42/— 3.42 re- 

compression, D = d -*/ — ; expansion, p = D -v/-^-- 



Example. A volume of air, of pressure, p = 15 pounds to the square ' 
inch, and of temperature, t = 62°, is to be compressed until the tempera- 
ture becomes T= 120°. Required the pressure, P, per square inch, at T = 
120°? 

d = 461 + 62 = 523, and D = 461 + 120 + 581. 

/ 581 \ s- 42 
Formula. P = 15 1 — ) = 21.49 pounds per square inch. 

Example. A volume of air, of pressure, P = 45 pounds to the square 
inch, and of temperature, T = 250° or D = 711°, is to be expanded to a 
pressure of p = 25 pounds. Required the temperature, t, of the expanded 
air? 

3.42/— 

Formula. d = 711 -yl^ = 598.72°, and 

t = 598.72 — 461 = 137.72°, the temperature required. 

Pressure and Volume. 

• *lj~y - 29 / .41 / .29/ 

\t- a/f and Vt= Vf 

1,4 /P 1A fo 

Expansion, V=v^—\ compression, v= V \^k- 

I F\ 14 / t> X 1 - 4 

Compression, P = pi — J ; expansion, p = PI — ) . 



Example. A volume, v = 50 cubic inches, and of pressure, P = 80 
pounds per square inch, is to be expanded until the pressure becomes p = 
15 pounds. Required the expanded volume, VI 






80 
Formula. V= 50 -%/— = 165 cubic inches. 

\1d 



Example. What will be the pressure of a volume of air expanded 
1.3308 times ? 

/ l \1.4 

Formula. p = ( ) = 0.5324 of the primitive pressure. 

\ l . 3308 / 



Air. 



503 



In the compression and expansion of air, as given in the following 
table, it is supposed that no heat is transmitted to or from the air operated 
upon. In compression, the temperature of the air rises ; and if the heat is 
allowed to be conducted through the sides of the vessel enclosing the air, 
the pressure will not correspond with the table. In expanding the air the 
temperature is lowered, as seen in the table. The primitive volume is 
assumed to be at 32° F. 



Compression and Expansion of Air. 



Compression of air. 


Expansion of air. 


Volume. 


Temper- 
ature, 
Fahr. 

Degrees. 


Pressure. 


Volume. 
v = 1. 


Temper- 
ature, 
Fahr. 

Degrees. 


Pressure. 


» = 1. 


Atmos- 
phere. 


Pounds per 
square inch. 


Atmos- 
phere. 


Pounds per 
square inch. 


7 


T 


A 


P 


V 


T 


A 


P 


1.000 


32.00 


1.0000 


14.700 


1.00 


+ 32.00 


1.00000 


14.7000 


.950 


42.43 


1.0297 


15.137 


1.10 


+ 13.20 


.87510 


12.8640 


.900 


53.66 


1.159 


17.036 


1.20 


— 3.30 


.77470 


11.3930 


.850 


65.81 


1.255 


18.456 


1.30 


— 18.06 


.69260 


10.1810 


.800 


79.01 


1.366 


20.090 


1.40 


— 31.26 


.62430 


9.1778 


.750 


93.43 


1.496 


21.991 


1.50 


— 39.65 


.58354 


8.5780 


.700 


109.26 


1.647 


24.215 


1.60 


— 54.06 


.5179 


7.6130 


.650 


126.77 


1.828 


26.561 


1.70 


— 64.00 


.4757 


6.9934 


.600 


146.30 


2.044 


30.054 


1.80 


— 73.16 


.4391 


6.4556 


.550 


168.25 


2.309 


33.948 


1.90 


— 82.34 


.4083 


6.0020 


.500 


193.20 


2.639 


38.792 


2.00 


— 89.47 


.3789 


5.5700 


.450 


221.96 


3.058 


44.547 


2.25 


—106.90 


.3213 


4.7235 


.400 


245.70 


3.607 


53.020 


2.50 


—121.83 


.2779 


4.0851 


.350 


295.73 


4.348 


63.917 


2.75 


—134.77 


.2426 


3.5666 


.330 


314.10 


4.721 


69.406 


3.00 


—146.15 


.2148 


3.1576 


.300 


344.87 


5.396 


79.313 


3.25 


—156.27 


.1920 


2.8228 


.250 


407.13 


6.964 


102.38 


3.50 , 


—167.29 


.1731 


2.5446 


.200 


489.91 


9.518 


139.92 


3.75 


—173.57 


.1572 


2.3103 


.150 


606.4 


14.240 


209.31 


4.0 


—181.00 


.1436 


2.1111 


.125 


691.0 


18.380 


270.17 


4.5 


—194.18 


.1218 


1.7900 


.10 


800.9 


25.120 


369.24 


5.0 


—205.40 


.1051 


1.5444 


.05 


1213.5 


66.289 


974.45 


6.0 


—223.74 


.0813 


1.1965 


.04 


1373.2 


90.60 


1331.8 


7.0 


—238.20 


.0656 


.9642 


.03 


1601.7 


135.53 


1992.3 


8.0 


—250.03 


.0544 


.7998 


.02 


1973.0 


239.09 


3514.6 


9.0 


—259.92 


.0461 


.6782 


.01 


4469.0 


794.33 


11676.0 


10.0 


—268.39 


.0355 


.5216 



The above table shows the necessity for taking into account the heat 
produced in air compressors. If the cylinders and valve chests are not 
sufficiently cooled there is danger of explosion from the air carburetted by 
the lubricant. High compression in gas engines is limited by the produc- 
tion of a temperature sufficient to cause a premature ignition of the charge. 
In the Diesel motor the air is compressed to about 30 atmospheres, this 
giving a temperature of about 875° F., supposing no cooling to occur. This 
is ample to ignite the heaviest oil injected into the cylinder. 



504 



Compressed Air. 



Table of Volumes of Air Transmitted, in Cubic Feet, 
per Minute in Pipes of Various Dimensions. 



oil 








Actual diameter of pipe 


, in inches. 










t 


2 


3 


4 


5 


6 


8 


10 


12 


16 


20 


24 


£><C a 


























1 


.33 


1.31 


2.95 


5.2 


8.2 


11.8 


20.9 


32.7 


47.1 


83.8 


131 


188 


2 


.65 


2.62 


5.89 


10.5 


16.4 


23.6 


41.9 


65.4 


94.2 


167.5 


262 


377 


3 


.98 


3.93 


8.84 


15.7 


24.5 


35.3 


62.8 


98.2 


141.4 


251.3 


393 


565 


4 


1.31 


5.24 


11.78 


20.9 


32.7 


47.1 


83.8 


131.0 


188.0 


335.0 


523 


754 


5 


1.64 


6.55 


14.7 


26.2 


41.0 


59.0 


104.0 


163.0 


235.0 


419.0 


654 


942 


6 


1.96 


7.85 


17.7 


31.4 


49.1 


70.7 


125.0 


196.0 


283.0 


502.0 


785 1131 


7 


2.29 


9.16 


20.6 


36.6 


57.2 


82.4 


146.0 


229.0 


330.0 


586.0 


916 1319 


8 


2.62 


10.50 


23.5 


41.9 


65.4 


94.0 


167.0 


262.0 


377.0 


670.0 


1047 1508 


9 


2.95 


11.78 


26.5 


47.0 


73.0 


106.0 


188.0 


294.0 


424.0 


754.0 


1178 1696 


10 


3.27 


13.1 


29.4 


52.0 


82.0 


118.0 


209.0 


327.0 


471.0 


838.0 


1309 1885 


12 


3.93 


15.7 


35.3 


63.0 


99.0 


141.0 


251.0 


393.0 


565.0 


1005.0 


1571 


2262 


15 


4.91 


19.6 


44.2 


78.0 


122.0 


177.0 


314.0 


491.0 


707.0 


1256.0 


1963 


2827 


18 


5.89 


23.5 


53.0 


94.0 


147.0 


212.0 


377.0 


589.0 


848.0 


1508.0 


2356 


3393 


20 


6.55 


26.2 


59.0 


105.0 


164.0 


235.0 


419.0 


654.0 


942.0 


1675.0 


2618 


3770 


24 


7.86 


31.4 


71.0 


125.0 


196.0 


283.0 


502.0 


785.0 


1131.0 


2010.0 


3141 


4524 


25 


8.18 


32.7 


73.0 


131.0 


204.0 


294.0 


523.0 


818.0 


1178.0 


2094.0 


3272 


4712 


28 


9.16 


36.6 


82.0 


146.0 


229.0 


330.0 


586.0 


916.0 


1319.0 


2346.0 


3665 


5278 


30 


9.80 


39.3 


88.0 


157.0 


245.0 


353.0 


628.0 


982.0 


1414.0 


2513.0 


3927 


5655 



Velocity of Escaping Compressed Air. 

(Hiscox.) 



Pressure, in 
atmos- 
pheres. 


Pressure, in 
inches, of 
mercury. 


Pressure, in 
pounds, 
per square 
inch. 


Theoretical 
velocity, 
in feet, 
per second. 


Pressure, in 
atmos- 
pheres. 


Pressure, in 
inches, of 
mercury. 


Pressure, in 
pounds, 
per square 
inch. 


Theoretical 
velocity, 
in feet, 
per second. 


.010 


.30 


.147 


94.4 


.680 


20.40 


10.0 


780 


.066 


2.10 


1.00 


246.0 


.809 


24.28 


12.0 


855 


.100 


3.00 


1.47 


299.0 


1.0 


30.0 


14.7 


946 


.136 


4.08 


2.00 


348.0 


2.0 


60.0 


29.4 


1094 


.204 


6.12 


3.00 


472.0 


5.0 


150.0 


73.5 


1219 


.272 


8.16 


4.00 


493.0 


10.0 


300.0 


147.0 


1275 


.340 


10.20 


5.00 


552.0 


20.0 


600.0 


294.0 


1304 


.408 


12.24 


6.00 


604.0 


40.0 


1200.0 


588.0 


1323 


.500 


15.00 


7.35 


673.0 


100.0 


3000.0 


1470.0 


1331 


.544 


16.32 


8.0 


697.0 


200.0 


6000.0 


2940.0 


1334 


.611 


18.34 


9.0 


741.0 











The theoretical velocities of efflux of compressed air. as given in the 
above table, are to be reduced by multiplying by the coefficient of actual 



Compressed Air. 



505 



discharge, the coefficient varying according to the nature of the orifice 
and the air pressure. 

The following coefficients will serve in practice : 

Coefficients of Air Discharge. 





Pressures, in atmospheres. 




.01 


.1 


.5 


1 


5 


10 


100 


Orifice in thin plate 


.65 
.834 


.64 

.82 


.57 
.71 


.54 
.67 


.45 
.53 


.436 
.51 


.423 


Orifice in short tube 


.487 







Thus, for a pressure of 5 atmospheres, or 73.5 pounds per square inch, 
the theoretical efflux would be 1219 feet per second. The actual efflux 
through a hole in a thin plate would be 

1219 X 0.45 = 548.55 feet per second ; 

and through a short tube, 

1219 X 0.53 = 646.07 feet per second. 

The work required to compress a cubic foot of air to any desired 
pressure may be obtained as follows : 
Let 

p = the initial pressure, usually 14.7 pounds ; 

p — final pressure required ; 

S = initial pressure per square foot (for 14.7 pounds per square 

inch = 2116.8 pounds per square foot) ; 
W== required work of compression, in foot-pounds; 

TF=5hyp. log.-^. 

This is true for isothermal compression only, in which the heat of com- 
pression is removed as rapidly as produced, so that a constant temperature 
is maintained. For adiabatic compression, in which all the heat is re- 
tained, the work is much greater. In actual practice the power is about 
midway between the two. 



Foot=pounds of Work Required to Compress Air. 

(Hiscox.) 
Initial pressure = 1 atmosphere. 



2 Z 

? K P 


Foot-pounds per cubic foot. 


O) K 3 


Foot-pounds per cubic foot. 


«i T3 cr 1 








Pressur 
pound 
per sq 
inch. 








Pressu 
poun 
per s 
inch. 


Isother- 
mal. 


Adia- 
batic. 


Actual. 


Isother- 
mal. 


Adia- 
batic. 


Actual. 


5 


619.6 


649.5 


637.5 


55 


3393.7 


4188.9 


3870.8 


10 


1098.2 


1192.0 


1154.6 


60 


3440.4 


4422.8 


4029.8 


15 


1488.3 


1661.2 


1592.0 


65 


3577.6 


4645.4 


4218.2 


20 


1817.7 


2074.0 


1971.4 


70 


3706.3 


4859.6 


4398.1 


25 


2102.6 


2451.6 


2312.0 


75 


3828.0 


5063.9 


4569.5 


30 


2353.6 


2794.0 


2617.8 


80 


3942.9 


5259.7 


4732.9 


35 


2578.0 


3111.0 


2897.8 


85 


4051.5 


5450.0 


4890.1 


40 


2780.8 


3405.5 


3155.6 


90 


4155.7 


5633.1 


5042.1 


45 


2966.0 


3681.7 


3395.4 


95 


4254.3 


5819.3 


5187.3 


50 


3136.2 


3942.3 


3619.8 


100 


4348.1 


5981.2 


5327.9 



506 



Compressed Air. 



Compressor Efficiencies at Different Altitudes. 

70 Pounds Pressure per Square Inch. 



Altitude. 


Barometric pressure. 


Yolumetric 

efficiency of 

compressor. 

Per cent. 


Loss of 
capacity. 
Per cent. 


Decreased 


Feet. 


Inches of 
mercury. 


Pounds per 
square inch. 


power. 
Per cent. 


Sea- level. 


30.00 


14.75 


100 






1000 


28.88 


14.20 


97 


3 


1.8 


2000 


27.80 


13.67 


93 


7 


3.5 


3000 


26.76 


13.16 


90 


10 


5.2 


4000 


25.76 


12.67 


87 


13 


6.9 


5000 


24.79 


12.20 


84 


16 


8.5 


6000 


23.86 


11.73 


81 


19 


10.1 


7000 


22.97 


11.30 


78 


22 


11.6 


8000 


22.11 


10.87 


76 


24 


13.1 


9000 


21.29 


10.46 


73 


27 


14.6 


10000 


20.49 


10.07 


70 


30 


16.1 


11000 


19.72 


9.70 


68 


32 


17.6 


12000 


18.98 


9.34 


65 


35 


19.1 


13000 


18.27 


8.98 


63 


37 


20.6 


14000 


17.59 


8.65 


60 


40 


22.1 


15000 


16.93 


8.32 


58 


42 


23.5 



The above table is computed for a delivery of air compressed to 70 
pounds per square inch. For pressures above 70 pounds the volumetric 
efficiency of the compressor may be decreased 3 per cent, for each 10 
pounds, and the power required diminished 10 per cent. 



Cubic Feet of Air Required, per Indicated Horse- 
power, in Motors. 

(Hiscox.) 



"s?" 


Gauge pressures. 


Ph 


30 


40 


50 


60 


70 


80 


90 


100 


110 


125 


150 


1 


23.30 


21.3 


20.2 


19.40 


18.80 


18.42 


18.10 


17.80 


17.62' 


17.40 


17.05 


% 


18.70 


17.1 


16.1 


15.47 


15.00 


14.60 


14.35 


14.15 


13.98 


13.78 


13.50 


% 


17.85 


16.2 


15.2 


14.50 


14.20 


13.75 


13.47 


13.28 


13.08 


12.90 


12.60 


V* 


16.4 


14.5 


13.5 


12.80 


12.30 


11.93 


11.70 


11.48 


11.30 


11.10 


10.85 


V* 


17.5 


15.2 


12.9 


11.85 


11.26 


10.80 


10.50 


10.21 


10.02 


9.78 


9.5 


y± 


20.6 


15.6 


13.4 


13.3 


11.4 


10.72 


10.31 


10.0 


75 


9.42 


9.1 



Compressed Air. 507 



In applying this table the amount must he increased to provide for the 
clearance hi the cylinder, this depending upon the construction of the 
motor. When the air is reheated the amount required will be diminished. 
The economy due to reheating will be proportional to the increase in abso- 
lute temperature. Thus, if T be the initial temperature of the air, and T' 
the reheated temperature, we have the amount of air required, equal to the 
tabular amount, multiplied by 

T + 461 

T' + 461 ' 

Thus, if the air be reheated from 60° to 300° F., the tabular value should 
be multiplied by 

60 + 461 
300 + 461 ==0 - 684 ' 

showing a gain of nearly 32 per cent., due to reheating. 

The flow of compressed air in pipes may be computed from the formula : 



IpdP 
JwL' 
in which 



>=4 



Q = flow, in cubic feet, per minute ; 

p = difference in pressure, in pounds, per square inch, by which 

the flow is caused ; 
d = the diameter of the pipe, in inches ; 
L = the length, in feet ; 

w = the density of the entering air, in pounds, per cubic foot ; and 
c = a constant coefficient. According to Halsey, the value of c 

may be taken as = 58. 



508 



Flow of Air. 



Table of Head or Additional Pressure Required to 

Deliver Air at 80 Pounds Gauge Pressure through 

1000 Feet of Pipe of Various Sizes. 

Pipe Sizes are Inside Diameters. 



"© 


£3 




© 


0/ • 




a> 


© . 
© © 




£ 


«M P 




Vh 


& p 




© 


«M J3 




c6 


o-§ 




PT3 


«•-. p 




fl*d 


<+- p 




*§ 


,© f~* 


p 2 


■- P 
^3 


© Vi 


p £ 


•* P 

£2 


»2 


c3 ©' 

P *H 


-^ © 


«« © 


O p 


£8 


Ch © 


o p 


+» © 


«£ © 


O P 


"3 <° 


o ft 


33 88 


o ft 


33 a! 


■g ■ 


o ft 


f* VI 

+* CO 


a% 


2 A 


§2 


o u 
~ © 


2.*j 


m g 


o t« 

-■=< © 


S.S 


£2 


® & 


3 o3 


T3 ft 


® ft 


P C3 


*5 ft 


« ft 


P e3 


T3 ft 


> 


o 


< 


> 


Q 


< 


k 


o 


< 




1 inch. 




4 inches. 




12 inches. 


3.07 


6 


.337 


3.07 


88 


.031 


3.07 


799 


.0067 


6.14 


12 


1.348 


6.14 


176 


.124 


6.14 


1598 


.0268 


9.20 


18 


3.033 


9.20 


264 


.279 


9.20 


2397 


.0603 


12.27 


24 


5.392 


12.27 


352 


.495 


12.27 


3196 


.1072 


15.34 


30 


8.425 


15.34 


440 


.775 


15.34 


3995 


.1675 


18.41 


36 


12.132 


18.41 


528 


1.116 


18.41 


4794 


.2412 


24.54 


48 


21.568 


24.54 


704 


1.984 


24.54 


6392 


.4288 


30.68 


60 


33.700 


30.68 


880 


3.100 


30.68 


7990 


.6700 




1% inches. 




5 inches. 




14 inches. 


3.07 


14 


.134 


3.07 


141 


.022 


3.07 


1087 


.0055 


6.14 


29 


.536 


6.14 


282 


.088 


6.14 


2174 


.0220 


9.20 


43 


1.206 


9.20 


423 


.198 


9.20 


3261 


.0495 


12.27 


57 


2.144 


12.27 


564 


.352 


12.27 


4348 


.0880 


15.34 


72 


3.350 


15.34 . 


705 


.550 


15.34 


5435 


.1375 


18.41 


86 


4.824 


18.41 


846 


.792 


18.41 


6522 


.1980 


24.54 


115 


8.576 


24.54 


1128 


1.408 


24.54 


8696 


.3520 


30.68 


144 


13.400 


30.68 


1410 


2.200 


30.68 


10870 


.5500 




2 inches. 




6 inches. 




16 inches. 


3.07 


23 


.100 


3.07 


204 


.018 


3.07 


1420 


.0047 


6.14 


47 


.400 


6.14 


408 


.072 


6.14 


2840 


.0188 


9.20 


70 


.900 


9.20 


612 


.162 


9.20 


4260 


.0423 


12.27 


94 


1.600 


12.27 


816 


.288 


12.27 


5680 


.0752 


15.34 


118 


2.500 


15.34 


1020 


.450 


15.34 


7100 


.1175 


18.41 


141 


3.600 


18.41 


1224 


.648 


18.41 


8520 


.1692 


24.54 


188 


7.200 


24.54 


1632 


1.152 


24.54 


11360 


.3009 


30.68 


235 


10.000 


30.68 


2040 


1.800 


30.68 


14200 


.4700 




2 1 ._. inches. 




8 inches. 




20 inches. 


3.07 


33 


.058 


3.07 


353 


.011 


3.07 


2219 


.0036 


6.14 


67 


.232 


6.14 


706 


.044 


6.14 


4438 


.0144 


9.20 


100 


.522 


9.20 


1059 


.099 


9.20 


6657 


.0324 


12.27 


134 


.928 


12.27 


1412 


.176 


12.27 


8876 


.0576 


15.34 


168 


1.450 


15.34 


1765 


.275 


15.34 


11095 


.0900 


18.41 


201 


2.088 


18.41 


2118 


.336 


18.41 


13314 


.1296 


24.54 


268 


3.712 


24.54 


2824 


.704 


24.54 


17752 


.2304 


30.68 


335 


5.800 


30.68 


3530 


1.100 


30.68 


22190 


.3600 




3 inches. 




10 inches. 




24 inches. 


3.07 


52 


.050 


3.07 


566 


.0087 


3.07 


3194 


.0029 


6.14 


104 


.200 


6.14 


1132 


.0348 


6.14 


6388 


.0116 


9.20 


156 


.450 


9.20- 


1698 


.0783 


9.20 


9582 


.0261 


12.27 


208 


.800 


12.27 


2264 


.1392 


12.27 


12776 


.0464 


15.34 


260 


1.250 


15.34 


2830 


.2175 


15.34 


15970 


.0725 


18.41 


312 


1.800 


18.41 


3396 


.3132 


18.41 


19164 


.1044 


24.54 


416 


3.200 


24.54 


4528 


.5568 


24.54 


25552 


.1856 


30.68 


520 


5.000 


30.68 


5660 


.8700 


30.68 


31940 


.2900 



Movement of Air. 509 



Movement of Air. 

When large volumes of air are to be moved at low pressures, as in 
ventilation, mechanical draft, etc., the following formulas, derived from 
Weisbach, by Snow, for the B. F. Sturtevant Company, may be used : 
Let 

d = diameter of pipe, in inches ; 
I = length of pipe, in feet ; 
v = velocity, in feet, per second ; 
p = loss of pressure, in ounces, per square inch, by friction. 

Then 



P = 
1 = 



25000d ' 
25000dp 



V 25000dp 



~ 25000p * 

If we call the area of the pipe = A, and take the weight of a cubic foot 
of aif as 0.08 pound, we have, for the loss in horse-power by friction in a 
length of 100 feet, 



IP = 



pAv 
8800' 



From these formulas the following tables have been computed for pipes 
of various diameters, all 100 feet long, the losses being directly propor- 
tioned for pipes of other lengths. Since the loss in pressure varies as the 
square of the velocity, the advantage of using large pipes and reducing 
the velocity is apparent. 

The whole subject of the compression, utilization, and movement of air 
is a most extensive one. For further details of the different departments 
of the subject reference may be had to the following works : 

" Mechanical Draft." By Walter B. Snow. 
Compressed Air and its Applications." By Gardner D. Hiscox. 
Compressed Air Information." By W. L. Saunders. 

11 Compressed Air." By Frank Richards. 

The monthly periodical, "Compressed Air," also contains current infor- 

,tion of value and interest. 



510 



Air Friction. 



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33 



514 Atmospheric Pressure. 



Determination of Difference in Level by Difference in 
Atmospheric Pressure. 

According to the law of Mariotte, also called Boyle's law, the volume 
of a given quantity of any gas varies inversely as the pressure which it 
bears, the temperature remaining constant. The average pressure of the 
atmosphere at the sea-level is 1.033 kilogrammes per square centimetre, 
corresponding to a column of mercury 760 millimetres in height. This is 
the same as 14.7 pounds per square inch, or a mercury column of 29.92 
inches. In English-speaking countries an atmosphere of pressure is under- 
stood to mean 14.7 pounds per square inch, but in countries in which the 
metric system is used an atmosphere means a pressure of 1 kilogramme 
per square centimetre = 14.22 pounds per square inch. This is sometimes 
called a metric atmosphere, and pressure gauges in France, Germany, and 
elsewhere on the Continent are generally graduated in metric atmospheres 
and tenths. 

In obedience to the law of Mariotte, the density of the air diminishes as 
we ascend, and the law of this reduction in pressure has been found to be 
proportional to the logarithms of the pressures at any two points under 
consideration. Thus, if b be the height of the barometer at any given 
point, and b' the height of the barometer at another point, and h the dif- 
ference in altitude between them, we have 

h = C log. A 

C being a constant. 

For a difference of height, in metres, we have 

h = 18429.1 log. A 
h = 60463.4 log. A 



and for feet, 



It is necessary to correct results obtained by these formulas for the 
effects of varying temperatures and other atmospheric conditions. If we 
assume the mean temperature of the air between two stations to be the 
half sum of the temperatures at these stations, we may take the coefficient 
of expansion of air, ^ = 0.00366 per degree centigrade, and hence have 
for a temperature correction factor, 

0.00366^-^ = o.00183(< + *'). 

in which t and V are the temperatures of the two stations, in degrees centi- 
grade. For the Fahrenheit thermometer, taking the coefficient of expan- 
sion at ¥ £ T = 0.002036, and remembering that the freezing-point is 32° above 
zero, we have 

0.002036 ( < ~ 32) + ( *'~ 32) = 0.00102(* + tf — 64). 

If we take the sea-level as a base station we may compute the altitudes 

for various barometric readings, and thus enable altitudes to be readily 4 

and accurately measured. With such tables the altitude of each station Jl 

above sea-level may be computed separately, and their difference taken. ^ 



Barometric Tables. 



515 



Barometric Table A. 

Metric. 

Normal Heights. 0° C. 

18429.1 log. -^-. 



400 
410 
420 
430 
440 
450 
460 
470 
480 
490 
500 
510 
520 
530 
540 
550 
560 
570 
580 
590 



Difference. 



5137.2 
4939.6 
4746.7 
4558.3 
4374.4 
4294.5 
4018.6 
3846.4 
3677.9 
3512.8 
3351.2 
3192.7 
3037.3 
2884.9 
2735.2 
2588.4 
2444.2 
2302.6 
2163.3 
2026.5 



—19.8 
-19.3 
—18.8 
—18.4 
-18.0 
—17.6 
—17.2 
-16.8 
-16.5 
—16.2 
—15.9 
-15.5 
-15.2 
-15.0 
-14.7 
—14.4 
—14.2 
-13.9 
—13.7 
—13.3 



B. mm. 



600 
610 
620 
630 
640 
650 
660 
670 
680 
690 
700 
710 
720 
730 
740 
750 
760 
770 
780 



H. metres. 



1892.0 

1759.8 

1629.6 

1501.5 

1375.5 

1251.4 

1129.1 

1008.8 

890.2 

773.3 

658.2 

544.6 

432.7 

322.4 

213.4 

105.9 

0.0 

—104.8 

—207.9 



Difference. 



—13.2 
-13.0 
-12.8 
—12.6 
-12.6 
-12.2 
—12.0 
-11.9 
-11.7 
-11.5 
-11.4 
-11.3 
-11.0 
-10.9 
-10.7 
-10.6 
-10.5 
-10.3 
—10.2 



In Table A the first column contains the reading of the barometer, in 
millimetres, for every 10 millimetres, and in the second column the corre- 
sponding heights above sea-level. The third column, headed ' l Difference," 
contains the difference for every millimetre, so that the height can be ob- 
tained very correctly for barometer readings as close as the hundredth of 
a millimetre. 

Suppose, now, that we have at one station a reading of 765 millimetres, 
and at another 732 millimetres, the air being at 0° C. We find in Table A, 
for a barometric weight of 760 millimetres, an altitude of 0.0 metres and 
5 X —10.5 = —52.5 metres. Also, for 732 millimetres, we have 730 milli- 
metres = 322.4 metres ; and 2 X —10.9 = —21.8 metres, so that for 732 milli- 
metres we have 322.4 — 21.8 = 300.6 metres; hence, one station is 300.6 
metres above sea-level, and the other is 52.5 metres below, and the differ- 
ence in altitude is 353.1 metres. It will be noticed that the differences are 
negative, because the altitude diminishes as the height of the barometer 
increases, and hence we multiply the number of millimetres in excess of 
the reading in the first column by the tabular difference and subtract the 
product from the tabular altitude. 

If the air had been at some other temperature than 0° a correction 
would have been necessary, and this may be readily applied by the follow- 
ing table. 



516 



Bakometric Tables. 



Barometric Table B. 

Temperature Correction Factors. 
Centigrade. 

1 + 0.00183 (t + V). 



t + V. 


Factor. 


t -f- /'. 


Factor. 


t + f. 


Factor. 


i + r. 


Factor. 


1 


1.0018 


15 


1.0275 


29 


1.0531 


43 


1.0787 


2 


1.0037 


16 


1.0293 


30 


1.0549 


44 


1.0805 


3 


1.0055 


17 


1.0311 


31 


1.0567 


45 


1.0823 


4 


1.0073 


18 


1.0329 


32 


1.0586 


46 


1.0842 


5 


1.0091 


19 


1.0348 


33 


1.0604 


47 


1.0860 


6 


1.0110 


20 


1.0366 


34 


1.0622 


48 


1.0878 


7 


1.0128 


21 


1.0384 


35 


1.0640 


49 


1.0897 


8 


1.0146 


22 


1.0403 


36 


1.0659 


50 


1.0915 


9 


1.0164 


23 


1.0421 


37 


1.0677 


51 


1.0933 


10 


1.0183 


24 


1.0439 


38 


1.0696 


52 


1.0952 


11 


1.0201 


25 


1.0458 


39 


1.0714 


53 


1.0970 


12 


1.0220 


26 


1.0476 


40 


1.0732 


54 


1.0988 


13 


1.0238 


27 


1.0495 


41 


1.0750 


55 


1.1006 


14 


1.0257 


28 


1.0513 


42 


1.0769 


56 


1.1025 



In this table t and V are the temperatures of the two stations. By 
taking the factor opposite their sum and multiplying by it the result 
obtained from Table A, the corrected difference in altitude between the 
two stations will be obtained. Thus, in the example just given, suppose 
that the temperature at the lower station had been 22° C., and at the upper 
station 16° C, we have 22 + 16 = 38 ; and opposite 38, in Table B. we find 
1.0696. The corrected altitude will then be 



353 X 1.0696 = 377.5 metres. 

Table C is computed for English measures, the barometer readings 
being given in inches and tenths and the corresponding heights in feet 
above sea-level, the sea-level reading of the barometer being assumed as 
30 inches. The column of differences here gives the differences in altitude 
for every hundredth of an inch, and so, if the difference be multiplied by 
the hundredths and thousandths, for any reading, and the product sub- 
tracted from the tabular altitude for the inches and tenths, it will give the 
precise altitude. An example will make this more readily understood. 

Suppose one reading to be 29.832 inches, and the other 26.636 inches, we 
have, from Table C, for 29.8 inches, 176 feet, and the difference —8.8 multi- 
plied by 32, the hundredths and thousandths, = —8.8 X 32 = —28.16, and 
176 — 28.16 = 147.84 feet. Likewise, we have for 26.5 inches, from the table, 
3257 feet, and the difference —9.9 multiplied by 36 = —35.64, whence 3257 — 
35.64 = 3221.36 feet, and the difference in altitude between the two sta- 
tions is 

3221.36 - 147.84 = 3073.36 feet. 



Barometric Tables. 



517 



Barometric Table C. 

English. 
Normal Heights. 32° F. 



60463.4 log. 



30 
b' 



H. feet. 



Differ- 
ence. 



18201 
18027 
17853 
17681 
17510 
17340 
17171 
17003 
16836 
16670 
16506 
16343 
16180 
16019 
15858 
15698 
15540 
15382 
15225 
15069 
14914 
14761 
14607 
14455 
14304 
14153 
14004 
13855 
13707 
13560 
13413 
13267 
13123 
12979 
12836 
12694 
12552 
12411 
12271 
12132 
11994 
11856 
11719 
11582 
11446 
11312 
11177 
11044 
10911 
10779 
10648 
10516 



—17.4 
—17.4 
—17.2 
-17.1 
-17.0 
-16.9 
-16.8 
-16.7 
—16.6 
—16.4 
-16.3 
-16.3 
—16.2 
-16.1 
—16.0 
-15.8 
—15.8 
-15.7 
—15.6 
—15.5 
—15.3 
—15.3 
—15.2 
—15.1 
—15.1 
-15.0 
—14.9 
-14.8 
—14.7 
-14.7 
—14.6 
—14.5 
-14.4 
—14.3 
—14.2 
—14.2 
—14.1 
—14.0 
-13.9 
—13.8 
—13.8 
—13.7 
—13.7 
—13.6 
—13.4 
—13.4 
—13.3 
—13.3 
—13.2 
—13.1 
—13.1 
—13.0 



B. 

inches. 



20.2 
20.3 
20.4 
20.5 
20.6 
20.7 
20.8 
20.9 
21.0 
21.1 
21.2 
21.3 
21.4 
21.5 
21.6 
21.7 
21.8 
21.9 
22.0 
22.1 
22.2 
22.3 
22.4 
22.5 
22.6 
22.7 
22.8 
22.9 
23.0 
23.1 
23.2 
23.3 
23.4 
23.5 
23.6 
23.7 
23.8 
23.9 
24.0 
24.1 
24.2 
24.3 
24.4 
24.5 
24.6 
24.7 
24.8 
24.9 
25.0 
25.1 
25.2 
25.3 



H. feet. 



10386 
10256 
10127 
9998 
9871 
9744 
9617 
9491 
9366 
9241 
9117 
8993 
8869 
8747 
8626 
8505 
8384 
8264 
8144 
8025 
7906 
7788 
7671 
7554 
7438 
7322 
7206 
7091 
6977 
6863 
6750 
6637 
6524 
6412 
6301 
6190 
6079 
5969 
5859 
5750 
5641 
5533 
5425 
5318 
5211 
5105 
4999 
4893 
4787 
4683 
4578 
4474 



Differ- 
ence. 



-13.0 
—12.9 
-12.9 
—12.8 
—12.7 
-12.7 
—12.6 
—12 6 
^-12.5 
-12.4 
—12.4 
—12.3 
—12.2 
-12.1 
-12.1 
-12.1 
-12.0 
—12.0 
-11.9 
-11.9 
-11.8 
-11.7 
-11.7 
-11.7 
-11.6 
-11.6 
—11.5 
—11.5 
-11.4 
—11.4 
-11.3 
-11.3 
—11.2 
—11.1 
—11.1 
—11.1 
-11.0 
-11.0 
-10.9 
-10.9 
-10.8 
—10.8 
—10.8 
—10.7 
—10.6 
—10.6 
—10.6 
—10.6 
—10.5 
—10.5 
—10.4 
—10.4 



B. 

inches. 



25.4 
25.5 
25.6 
25.7 
25.8 
25.9 
26.0 
26.1 
26.2 
26.3 
26.4 
26.5 
26.6 
26.7 
26.8 
26.9 
27.0 
27.1 
27.2 
27.3 
27.4 
27.5 
27.6 
27.7 
27.8 
27.9 
28.0 
28.1 
28.2 
28.3 
28.4 
28.5 
28.6 
28.7 
28.8 
28.9 
29.0 
29.1 
29.2 
29.3 
29.4 
29.5 
29.6 
29.7 
29.8 
29.9 
30.0 
30.1 
30.2 
30.3 
30.4 
30.5 



H. feet. 



4371 

4268 

4165 

4062 

3960 

3859 

3758 

3657 

3556 

3456 

3356 

3257 

3158 

3060 

2962 

2864 

2767 

2670 

2573 

2476 

2380 

2285 

2190 

2095 

2000 

1906 

1812 

1718 

1625 

1532 

1439 

1347 

1255 

1163 

1072 

981 

890 

800 

710 

620 

530 

441 

352 

264 

176 

88 



— 87 

—174 

—261 

—348 

—434 



518 



Barometric Tables. 



For the temperature correction the following may be used : 
Barometric Table D. 

Temperature Correction Factors. 
Fahrenheit. 

1 + 0.00102 (t + V — 64). 



t + V. 


Factor. 


t + t>. 


Factor. 


t + K 


Factor. 


t + t f . 


Factor. 


32 


.9673 


68 


1.0041 


102 


1.0388 


136 


1.0735 


34 


.9697 


70 


1.0061 


104 


1.0408 


138 


1.0755 


36 


.9714 


72 


1.0082 


106 


1.0429 


140 


1.0776 


38 


.9735 


74 


1.0102 


108 


1.0450 


142 


1.0796 


40 


.9755 


76 


1.0122 


110 


1.0470 


144 


1.0817 


42 


.9776 


78 


1.0143 


112 


1.0490 


146 


1.0837 


44 


.9796 


80 


1.0163 


114 


1.0511 


148 


1.0858 


46 


.9816 


82 


1.0183 


116 


1.0531 


150 


1.0878 


48 


.9837 


84 


1.0204 


118 


1.0552 


152 


1.0898 


50 


.9857 


86 


1.0224 


120 


1.0572 


154 


1.0919 


52 


.9878 


88 


1.0245 


122 


1.0592 


156 


1.0939 


54 


9898 


90 


1.0265 


124 


1.0612 


158 


1.0960 


56 


.9918 


92 


1.0286 


126 


1.0633 


160 


1.0980 


58 


.9938 


94 


1.0306 


128 


1.0653 


162 


1.1000 


60 


.9959 


96 


1.0326 


130 


1.0674 


164 


1.1019 


62 


.9980 


98 


1.0347 


132 


1.0694 


166 


1.1039 


64 


1.0000 


100 


1.0368 


134 


1.0714 


168 


1.1060 


66 


1.0021 















Thus, if in the preceding example the temperatures at the two stations 
had been 65° F. and 43° F., we have 65 + 43 = 108, and opposite 108, in 
Table D, we find the correction factor, 1.045. 

The corrected altitude will then be 



3073.36 X 1.045 = 3211.66, 

an increase of more than 38 feet. 

When observations are taken simultaneously, the preceding tables will 
enable altitudes to be computed with much accuracy. When but single 
observations arc possible, the date and hour of the day should always be 
noted, as the simultaneous reading of the nearest weather bureau station 
may then be subsequently obtained, as well as its altitude, and tho desired 
height thus computed. 

For field work the aneroid barometer is undoubtedly the best. It should 
be carefully compared with the standard at the base station, both on leav- 
ing and returning, and the mean of the difference used as a base correction. 

Aneroids are often marked "compensated, meaning that they are so | 
constructed as to be unaffected by changes in their own temperature. 
This is rarely perfectly accomplished, as may be seen by warming or cool- 
ing the instrument. The best plan is to set the instrument, by means of 
the adjusting-screw at the back, so that it agrees with a standard mercurial 
barometer at 32°, and then warm the aneroid carefully up to about 70°, 
taking readings at every io°. A correction table can then be prepared for 
use on subsequent occasions. 



Water. 510 

The complete barometric formula of Laplace includes corrections for 
atmospheric humidity and for the variations in the action of gravity, but 
these need be considered only in precise work for great differences in alti- 
tude. Full details of this work will be found in the Smithsonian Meteoro- 
logical Tables. 

The altitude scales engraved on the dials of some aneroid barometers 
are of little use, except for rough approximate work, and their use has 
done much to bring the barometric method into undeserved discredit. 



WATER. 

Water is composed of 1 part of hydrogen combined with 8 parts of 
oxygen, or more nearly, according to the determinations of Morley and of 
Rayleigh, its composition by weight is 

Hydrogen, 2 atoms 2.00 or 11.186 

Oxygen, 1 atom 15.88 or 88.814 



17.88 100.000 

This gives 17.88 for the molecular weight in the gaseous state, but in the 
liquid state it is probably a multiple of this. 

In the production of 1 kilogramme of water by the burning of hydro- 
gen and oxygen 3830 calories are evolved. 

Its specific heat is taken as unity, being the basis upon which the 
specific heats of solids and liquids are computed ; but this specific heat is 
not constant, but varies with the temperature. 

According to Dieterici, the specific heat at various temperatures, taking 
the specific heat at 0° C. as unity, varies as follows : 

Specific Heat of Water. 

0°C. 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 
1.000 0.9943 0.9893 0.9872 0.9934 0.9995 1.0057 1.0120 1.0182 1.0244 1.0306 

In regard to the density of water at various temperatures there has 
been a material difference of opinion among various authorities. The 
temperature of maximum density is 4° C. or 39.1° F., but the actual weight 
of a unit of volume at this temperature ranges, according to different 
authorities, from 62.379 pounds to 62.425 pounds. The tables on pages 520- 
523 are those computed by Nystrom from the experiments of Kopp, and 
may be accepted as being' as accurate as any. In the metric system the 
litre is usually made equal to a kilogramme of water by weighing, thus 
practically determining the volume from the weight. 



520 Properties of Water from Freezing- to ] 


Boiling-point. 


Temp. 


Volume 


Units of heat. 


Pounds per 


Cubic feet 


Temp. 






Fahr. 


1 at 39°. 


Per pound. 


Per cubic 
foot. 


cubic foot. 


per pound. 


Cent. 


32 


1.000 109 


.000 000 000 


.000 


62.381 000 


.016 03046 


.000 


33 


1.000 077 


1.000 000 867 


62.383 


62.383 000 


.016 02994 


.555 


34 


1.000 055 


2.000 000 545 


124.77 


62.384 000 


.016 02956 


1.111 


35 


1.000 035 


3.000 001 609 


187.16 


62.385 871 


.016 02927 


1.666 


36 


1.000 020 


4.000 034 680 


249.55 


62.386 791 


.016 02904 


2.222 


37 


1.000 009 


5.000 062 940 


311.99 


62.387 493 


.016 02886 


2.777 


38 


1.000 002 


6.000 102 410 


374.33 


82.387 930 


.016 02874 


3.333 


39 


1.000 000 


7.000 154 550 


436.72 


62.388 055 


.016 02871 


3.888 


40 


1.000 002 


8.000 220 760 


499.12 


62.387 930 


.016 02874 


4.444 


41 


1.000 009 


9.000 302 340 


561.51 


62.387 493 


.016 02886 


5.000 


42 


1.000 019 


10.000 400 560 


623.89 


62.386 869 


.016 02902 


5.555 


43 


1.000 034 


11.000 516 630 


686.28 


62.385 933 


.016 02926 


6.111 


44 


1.000 053 


12.000 651 750 


748.66 


62.384 748 


.016 02956 


6.666 


45 


1.000 077 


13.000 807 040 


811.03 


62.383 251 


.016 02994 


7.222 


46 


1.000 104 


14.000 983 620 


873.40 


62.381 567 


.016 03038 


7.777 


47 


1.000 136 


15.001 326 000 


935.70 


62.379 571 


.016 03088 


8.333 


48 


1.000 171 


16.001 405 000 


997.77 


62.377 388 


.016 03146 


8.888 


49 


1.000 211 


17.001 (551 800 


1060.G 


62.374 893 


.016 03210 


9.444 


50 


1.000 254 


18.001 924 200 


1122.8 


62.372 212 


.016 03278 


10.000 


51 


1.000 302 


19.002 223 000 


1185.1 


62.369 219 


.016 03355 


10.555 


52 


1.000 353 


20.002 549 300 


1248.0 


62.366 039 


.016 03437 


11.111 


53 


1.000 408 


21.002 924 100 


1310.1 


62.362 611 


.016 03525 


11.666 


54 


1.000 468 


22.003 288 000 


1372.3 


62.358 871 


.016 03621 


12.222 


55 


1.000 531 


23.003 702 400 


1434.3 


62.354 944 


.016 03723 


12.777 


56 


1.000 597 


24.004 147 900 


1496.4 


62.350 831 


.016 03828 


13.333 


57 


1.000 668 


25.004 625 600 


1558.6 


62.346 407 


.016 03942 


13.888 


58 


1.000 740 


26.005 136 200 


1620.9 


62.341 921 


.016 04057 


14.444 


59 


1.000 819 


27.005 680 800 


1683.2 


62.337 000 


.016 04184 


15.000 


60 


1.000 901 


28.006 260 000 


1745.5 


62.331 893 


.016 04316 


15.555 


61 


1.000 986 


29.006 874 900 


1807.8 


62.326 620 


.016 04451 


16.111 


62 


1.001 075 


30.007 526 300 


1870.1 


62.321 059 


.016 04594 


16.666 


63 


1.001 167 


31.008 214 900 


1932.4 


62.315 333 


.016 04741 


17.222 


64 


1.001 262 


32.008 941 600 


1994.4 


62.309 420 


.016 04894 


17.777 


65 


1.001 362 


33.009 707 300 


2056.6 


62.303 198 


.016 05054 


18.333 


66 


1.001 464 


34.010 513 


2118.7 


62.296 852 


.016 05218 


18.888 


67 


1.001 570 


35.011 359 


2180.8 


62.290 259 


.016 05388 


19.444 


68 


1.001 680 


36.012 246 


2242.9 


62.283 418 


.016 05564 


20.000 


69 


1.001 793 


37.013 175 


2305.0 


62.276 293 


.016 05748 


20.555 


70 


1.001 909 


38.014 148 


2367.1 


62.269 183 


.016 05921 


21.111 


71 


1.002 028 


39.015 164 


2429.2 


62.261 788 


.016 06122 


21.666 


72 


1.002 151 


40.016 224 


2491.2 


62.254 146 


.016 06318 


22.222 


73 


1.002 277 


41.017 330 


2553.2 


62.246 320 


.016 06521 


22.777 


74 


1.002 406 


42.018 482 


2615.2 


62.238 309 


.016 06728 


23.333 


75 


1.002 539 


43.019 680 


2677.1 


62.230 052 


.016 06941 


23.888 


76 


1.002 675 


44.020 926 


2739.2 


62.221 612 


.016 07158 


24.444 


77 


1.002 814 


45.022 220 


2801.0 


62.212 987 


.016 07382 


25.000 


78 


1.002 956 


46.023 563 


2862.8 


62.204 179 


.016 07610 


25.555 


79 


1.003101 


47.024 956 


2924.6 


62.195 187 


.016 07841 


26.111 


80 


1.003 249 


48.026 398 


2985.4 


62.186 012 


.016 08078 


26.666 


81 


1.003 400 


49.027 893 


3048.2 


62.176 654 


.016 08321 


27.222 


82 


1.003 654 


50.029 438 


3111.0 


62.167 113 


.016 08567 


27 777 


83 


1.003 711 


51.031 039 


3172.8 


62.157 388 


.016 08820 


28.333 


84 


1.00", 872 


52.032 688 


3234.4 


(12.1 17 420 


.016 09077 


28.888 


85 


L.004 035 


53.034 394 


3296.2 


62.137 330 


.016 09338 


29.444 


86 


1.004 199 


64.086154 


3358.2 


62.127 182 


.016 09601 


30.000 


87 


1.004 370 


55.037 969 


341 8. 7 


62.116 605 


.016 09875 


30.555 


88 


1.004 542 


56.039 841 


3480.4 


62.105 969 


.016 10151 


31.111 


89 


1.004 717 


57.041 769 


3542.1 


62.095 152 


.016 10432 


31.666 


90 


1.004 894 


58.043 754 


3603.8 


62.084 214 


.016 10715 


32.222 



Properties of Water from Freezing- to Boiling-point. 521 



Temp. 




Units of heat. 


Pounds per 


Cubic feet 


Temp. 


"V olunie 






Fahr. 


1 at 39°. 


Per pound. 


Per cubic 
foot. 


cubic foot. 


per pound. 


Cent. 


91 


1.005 094 


59.045 797 


3665.0 


62.071 860 


.016 11036 


32.777 


92 


1.005 258 


60.047 899 


3726.6 


62.061 734 


.016 11298 


33.333 


93 


1.005 444 


61.050 061 


3788.2 


62.050 252 


.016 11597 


33.888 


94 


1.005 633 


62.052 282 


3849.8 


62.038 591 


.016 11900 


34.444 


95 


1.005 825 


63.054 564 


3911.2 


62.026 749 


.016 12208 


35.000 


96 


1.006 019 


64.056 907 


3972.6 


62.014 787 


.016 12519 


35.555 


97 


1.006 216 


65.059 312 


4033.9 


62.002 646 


.016 12834 


36.111 


98 


1.006 415 


66.061 780 


4095.2 


61.990 386 


.016 13153 


36.666 


99 


1.006 618 


67.064 311 


4156.5 


61.977 885 


.016 13478 


37.222 


100 


1.006 822 


68.066 906 


4217.7 


61.965 322 


.016 13806 


37.777 


101 


1.007 030 


69.069 565 


4278.9 


61.952 528 


.016 14140 


38.333 


102 


1.007 240 


70.072 290 


4340.1 


61.939 612 


.016 14475 


38.888 


103 


1.007 553 


71.075 080 


4401.3 


61.920 370 


.016 14944 


39.444 


104 


1.007 668 


72.077 937 


4462.5 


61.913 303 


.016 15161 


40.000 


105 


1.007 905 


73.080 861 


4523.0 


61.898 745 


.016 15541 


40.555 


106 


1.008 106 


74.083 852 


4585.0 


61.886 403 


.016 15863 


41.111 


107 


1.008 328 


75.086 912 


4645.9 


61.872 778 


.016 16220 


41.666 


108 


1.008 554 


76.090 044 


4706.8 


61.858 913 


.016 16581 


42.222 


109 


1.008 781 


77.093 239 


4767.7 


61.844 994 


.016 16946 


42.777 


110 


1.009 032 


78.096 509 


4828.6 


61.829 609 


.016 17348 


43.333 


111 


1.009 244 


79.099 846 


4889.5 


61.816 622 


.016 17677 


43.888 


112 


1.009 479 


80.103 255 


4950.4 


61.802 231 


.016 18064 


44.444 


113 


1.009 718 


81.106 740 


5011.3 


61.787 602 


.016 18447 


45.000 


114 


1.009 956 


82.110 290 


5072.2 


61.773 042 


.016 18829 


45.555 


115 


1.010 197 


83.113 920 


5133.0 


61.758 305 


.016 19216 


46.111 


116 


1.010 442 


84.117 620 


5193.7 


61.743 331 


.016 19608 


46.666 


117 


1.010 688 


85.121 400 


5254.3 


61.728 302 


.016 20003 


47.222 


118 


1.010 938 


86.125 250 


5314.9 


61.713 037 


.016 20403 


47.777 


119 


1.011 189 


87.129 180 


5375.5 


61.697 719 


.016 20806 


48.333 


120 


1.011 442 


88.133 180 


5436.1 


61.682 286 


.016 21211 


48.888 


121 


1.011 698 


89.137 260 


5496.6 


61.666 678 


.016 21621 


49.444 


122 


1.011 956 


90.141 410 


5557.1 


61.650 956 


.016 22034 


50.000 


123 


1.012 216 


91.145 650 


5617.6 


61.635 123 


.016 22451 


50.555 


124 


1.012 478 


92.149 960 


5678.1 


61.619 170 


.016 22871 


51.111 


125 


1.012 743 


93.154 350 


5738.6 


61.603 047 


.016 23296 


51.666 


126 


1.013 010 


94.158 820 


5798.9 


61.586 810 


.016 23724 


52.222 


127 


1.013 278 


95.163 380 


5859.2 


61.570 516 


.016 24153 


52.777 


128 


1.013 550 


96.168 010 


5919.5 


61.553 998 


.016 24590 


53.333 


129 


1.013 823 


97.172 720 


5979.7 


61.537 423 


.016 25027 


53.888 


130 


1.014 098 


98.177 520 


6040.0 


61.520 735 


.016 25468 


54.444 


131 


1.014 358 


99.182 390 


6100.2 


61.504 966 


.016 25884 


55.000 


135 


1.015 505 


103.202 740 


6340.3 


61.435 497 


.016 27724 


57.222 


140 


1.016 962 


108.230 090 


6639.6 


61.347 282 


.016 30064 


60.000 


145 


1.018 468 


113.259 650 


6937.9 


61.256 765 


.016 32473 


62.777 


150 


1.020 021 


118.291 470 


7215.1 


61.163 500 


.016 34961 


65.555 


155 


1.021 619 


123.325 620 


7531.2 


61.067 829 


.016 37523 


68.333 


160 


1.023 262 


128.362 170 


7826.2 


60.969 776 


.016 40156 


71.111 


165 


1.024 947 


133.401 190 


8098.1 


60.869 542 


.016 42857 


73.888 


170 


1.026 672 


138.442 730 


8412.8 


60.767 270 


.016 45623 


76.666 


175 


1.028 438 


143.486 870 


8704.2 


60.662 047 


.016 48477 


79.444 


180 


1.030 242 


148.536 660 


8994.9 


60.556 699 


.016 51345 


82.222 


185 


1.032 083 


153.583 160 


9281.9 


60.448 679 


.016 54296 


85.000 


190 


1.033 960 


158.635 450 


9571.6 


60.338 944 


.016 57305 


87.777 


195 


1.035 873 


163.690 570 


9858.5 


60.227 513 


.016 60370 


90.555 


200 


1.037 819 


168.748 580 


10318.0 


60.114 581 


.016 63489 


93.333 


205 


1.039 798 


173.809 560 


10428.0 


60.000 168 


.016 66662 


96.111 


210 


1.041 809 


178.873 550 


10712.0 


59.884 350 


.016 69885 


98.888 


212 


1.042 622 


180.900 000 


10824.0 


59.837 654 


.016 71160 


100.000 



522 



Properties of Water. 



Indicated 


pressure. 1 




Water. ; > 


Atmos. 

excluded. 

Lb. per 

sq. in. 


Atmos. 
excluded. 
Inches of 
mercury. 


Temp., 
Fahr. 
scale. 


Units of heat. 


Bulk, 
cub. ft. 
per lb. 


Weight, Volume r 


Cemp., ; 
Cent. ] 
scale, i r , 


Per 
cub. ft. 


Per 

pound. 


lbs., per i 
cub. ft. 


svat. — 1 
at 39°. 


—14 


—28.52 


101.36 


4301 


69.430 


.01617 


61.848 


1.0071 


30.83 


—13 


—26.48 


126.21 


5631 


94.369 


.01624 


61.583 


1.0130 


41.87 ' 


—12 


—24.44 


141.67 


6583 


109.91 


.01630 


61.317 


1.0174 


48.74 


—11 


—22.41 


153.27 


7331 


121.58 


.01637 


61.101 


1.0210 


53.90 


—10 


—20.37 


162.51 


7974 


130.89 


.01638 


60.920 


1.0241 


58.00 


— 9 


—18.33 


170.25 


8421 


138.69 


.01644 


60.762 


1.0267 


61.44 


— 8 


—16.29 


176.97 


8812 


145.46 


.01647 


60.657 


1.0288 


64.43 


— 7 


—14.26 


182.96 


9203 


151.52 


.01652 


60.514 


1.0309 


67.09 


— 6 


—12.22 


188.36 


9531 


156.97 


.01656 


60.372 


1.0333 


69.49 


— 5 


—10.18 


193.20 


9755 


161.87 


.01659 


60.282 


1.0359 


71.64 


— 4 


— 8.149 


197.60 


9975 


166.32 


.01663 


60.169 


1.0369 


73.60 


— 3 


— 6.111 


201.90 


10183 


170.67 


.01666 


60.072 


1.0385 


75.51 


— 2 


— 4.074 


205.77 


10398 


174.59 


.01669 


59.973 


1.0401 


77.23 


— 1 


— 2.037 


209.55 


10613 


178.42 


.01672 


59.896 


1.0416 


78.91 





.0000 


212.00 


10824 


180.95 


.01674 


59.838 


1.0426 


100.00 


.3125 


.6365 


213.04 


10883 


181.95 


.01675 


59.814 


1.0430 


100.58 


+ 1 


+ 2.037 


216.33 


11047 


185.29 


.01677 


59.735 


1.0444 


102.45 


+ 2 


+ 4.074 


219.45 


11225 


188.45 


.01679 


59.659 


1.0457 


104.36 


+ 3 


+ 6.111 


222.40 


11389 


191.44 


.01680 


59.592 


1.0469 


105.78 i 


+ 4 


+ 8.149 


225.25 


11550 


194.33 


.01681 


59.523 


1.0481 


107.35 


+ 5 


+10.18 


227.95 


11718 


197.08 


.01684 


59.459 


1.0492 


108.86 i 


+ 6 


+12.22 


230.60 


11868 


199.77 


.01686 


59.389 


1.0503 


110.33 


+ v 


+14.26 


233.10 


12012 


202.40 


.01688 


59.329 


1.0514 


111.50 


+ 8 


+16.29 


235.49 


12150 


204.73 


.01690 


59.270 


1.0524 


113.05 


+ 9 


+18.33 


237.81 


12282 


207.10 


.01692 


59.212 


1.0534 


114.00 


+10 


+20.37 


240.07 


12408 


209.39 


.01693 


59.154 


1.0545 


115.59 


+11 


+22.41 


242.24 


12528 


211.57 


.01695 


59.097 


1.0555 


116.80 


+12 


+24.44 


244.32 


12642 


213.72 


.01696 


59.057 


1.0564 


117.95 


+13 


+26.48 


246.35 


12750 


215.78 


.01697 


59.006 


1.0573 


119.08 


+14 


+28.52 


248.33 


12852 


217.80 


.01698 


58.953 


1.0589 


120.18 , 


+15 


+30.55 


250.26 


12946 


219.76 


.01699 


58.901 


1.0590 


121.25 


+16 


+32.59 


252.13 


13053 


221.67 


.01700 


58.851 


1.0599 


122.29 


+17 


+34.63 


253.98 


13157 


223.55 


.01701 


58.803 


1.0607 


123.32 ' 


+18 


+36.67 


255.77 


13258 


225.38 


.01702 


58.757 


1.0615 


124.32 


+19 


+38.71 


257.52 


13336 


227.16 


.01703 


58.713 


1.0623 


125.29 : 


+20 


+40.74 


259.22 


13430 


228.89 


.01704 


58.671 


1.0631 


126.23 


+21 


+42.78 


260.88 


13520 


230.59 


.01705 


58.631 


1.0639 


127.15 


+22 


+44.82 


262.50 


13608 


232.24 


.01707 


58.592 


1.0646 


128.05 


+23 


+46.85 


264.09 


13694 


233.86 


.01708 


58.560 


1.0654 


128.94 


+24 


+48.89 


265.65 


13778 


235.45 


.01709 


58.517 


1.0661 


129.80 


+25 


+50.93 


267.17 


13860 


237.00 


.01710 


58.481 


1.0668 


130.65 


+26 


+52.97 


268.66 


13940 


238.52 


.01711 


58.435 


1.0675 


131.48 ! 


+27 


+55.00 


270.12 


14018 


240.02 


.01712 


58.400 


■ 1.0684 


132.29 


+28 


+57.04 


271.55 


14094 


241.48 


.01713 


58.366 


1.0688 


133.05 


+29 


+59.08 


272.96 


14168 


242.92 


.01714 


58.332 


1.0695 


133.86 


+30 


+61.11 


274.33 


14241 


244.32 


.01715 


58.298 


1.0701 


134.63 


+31 


+63.15 


275.68 


14314 


245.70 


.01716 


58.264 


1.0708 


135.38 


+32 


+65.19 


277.01 


14385 


247.06 


.01717 


58.230 


1.0714 


136.12 


+33 


+67.23 


278.32 


14454 


248.40 


.01718 


58.197 


1.0720 


136.84 


+34 


+69.20 


279.62 


14522 


249.73 


.01719 


58.164 


1.0726 


137.56 


+35 


+71.30 


280.89 


14592 


251.03 


.01720 


58.131 


1.0732 


138.27 


+36 


+73.34 


282.14 


1 1659 


252.30 


.01721 


58.098 


1.0738 


138.96 


+37 


+75.38 


283.39 


14725 


253.58 


.01722 


58.066 


1.0744 


139.66 


+38 


+77.41 


284.58 


14789 


254.80 


.01723 


58.035 


1.0750 


140.33 


+39 


+79.45 


285.76 


14852 


256.01 


.01724 


58.004 


1.0756 


140.98 


+40 


+81.49 


286.96 


14913 


257.24 


.01725 


57.972 


1.0761 


141.64 


+41 


| 88.52 


288.06 


14973 


258.38 


.01726 


57.941 


1.0767 


142.27 


+42 


+85.56 


289.24 


15032 


259.67 


.01727 


57.910 


1.0773 


142.91 


+43 


+87.61 


290.37 


15091 


260.71 


.01728 


57.879 


1.0778 


143.54 


+44 


+89.64 


29 1. is 


15149 


261.87 


.01729 


57.848 


1.0783 


144.15 


+45 


+91.67 


292.58 


15208 


262.99 


.01730 


57.817 


1.0789 


144.76 



Propekties of Water. 



5^3 



Indicated 


pressure. 


Temp., 
Fahr. 
scale. 






Water. 




Atmos. 

excluded. 

Lb. per 

sq. in. 


Atmos. 
excluded. 
Inches of 
mercury. 


Units of heat. 


Bulk, 
cub. ft. 
per lb. 


Weight, 
lbs., per 
cub. ft. 


Volume 
wat. = 1 
at 39°. 


Temp., 


Per 
cub. ft. 


Per 
pound. 


Cent, 
scale. 


+ 46 


+ 93.71 


293.66 


15265 


264.10 


.01731 


57.786 


1.0794 


145.37 


+ 47 


+ 95.75 


294.73 


15321 


265.20 


.01732 


57.769 


1.0799 


145.96 


+ 48 


+ 97.78 


295.78 


15377 


266.27 


.01733 


57.742 


1.0804 


146.54 


+ 49 


+ 99.82 


296.82 


15432 


267.34 


.01734 


57.714 


1.0809 


147.12 


+ 50 


+101.8 


297.84 


15485 


268.39 


.01735 


57.687 


1.0814 


147.69 


+ 51 


+103.9 


298.85 


15536 


269.42 


.01735 


57.660 


1.0820 


148.25 


+ 52 


+105.9 


299.85 


15588 


270.45 


.01736 


57.633 


1.0825 


148.80 


+ 53 


+108.0 


300.84 


15639 


271.46 


.01737 


57.606 


1.0830 


149.34 


+ 54 


+110.0 


301.81 


15690 


272.46 


.01737 


57.580 


1.0835 


149.89 


+ 55 


+112.0 


302.77 


15739 


273.44 


.01738 


57.554 


1.0840 


150.43 


+ 56 


+114.1 


303.72 


15789 


274.42 


.01739 


57.529 


1.0844 


150.95 


+ 57 


+116.1 


304.69 


15839 


275.40 


.01739 


57.504 


1.0849 


151.48 


+ 58 


+118.1 


305.60 


15888 


276.35 


.01740 


57.480 


1.0854 


152.00 


+ 59 


+120.2 


306.52 


15936 


277.30 


.01741 


57.456 


1.0859 


152.51 


+ 60 


+122.2 


307.42 


15983 


278.22 


.01741 


57.432 


1.0863 


153.01 


+ 61 


+124.3 


308.38 


16029 


279.14 


.01742 


57.410 


1.0867 


153.51 


+ 62 


+126.3 


309.22 


16075 


280.07 


.01743 


57.388 


1.0871 


154.01 


+ 63 


+128.3 


310.11 


16120 


280.98 


.01743 


57.364 


1.0875 


154.50 


+ 64 


+130.4 


310.99 


16165 


281.87 


.01744 


57.344 


1.0880 


154.99 


+ 65 


+132.4 


311.86 


16209 


282.78 


.01745 


57.322 


1.0884 


155.48 


+ 66 


+134.4 


312.72 


16254 


283.66 


.01745 


57.300 


1.0888 


155.95 


+ 67 


+136.5 


313.57 


16298 


284.54 


.01746 


57.278 


1.0892 


156.42 


+ 68 


+138.5 


314.42 


16342 


285.41 


.01746 


57.254 


1.0897 


156.90 


+ 69 


+140.5 


315.25 


16384 


286.27 


.01747 


57.232 


1.0901 


157.36 


+ 70 


+142.6 


316.08 


16426 


287.12 


.01748 


57.210 


1.0905 


157.82 


+ 71 


+144.6 


316.90 


16467 


287.96 


.01748 


57.188 


1.0909 


158.28 


+ 72 


+146.7 


317.71 


16507 


288.80 


.01749 


57.166 


1.0913 


158.73 


+ 73 


+148.7 


318.51 


16547 


289.62 


.01750 


57.144 


1.0918 


159.17 


+ 74 


+150.7 


319.31 


16587 


290.44 


.01751 


57.122 


1.0921 


159.62 


+ 75 


+152.8 


320.10 


16637 


291.26 


.01752 


57.101 


1.0926 


160.05 


+ 76 


+154.8 


320.88 


16677 


292.06 


.01752 


57.080 


1.0929 


160.49 


+ 77 


+156.8 


321.66 


16717 


292.85 


.01753 


57.059 


1.0935 


160.92 


+ 78 


+158.9 


322.42 


16756 


293.65 


.01753 


57.038 


1.0937 


161.34 


+ 79 


+160.9 


323.18 


16795 


294.43 


.01754 


57.017 


1.0941 


161.76 


+ 80 


+163.0 


323.94 


16834 


295.21 


.01755 


56.996 


1.0945 


162.17 


+ 81 


+165.0 


324.67 


16872 


295.96 


.01756 


56.975 


1.0949 


162.59 


+ 82 


+167.0 


325.43 


16910 


296.75 


.01756 


56.954" 


1.0953 


163.02 


+ 83 


+169.1 


326.17 


16947 


297.51 


.01757 


56.933 


1.0956 


163.43 


+ 84 


+171.1 


326.90 


16984 


298.26 


.01757 


56.912 


1.0960 


163.83 


+ 85 


+173.1 


327.63 


17020 


299.01 


.01758 


56.891 


1.0964 


164.24 


+ 86 


+175.2 


328.35 


17056 


299.75 


.01759 


56.871 


1.0968 


164.64 


+ 87 


+177.2 


329.07 


17092 


300.50 


.01759 


56.862 


1.0972 


165.04 


+ 88 


+179.2 


329.78 


17127 


301.23 


.01760 


56.844 


1.0975 


165.43 


+ 89 


+181.3 


330.48 


17162 


301.95 


.01761 


56.826 


1.0979 


165.82 


+ 90 


+183.3 


331.18 


17197 


302.67 


.01761 


56.808 


1.0982 


166.21 


+ 91 


+185.4 


331.87 


17231 


303.38 


.01762 


56.790 


1.0986 


166.59 


+ 92 


+187.4 


332.56 


17265 


304.10 


.01763 


56.772 


1.0989 


166.98 


+ 93 


+189.4 


333.24 


17299 


304.80 


.01763 


56.754 


1.0993 


167.35 


+ 94 


+191.5 


333.92 


17333 


305.50 


.01764 


56.735 


1.0996 


167.77 


+ 95 


+193.5 


334.59 


17366 


306.19 


.01765 


56.716 


1.0999 


168.10 


+ 96 


+195.5 


335.26 


17399 


306.88 


.01765 


56.699 


1.1003 


168.47 


+ 98 


+199.6 


336.58 


17465 


308.34 


.01767 


56.664 


1.1010 


169.21 


+ 99 


+201.6 


337.23 


17497 


308.91 


.01768 


56.647 


1.1013 


169.57 


+100 


+203.7 


337.89 


17529 


309.60 


.01769 


56.629 


1.1017 


169.94 


+105 


+213.9 


341.0 


17688 


312.87 


.01772 


56.549 


1.1035 


171.70 


+110 


+224.1 


344.1 


17840 


316.04 


.01775 


56.469 


1.1050 


173.40 


+115 


+234.2 


347.1 


17993 


319.12 


.01778 


56.389 


1.1065 


175.06 


+120 


+244.4 


350.0 


18136 


322.13 


.01781 


56.309 


1.1080 


176.68 


+125 


+254.6 


352.8 


18278 


325.06 


.01784 


56.220 


1.1095 


178.25 


+130 


+264.8 


355.6 


18413 


327.91 


.01786 


56.146 


1.1110 


179.78 


+135 


+275.0 


358.4 


18549 


330.75 


.01788 


56.073 


1.1124 


181.35 



524 



Peoperties of Water. 



Density and Volume of Water. 

Centigrade Temperatures. 



Temp. 
Cent. 


Density. 


Volume. 


Temp. 
Cent. 


Density. 


Volume. 


Temp. 
Cent. 


Density. 


Volume. 


—10 


.99814 


1.00186 


14 


.99930 


1.00070 


38 


.99310 


1.00694 


— 9 


.99843 


1.00157 


15 


.99916 


1.00084 


39 


.99273 


1.00732 


— 8 


.99868 


1.00132 


16 


.99900 


1.00100 


40 


.99235 


1.00770 


- 7 


.99891 


1.00109 


17 


.99884 


1.00116 


41 


.99197 


1.00809 


— 6 


.99912 


1.00088 


18 


.99865 


1.00135 


42 


.99158 


1.00849 


- 5 


.99930 


1.00070 


19 


.99846 


1.00154 


43 


.99118 


1.00889 


— 4 


.99945 


1.00054 


20 


.99826 


1.00174 


44 


.99078 


1.00929 


— 3 


.99959 


1.00041 


21 


.99805 


1.00196 


45 


.99037 


1.00971 


— 2 


.99970 


1.00030 


22 


.99783 


1.00218 


46 


.98996 


1.01014 


— 1 


.99980 


1.00020 


23 


.99760 


1.00240 


47 


.98954 


1.01057 





.99987 


1.00013 


24 


.99737 


1.00264 


48 


.98910 


1.01101 


1 


.99993 


1.00007 


25 


.99712 


1.00289 


49 


.98865 


1.01148 


2 


.99997 


1.00003 


26 


.99687 


1.00314 


50 


.98820 


1.01195 


3 


.99999 


1.00001 


27 


.99660 


1.00341 


55 


.98582 


1.01439 


4 


1.00000 


1.00000 


28 


.99633 


1.00368 


60 


.98338 


1.01691 


5 


.99999 


1.00001 


29 


.99605 


1.00396 


65 


.98074 


1.01964 


6 


.99997 


1.00003 


30 


.99577 


1.00425 


70 


.97794 


1.02256 


7 


.99993 


1.00007 


31 


.99547 


1.00455 


75 


.97498 


1.02566 


8 


.99989 


1.00011 


32 


.99517 


1.00486 


80 


.97194 


1.02887 


9 


.99983 


1.00018 


33 


.99485 


1.00518 


85 


.96879 


1.03221 


10 


.99975 


1.00025 


34 


.99452 


1.00551 


90 


.96556 


1.03567 


11 


.99965 


1.00034 


35 


.99418 


1.00586 


95 


.96219 


1.03931 


12 


.99955 


1.00045 


36 


.99383 


1.00621 


100 


.95865 


1.04312 


13 


.99943 


1.00057 


37 


.99347 


1.00657 









Head of Water. 



525 



Table of Water=heads, Equivalent Pressures, Work, 
and Horse=power. 

Pelton Water-wheel Company. 







d 




! 




d 






W n & 


"S3 ^ -m 






to -,d* 


'to J-i -*3 






*- a c 


•s'SS..S = 


fcfi , 




s- a 9 


<^'£ CD « W) 

O 2 ft.2 c 


hJO , 


"S 




to fl » ^''B 


.2 S 

T3 *-> 




ftg" - 
+= o 2 


-d S ci ^H 


•-s 2 


<•-. 


S ft « 


c^o^o 


flO 


**H 


£ ft «8 


fl^j o ««o 


a 2 


.2 


* d S 


d > f-h g a 

O •* flS -*-> so -A 
£uM bO d ©^ 


O A . 


_d 


-2 a ss 


d > — © ft 


O'-i '". 




a .5 oh 


ft u ~ 






O ^ si +* to r A 
AM bed ©,§ 


i 1 © ® 


73 


P ti 01 

o<d ft 




Ego 

o £ ft 




d s-, <u 
c< d ft 




Ego 

O ? ft 


w 


H 


ft 


Q 


W 


H 


p=l 


o 


1 


.43 


834 


.03 


500 


216.50 


417 000 


12.64 


2 


.87 


1668 


.05 


525 


227.33 


437 850 


13.27 


3 


1.30 


2 502 


.08 


550 


238.15 


458 700 


13.90 


4 


1.73 


3 336 


.10 


575 


248.98 


479 550 


14.53 


5 


2.17 


4 170 


.13 










6 


2.60 


5 004 


.15 


600 


259.80 


500 400 


15.16 


7 


3.03 


5 838 


.18 


625 


270.63 


521 250 


15.79 


8 


3.46 


6 672 


.20 


650 


281.45 


542 100 


16.42 


9 


3.90 


7 506 


.23 


675 


292.28 


562 950 


17.05 


10 


4.33 


8 340 


.25 


700 


303.10 


583 800 


17.68 


11 


4.76 


9 174 


.28 


725 


313.93 


604 650 


18.31 


12 


5.20 


10 008 


.30 


750 


324.75 


625 500 


18.95 


13 


5.63 


10 842 


.33 


775 


335.58 


646 350 


19.58 


14 


6.06 


11676 


.35 










15 
16 


6.50 
6.93 


12 510 

13 344 


.38 
.40 


800 


346.40 


667 200 
688 050 


20.20 

20.85 


825 


357.23 


17 


7.36 


14 178 


.43 


850 


368.05 


708 900 


21.48 


18 


7.79 


15 012 


.46 


875 


378.88 


729 750 


22.11 


, 19 


8.23 


15 846 


.48 


















900 


389.70 


750 600 


22.74 


20 


8.66 


16 680 


.50 


925 


400.53 


771 450 


23.38 


30 


12.99 


25 020 


.76 


950 


411.35 


792 300 


24.01 


40 


17.32 


33 360 


1.01 


975 


422.18 


813 150 


24.64 


50 


21.65 


41700 


1.26 










60 


25.98 


50 040 


1.52 


1000 


433.00 


834 000 


25.27 


70 


30.31 


58 380 


1.77 


1025 


443.83 


854 850 


25.90 


80 


34.64 


66 720 


2.02 


1050 


454.65 


875 700 


26.53 


90 


38.97 


75 060 


2.27 


1075 


465.48 


896 550 


27.17 


100 


43.30 


83 400 


2.53 


1100 


476.30 


917 400 


27.80 


125 


54.13 


104 250 


3.16 


1125 


487.13 


938 250 


28.43 


150 


64.95 


125 100 


3.79 


1150 


497.95 


959 100 


29.06 


175 


75.78 


145 950 


4.42 


1175 


508.78 


979 950 


29.69 


200 


86.60 


166 800 


5.05 


1200 


519.60 


1 000 800 


30.33 


225 


97.43 


187 650 


5.68 


1225 


530.43 


1 021 650 


30.96 


250 


108.25 


208 500 


6.31 


1250 


541.25 


1 042 500 


31.59 


275 


119.08 


229 350 


6.94 


1275 


552.08 


1 063 350 


32.23 


300 


129.90 


250 200 


7.57 


1300 


562.90 


1 084 200 


32.86 


325 


140.73 


271 050 


8.22 


1325 


573.73 


1 105 050 


33.49 


350 


151.55 


291 900 


8.85 


1350 


584.55 


1 125 900 


34.12 


375 


162.38 


312 750 


9.48 


1375 


595.38 


1 146 750 


34.75 


400 


173.20 


333 600 


10.11 


1400 


606.20 


1 167 600 


35.38 


425 


184.03 


354 450 


10.74 


1425 


617.03 


1 188 450 


36.01 


450 


194.85 


375 300 


11.38 


1450 


627.85 


1 209 300 


36.64 


475 


205.68 


396 150 


12.01 


1475 


638.68 


1 230 150 


37.28 



526 



Head of Water. 



Table of Water=heads, etc. — continued. 







b£ 








bp 






w ~ji 


'§ ft -£ 






o5 x "** 


"53 p -w 






2|§ 


o 2 ft.9 d 


b£ , 




<S T~) 2 


o ^ft.S a 


&c . 


5® 




nds 
hen 
Ions 
agai 
ondi 


.2 $ 




ft 5-~ 


nj - DO ^S 

C^ O ed § 


.S $ 




-2 c s 


S fe- » ft . 


o^ . 


n 


® d 3 


d te- ~- flj ft 


O iP . 


""' 




ft±d &Q ^ » ,2 


ft i-, S-> 


'** 




oot-po 
work a 
100 ga 
minut 
eorres 
heads. 


ft kl '— 
5? q) a; 


0> 


P h is 

r 9T3 ft 


oot- 
wor 
100 
min 

con 
heai 


P +* ^ 
O > ft 


•a 

w 


'2 © Sh 
ft ^ ® 
C«d ft 


£ " £ 

III 


X 


H 


N 


O 


S 


£H 


O 


1500 


649.50 


1 251 000 


37.91 


2300 


995.90 


1 918 200 


58.12 


1525 


660.33 


1 271 850 


38.54 


2325 


1006.73 


1 939 050 


58.75 


1550 


671.15 


1 292 700 


39.17 


2350 


1017.55 


1 959 900 


59.39 


1575 


681.98 


1 313 550 


39.80 


2375 


1028.38 


1980 750 


60.02 


1600 


692.80 


1 334 400 


40.44 


2400 


1039.20 


2 001 600 


60.65 


1625 


703.63 


1 355 250 


41.07 


2425 


1050.03 


2 022 450 


61.28 


1650 


714.45 


1 376 100 


41.70 


2450 


1060.85 


2 043 300 


61.91 


1675 


725.28 


1 396 950 


42.33 


2475 


1071.68 


2 064 150 


62.55 


1700 


736.10 


1 417 800 


42.96 


2500 


1082.50 


2 085 000 


63.18 


1725 


746.93 


1 438 650 


43.59 


2525 


1093.33 


2 105 850 


63.81 


1750 


757.75 


1 459 500 


44.22 


2550 


1104.15 


2 126 700 


64.44 


1775 


768.58 


1 480 350 


44.85 


2575 


1114.98 


2 147 550 


65.07 


1800 


779.40 


1 501 200 


45.49 


2600 


1125.80 


2 168 400 


65.70 


1825 


790.23 


1 522 050 


46.13 


2625 


1136.63 


2 189 250 


66.34 


1850 


801.05 


1 542 900 


46.76 


2650 


1147.45 


2 210 100 


66.97 


1875 


811.88 


1 563 750 


47.39 


2675 


1158.28 


2 230 950 


67.60 


1900 


822.70 


1 584 600 


48.02 


2700 


1169.10 


2 251 800 


68.23 


1925 


833.53 


1 605 450 


48.65 


2725 


1179.93 


2 272 650 


68.85 


1950 


844.35 


1 626 300 


49.29 


2750 


1190.75 


2 293 500 


69.49 


1975 


855.18 


1 647 150 


49.92 


2775 


1201.58 


2 314 350 


70.12 


2000 


866.00 


1 668 000 


50.55 


2800 


1212.40 


2 335 200 


70.75 


2025 


876.83 


1 688 850 


51.18 


2825 


1223.23 


2 356 050 


71.39 


2050 


887.65 


1 709 700 


51.81 


2850 


1234.05 


2 376 900 


72.02 


2075 


898.48 


1 730 550 


52.44 


2875 


1244.88 


2 397 750 


72.65 


2100 


909.30 


1 751 400 


53.07 


2900 


1255.70 


2 418 600 


73.28 


2125 


920.13 


1 772 250 


53.70 


2925 


1266.53 


2 439 450 


73.92 


2150 


930.95 


1 793 100 


54.33 


2950 


1277.35 


2 460 300 


74.55 


2175 


941.78 


1 813 950 


54.96 


2975 


1288.18 


2 481 150 


75.18 


2200 


952.60 


1 834 800 


55.60 


3000 


1299.00 


2 502 000 


75.82 


2225 


963.43 


1 855 650 


56.23 










2250 


974.25 


1 876 500 


56.86 










2275 


985.08 


1 897 350 


57.49 











The head,— vertical distance to which water is pumped above level of 
supply. Constant used for equivalent pressure = 0.433, which is the press- 
ure per square inch of 1 foot-head of water at 62° F. 

1 gallon of water at 62° F. weighs 8.34 pounds. 

1 horse-power = 33,000 foot-pounds per minute. 

If equivalents of heads that are not tabulated are desired, divide the 
head into heads that are given, and add their equivalents. 

E.g., to find the equivalent pressure for a head of 129 feet, 129 = 
125 + 4. 

125 feet - 54.13 pounds \ H _ 

" 129 feet. 



1 feet = Til pounds j sum ^jt== P° unds = e <l uivalent P ressure of 



Head of Water. 527 



The pressure of 1 foot-head of water, taking the density at the average 
temperature of 62° F., is 0.433 pound per square inch. The head corre- 
sponding to a pressure of 1 pound per square inch is 2.3095 feet. 

The pressure within a vessel is the same upon every square inch of its 
♦Surface, regardless of the shape or size of the vessel, and is that due to 
the head of water upon it. The horizontal pressure against a wall or dam 
varies as the square of the height. If h be the height of the dam, and w 
the weight of a cubic foot of water, the pressure per foot-width will be 
%ivh 2 , and its point of application will be two-thirds of the distance from 
the top. 

The theoretical velocity of issuing from an orifice is the same as that 
which would be acquired by a body falling from the height of the head 
of water above the orifice. This is 

V = j/ 2gh , 

in which h is the head of water ; g, the acceleration of gravity = 32.2 ; and 
V, the velocity, in feet, per second. In practice, this theoretical velocity is 
not attained, owing to various resistances, but the principle should always 
be borne in mind. If the water is under a pressure other than that due 
to its own weight, the head corresponding to that pressure may be found, 
taking 2.3095 feet to the pound, and this value used in the formula. 

If a be the area of a jet, in square inches; v, its velocity, in feet, per 
second ; and iu, the weight of a cubic foot of water, the energy, in foot- 
pounds, per second will be 

ivav 2 

The coefficient of discharge of a jet of water is the proportion of the full 
theoretical discharge which is realized in practice. As a result of many 
experiments this coefficient may be given a mean value of 0.61. If, there- 
fore, the area of an opening be multiplied by the theoretical velocity = 
\/2gh, and 61 per cent, of this taken, the actual discharge will be found. 
.This is true for orifices in the comparatively thin wall or bottom of the 
vessel containing the water ; the area of the orifice being small compared 
with the size of the reservoir, and the edges having a definite square 
corner. 

When, instead of a mere orifice, a short tube or nozzle is used, having 
a length of about three times its diameter, the coefficient of discharge is 
about 80 per cent, of the theoretical. By using smooth conveying nozzles, 
with the inner edges rounded, the coefficient may be raised to about 97 
per cent. 

In computing the flow of water through long pipes the principal loss to 
be provided for is that due to friction between the water and the surface 
of the pipe. The resistance due to friction may be computed in terms of 
feet of head, — that is, the number of feet of head necessary to overcome 
the resistance of friction may be found and deducted from the actual total 
! head available. 



528 



Water-heads and Velocities. 



Theoretical Velocity of Water Due to Given Heads. 





a 


d 




d 


d 




.s 


^d 




















O U 


o3 ^ ^ 

Fo „ a 
o o+- ° 


8js 


d 

o f-T 
<x> . 

t3 -*-> +* 


e3 ... 

.2 >»© • 
o o *ro 


o3 „ 

•Js is* ® ©* 


O *H 


S o Jo 


o O +j q 


&t* 


§© £ s 


-3£<8g 


e3 o3 © 


2 $ 0) © 


-d t>cS a 


* £ $ 




Jill 


H 


H 


H 


H 


W 


H 


H 


1 


8.205 


481.5 


51 


57.309 


3438.5 


105 


82.231 


4933.9 


2 


11.345 


681.7 


52 


57.869 


3472.1 


110 


84.166 


5050.0 


3 


13.899 


833.9 


53 


58.422 


3505.3 


115 


86.058 


5163.5 


4 


16.050 


963.0 


54 


58.971 


3538.2 


120 


• 87.909 


5274.5 


5 


17.944 


1076.6 


55 


59.515 


3570.9 


125 


89.722 


5383.3 


6 


19.657 


1179.4 


56 


60.053 


3603.2 


130 


91.499 


5489.9 


7 


21.232 


1273.6 


57 


60.587 


3635.2 


135 


93.242 


5594.5 


8 


22.698 


1361.8 


58 


61.116 


3666.9 


140 


94.953 


5697.2 


9 


24.075 


1444.5 


59 


61.641 


3698.4 


145 


96 633 


5798.0 


10 


25.377 


1522.6 


60 


62.161 


3729.6 


150 


98.285 


5897.1 


11 


26.615 


1596.9 


61 


62.677 


3760.6 


155 


99.909 


5994.5 


12 


27.799 


1667.9 


62 


63.188 


3791.3 


160 


101.50 


6090.5 


13 


28.934 


1736.0 


63 


63.696 


3821.7 


165 


103.08 


6184.9 


14 


30.026 


1801.6 


64 


64.200 


3852.0 


170 


104.63 


6277.9 


15 


31.080 


1864.8 


.65 


64.699 


3881.9 


175 


106.16 


6369.6 


16 


32.100 


1926.0 


66 


65.195 


3911.7 


180 


107.66 


6460.0 


17 


33.087 


1985.2 


67 


65.687 


3941.2 


185 


109.15 


6549.1 


18 


34.047 


2042.8 


68 


66.175 


3970.3 


190 


110.61 


6637.0 


19 


34.980 


2098.8 


69 


66.660 


3999.6 


195 


112.06 


6723.7 


20 


35.888 


2153.3 


70 


67.141 


4028.5 


200 


113.49 


6809.4 


21 


36.775 


2206.5 


71 


67.619 


4057.1 


205 


114.90 


6894.0 


22 


37.640 


2258.4 


72 


68.094 


4085.6 


210 


116.29 


6977.6 


23 


38.486 


2309.1 


73 


68.565 


4113.9 


215 


117.66 


7060.1 


24 


39.314 


2358.8 


74 


69.033 


4142.0 


220 


119.03 


7141.8 


25 


40.125 


2407.5 


75 


69.498 


4169.9 


225 


120.00 


7222.5 


26 


40.919 


2455.1 


76 


69.960 


4197.6 


230 


121.70 


7302.3 


27 


41.699 


2501.9 


77 


70.419 


4225.1 


235 


123.02 


7381.2 


28 


42.464 


2547.8 


78 


70.874 


4252.4 


240 


124.32 


7459.3 


29 


43.215 


2592.9 


79 


71.327 


4279.6 


245 


125.60 


7536.6 


30 


43.954 


2637.2 


80 


71.777 


4306.6 


250 


126.88 


7613.1 


31 


44.681 


2680.8 


81 


72.225 


4333.5 


255 


128.15 


7648.8 


32 


45.396 


2723.7 


82 


72.673 


4360.4 


260 


129.39 


7763.9 


33 


46.100 


2766.0 


83 


73.111 


4386.6 


265 


130.63 


7837.6 


34 


46.793 


2783.0 


84 


73.550 


4413.0 


270 


131.86 


7911.8 


35 


47.476 


2848.5 


85 


73.986 


4439.2 


275 


133.08 


7984.8 


36 


48.150 


2889.0 


86 


74.420 


4465.2 


280 


134.28 


8057.0 


37 


48.814 


2928.8 


87 


74.852 


4491.1 


285 


135.48 


8128.6 


38 


49.469 


2968.1 


88 


75.281 


4516.8 


290 


136.66 


8199.6 


39 


50.116 


3006.9 


89 


75.707 


4542.4 


295 


137.83 


8270.1 


40 


50.754 


3045.2 


90 


76.131 


4567.9 


300 


138.99 


8339.8 


41 


51.385 


3083.1 


91 


76.553 


4593.2 


305 


140.15 


8409.0 


42 


52.007 


3120.4 


92 


76.973 


4618.3 


310 


141.29 


8477.6 


43 


52.623 


3157.4 


93 


77.390 


4643.4 


315 


142.42 


8545.6 


44 


53.231 


3193.9 


94 


77.805 


4668.3 


320 


143.55 


8613.3 


45 


53.833 


3229.9 


95 


78.217 


4693.0 


325 


144.67 


8690.4 


46 


54.427 


3265.6 


96 


78.628 


4717.7 


330 


145.78 


8760.9 


47 


55.016 


3301.0 


97 


79.037 


4742.2 


335 


146.88 


8812.9 


48 


55.598 


3335.8 


98 


79.443 


4766.6 


340 


147.97 


8878.4 


49 


56.175 


3370.5 


99 


79.847 


4790.8 


345 


149.06 


8943.5 


50 


56.745 


3404.7 


100 


80.250 


4815.0 


350 


150.13 


9007.9 



Plow of Watek. 529 



Flow of Water Through Pipes. 

L 

The quantity of water which flows through a pipe is measured by the 
product of the area of its cross-section and by the velocity of the flow. 

The velocity is not uniform over the entire cross-section, but a mean 
velocity may be computed which will serve for purposes of computation. 
In order to compute the velocity two elements must be given: the slope 
and the hydraulic radius. The slope is the sine of the angle of inclination 
of the pipe, or the head divided by the length ; the hydraulic radius is the 
area divided by the wetted perimeter. The slope is called s, and the 
hydraulic radius r. For pipes of circular cross-section running full, r = 

— - — , the same being true when half-full. ' 

The first attempt to express the relations between these elements was 
that of Chezy, in 1775, his formula being 

v = C\/rs, 

v being the velocity, in feet or metres, per second, and C being a constant 
coefficient. A vast number of experiments have been made to determine 
the value of the coefficient, C, with the result of showing it to varv with 
different slopes and diameters of pipes. In 1896 Tutton collected the 
results of more than 1000 experiments and suggested a modification of the 
formula, which appears to be the most reliable one available, and which 
we shall use in preference to any other. 

Instead of placing the two quantities, r and s, under the radical sign, 
Tutton gives them independent exponents, writing the formula 



By comparing the results of many experiments it appears that if the expo- 
nents are made z = %, y = %, the coefficient, C, remains practically con- 
stant for any one kind of pipe, regardless of slope or diameter. The 
formula then reads, 

v = C rM, 

so that the cube root of the square of the hydraulic radius is taken and 
the square root of the slope, and the product of tfhese, by a constant 
depending only upon the character of the pipe, gives the velocity. 
The following values for Care given for different surfaces : 

Values of C for Pipe Flow. 

C 

Wrought-iron pipe 160 

Cast-iron pipe, new 130 

Cast-iron pipe, in service 104 

Lap riveted pipe 115 

Wrought-iron pipe, asphalted 170 

Wood-stave pipe 125 

Tubercula.ted pipe 30 to 80 

Brick conduits 110 

In order to facilitate the use of the formula the following tables are 

appended, giving values of r* and s*. Other values may be taken from 
the tables of power and roots, the % power being the square of the cube 
root, and the % power being the square root. 



34 



530 



Flow of Water. 



Values of rl from 0.01 to 1. 



r 


rl 


r 


rl 


r 


rl 


r 


r§ 


r 


rl 


.01 


.0464 


.21 


.3533 


.41 


.5519 


.61 


.7193 


.81 


.8690 


.02 


.0737 


.22 


.3644 


.42 


.5608 


.62 


.7271 


.82 


.8761 


.03 


.0965 


.23 


.3754 


.43 


.5697 


.63 


.7349 


.83 


.8832 


.04 


.1169 


.24 


.3861 


.44 


.5785 


.64 


.7427 


.84 


.8902 


.05 


.1357 


.25 


.3969 


.45 


.5872 


.65 


.7503 


.85 


.8974 


.06 


.1533 


.26 


.4073 


.46 


.5958 


.66 


.7581 


.86 


.9044 


.07 


.1698 


.27 


.4177 


.47 


.6045 


.67 


.7656 


.87 


.9111 


.08 


.1857 


.28 


.4280 


.48 


.6131 


.68 


.7733 


.88 


.9183 


.09 


.2008 


.29 


.4381 


.49 


.6216 


.69 


.7809 


.89 


.9252 


.10 


.2155 


.30 


.4481 


.50 


.6300 


.70 


.7884 


.90 


.9322 


.11 


.2295 


.31 


.4580 


.51 


.6384 


.71 


.7958 


.91 


.9390 


.12 


.2432 


.32 


.4679 


.52 


.6465 


.72 


.8033 


.92 


.9459 


.13 


.2566 


.33 


.4775 


.53 


.6550 


.73 


.8107 


.93 


.9528 


.14 


.2696 


.34 


.4871 


.54 


.6631 


.74 


.8181 


.94 


.9596 


.15 


.2823 


.35 


.4966 


.55 


.6712 


.75 


.8255 


.95 


.9663 


.16 


.2947 


.36 


.5061 


.56 


.6795 


.76 


.8328 


.96 


.9732 


.17 


.3069 


.37 


.5154 


.57 


.6874 


.77 


.8401 


.97 


.9799 


.18 


.3188 


.38 


.5246 


.58 


.6955 


.78 


.8473 


.98 


.9866 


.19 


.3305 


.39 


.5338 


.59 


.7034 


.79 


.8545 


.99 


.9932 


.20 


.3420 


.40 


.5429 


.60 


.7113 


.80 


.8617 


1.00 


1.0000 



Values of s* for Slopes from .000 025 to 1. 



s 


si 


S 


si 


S 


sh 


.000 025 


.00500 


.000 275 


.01658 


.006 


.07746 


.000 030 


.00547 


.000 300 


.01732 


.007 


.08366 


.000 035 


.00592 


.000 325 


.01803 


.008 


.08944 


.000 040 


.00632 


.000 350 


.01871 


.009 


.09487 


.000 045 


.00671 


.000 375 


.01936 


.01 


.1000 


.000 050 


.00707 


.000 400 


.02000 


.02 


.1414 


.000 055 


.00742 


.000 450 


.02121 


.03 


.1732 


.000 060 


.00775 


.000 500 


.02236 


.04 


.2000 


.000 065 


.00806 


.000 550 


.02345 


.05 


.2236 


.000 070 


.00837 


.000 600 


.02449 


.06 


.2449 


.000 075 


.00866 


.000 650 


.02549 


.07 


.2646 


.000 080 


.00894 


.000 700 


.02646 


.08 


.2828 


.000 085 


.00921 


.000 750 


.02739 


.09 


.3000 


.000 090 


.00949 


.000 800 


.02828 


.1 


.3162 


.000 095 


.00975 


.000 850 


.02915 


.2 


.4472 


.000 100 


.01000 


.000 900 


.03000 


.3 


.5477 


.000 125 


.01118 


.000 950 


.03082 


.4 


.6324 


.000 150 


.01225 


.001 


.03162 


.5 


.7071 


.000 175 


.01323 


.002 


.04472 


.6 


.7746 


.000 200 


.01414 


.003 . 


.05477 


.7 


.8367 


.000 225 


.01500 


.004 


.06324 


.8 


.8944 


.000 250 


.01581 


.005 


.07071 


.9 


.9487 



Flow of Water. 



531 



Example. A wrought-iron pipe 3 inches diameter, = 0.25 foot, and 1000 
feet long, has a head of water of 20 feet. Required the velocity ? 
We have 

r = ^ = 0.06; 

4 



and the formula 



becomes 



v -- 



s = 0.02 ; 

: Crh? 

■■ 160 X 0.1533 X 0.1414 = 



■■ 3.47 feet per second. 



Again : A brick conduit is 7.5 feet in diameter, with a slope, s = 
Required the velocity ? 

Here 7 * 

r = ^= 1.875, 
*•» 4 
and we have „ x 

v = Cr s 6' 5 
=? 110 X 1.52 X 0.024 = 4.01 feet per second. 
The measured velocity in this conduit was 3.929 feet per second. 



= 0.00058. 



Discharge of Water from Smooth Wrought=iron Pipes. 

2. 1 

v = 160r 3 s^, times area. 
Cubic Feet per Second. 

For Cast-iron, multiply by 0.81; Lap Riveted, 0.72; Wood Stave, 0.78; 

Brick, 0.68. 

Diameter, in Inches. 



Slope = 
















head 


2 


4 


6 


8 . 


10 


12 


14 


length' 




^ 












.001 


.014 


.083 


.249 


.532 


.970 


1.591 


2.390 


.002 


.019 


.119 


.351 


.750 


1.370 


2.240 


3.375 


.003 


.023 


.145 


.429 


.936 


1.680 


2.752 


4.135 


.004 


.026 


.167 


.494 


1.062 


1.935 


3.180 


4.780 


.005 


.029 


.186 


.549 


1.179 


2.175 


3.555 


5.341 


.006 


.032 


.205 


.612 


1.280 


2.372 


3.900 


5.850 


.007 


.035 


.222 


.657 


1.405 


2.565 


4.200 


6.320 


.008 


.037 


.236 


.700 


1.500 


2.740 


4.500 


6.755 


.009 


.040 


.252 


.742 


1.590 


2.910 


4.770 


7.170 


.01 


.043 


.265 


.784 


1.675 


3.061 


5.026 


7.544 


.02 


.059 


.375 


1.080 


2.375 


4.330 


7.110 


10.670 


.03 


.069 


.458 


1.357 


2.812 


5.310 


8.720 


13.100 


.04 


.081 


.530 


1.567 


3.350 


6.122 


10.052 


15.088 


.05 


.094 


.593 


1.735 


3.650 


6.850 


11.250 


16.870 


.06 


.103 


.648 


1.920 


4.110 


7.490 


12.310 


18.500 


.07 


.111 


.695 


2.072 


4.340 


8.100 


13.300 


20.000 


.08 


.118 


.750 


2.220 


4.740 


8.660 


14.230 


21.700 


.09 


.125 


.795 


2.350 


5.022 


9.183 


15.078 


22.632 


.1 


.133 


.838 


2.480 


5.287 


9.70 


15.920 


23.90 


.2 


.187 


1.185 


3.505 


7.50 


13.71 


22.510 


33.80 


.3 


.230 


1.453 


4.290 


8.78 


16.77 


27.530 


41.35 


.4 


.265 


1.680 


4.805 


10.50 


19.38 


31.820 


47.80 


.5 


.293 


1.875 


5.523 


11.87 


21.65 


35.054 


53.20 


.6 


.325 


2.055 


6.08 


13.1 


23.42 


38.95 


58.35 



532 



Flow of Water. 



Discharge of Water from Smooth Wrought>iron Pipes, j 

2. 1 

v = 160r 3 s 2 , times area. J 

Cubic Feet per Second. -A 

For Cast-iron multiply by 0.81; Lap Riveted, 0.72; Wood Stave, 0.78; 

Brick. 0.68. 
Diameter, in Inches. 



Slope = 
















head 


16 


18 


20 


22 


24 


26 


28 


length' 
















.001 


3.390 


4.650 


6.230 


7.935 


10.000 


12.420 


15.130 


.002 


4.790 


6.575 


8.800 


11.240 


14.075 


17.550 


21.390 


.003 


5.865 


8.045 


10.770 


13.740 


17.250 


21.470 


26.170 


.004 


6.780 


9.310 


12.450 


15.875 


20.000 


24.750 


30.250 


.005 


7.580 


10.400 


13.920 


17.760 


22.700 


26.300 


33.750 


.006 


8.310 


11.400 


15.350 


19.450 


24.500 


30.400 


37.000 


.007 


8.970 


12.300 


16.460 


21.000 


26.475 


32.750 


39.900 


.008 


9.590 


13.150 


17.600 


22.450 


28.60 


35.050 


42.700 


.009 


10.175 


13.950 


18.670 


23.800 


30.00 


37.200 


44.300 


.01 


10.721 


14.701 


19.669 


25.091 


31.67 


39.163 


47.746 


.02 


15.150 


20.770 


27.80 


35.450 


44.70 


55.400 


67.40 


.03 


18.575 


25.470 


34.05 


43.450 


54.80 


67.900 


82.70 


.04 


21.442 


29.402 


39.34 


50.182 


63.34 


78.326 


95.50 


.05 


23.950 


32.870 


43.95 


56.100 


70.50 


87.50 


106.75 


.06 


26.230 


36.000 


48.15 


61.400 


77.40 


94.90 


117.00 


.07 


28.350 


38.850 


52.05 


66.400 


83.70 


103.60 


126.50 


.08 


30.300 


41.500 


55.60 


70.950 


89.40 


110.75 


135.00 


.09 


32.163 


44.103 


59.01 


75.274 


95.00 


115.50 


143.24 


.1 


33.800 


46.500 


62.05 


79.35 


100.00 


124.20 


151.3 


.2 


47.950 


65.750 


88.10 


112.40 


140.75 


175.50 


213.9 


.3 


58.700 


80.500 


107.75 


137.40 


172.50 


214.70 


261.7 


.4 


67.800 


93.000 


124.70 


158.75 


200.00 


247.50 


302.5 


.5 


75.800 


104.000 


138.25 


177.60 


227.00 


263.00 


337.5 


.6 


83.100 


113.900 


152.50 


194.50 


245.00 


304.00 


370.0 


- 


30 


32 


34 


36 


38 


40 


42 


.001 


18.175 


21.650 


25.35 


29.52 


34.10 


39.15 


42.4 


.002 


25.70 


30.650 


35.85 


41.75 


48.20 


55.30 


60.1 


.003 


31.45 


37.475 


43.80 


51.11 


59.00 


67.75 


73.5 


.004 


36.70 


43.35 


50.70 


59.00 


68.20 


78.3 


85.0 


.005 


40.60 


48.40 


57.65 


66.00 


76.20 


87.5 


95.0 


.006 


44.30 


53.00 


62.10 


72.30 


83.50 


95.9 


104.1 


.007 


48.10 


57.25 


67.00 


78.10 


90.20 


103.5 


112.3 


.008 


51.40 


61.25 


71.60 


83.50 


96.40 


110.6 


120.1 


.009 


54.50 


65.00 


76.00 


87.60 


102.25 


117.5 


127.4 


.01 


57.43 


68.15 


80.13 


93.35 


107.75 


123.7 


134.2 


.02 


81.20 


96.80 


113.25 


132.00 


152.50 


175.0 


190.0 


.03 


99.50 


110.86 


138.75 


161.75 


186.75 


214.5 


232.5 


.04 


114.85 


136.30 


160.26 


186.70 


215.50 


247.4 


268.4 


.05 


128.50 


153.20 


178.00 


208.70 


241.00 


276.5 


300.0 


.06 


140.75 


167.50 


196.25 


228.50 


264.00 


303.0 


328.5 


.07 


152.00 


181.20 


212.00 


247.00 


286.20 


327.0 


355.0 


.08 


162.50 


193.70 


220.65 


263.70 


304.70 


349.5 


379.5 


.09 


L72.28 


204.45 


240.4 


280.05 


323.25 


371.0 


402.6 


.1 


181.75 


216.50 


253.5 


295.2 


341.0 


391.5 


424.0 


,2 


257.0 


306.50 


358.5 


417.5 


482.0 


553.0 


601.0 


'.a 


314.5 


374.75 


438.0 


511.1 


590.0 


677.5 


735.0 


A 


367.0 


433.5 


507.0 


590.0 


682.0 


783.0 


850.0 


.5 


406.0 


484.0 


576.5 


660.0 


762.0 


875.0 


950.0 


.6 


443.0 


530.0 


621.0 


723.0 


835.0 


959.0 


1041.0 



Flow of Water. 



533 



be to m I-i to "to co lu b ^ b "to "^ ^t b '*> bo co ^j 'to bo co 



<I OS Cn C" cn Cn Cn jfe- ►&» rf*. ^ rf*. CO CO CO co co to to to to to 

b b bo b ^ "to b bo b Ip- "to b bo b Jp> to b bo b '►£■• to © 



JpOtOfDOOvKIOi 



^cooo<wqascncn^HP.^cococotot>otOh 

. ^ O) O CB H O) ^ <I C 
OiHOOOOtOMtOC 



Velocity, in feet, 
per second. 



Loss of head, 
in feet. 



Cubic feet 
per minute. 



Loss of head. 



CO^^<I<ia:asq©CnCnCn^^>P.Hfc>.COCOCOCOtOtO 



Cubic feet 
per minute. 



^CnCnCn^^^COCOCOtOtCtOtO ^ t -1 ! -1 ^ 1 ^^ 

b<i" 

to rfs* 



©qcoi-»i-'i-*cocnocco«oo5>e>-toto>^as<oco«o 

COOM 



Loss of head, 
in feet. 



p^^poi^^^wwbOHHop«opooo<i<ipiai 
b<j" 



O CO CO co to to to to to J- 

a en co b bo b '^ to b bo b en co to b b bo <i en 

:>Cnh-iOOCn>4^COkP.cn^I?OtO^JtOOO^>-'©cO 
en co toco 



Cubic feet 
per minute 



Loss of head 
in feet. 



gs^cocotototoi-* p_o_o_o <r> oooo<i^3asascncn4 



Cubic feet 
per minute, 



^MMMWtOtOMtOHMHH'MHH i 

b lu "to b bo as ^ to m b «^i b '►&»• co to b b bo ^a b en Ju 

^cn^co^cnas^i— 'cn«o*>-<ocnioeo--iascncnas^4 

00 00 ►*». ^J © H 1 ►£»• 



Loss of head 
in feet. 



Cn^hP-^^^^COCOCOCOCOCOtOtOtOtOtOtOl-'MI-- 1 
<j^^p»4^WOjO^OSJ^tOJ- l «0<IOJrf>".tOH»COQDas 

to f- 1 ^ bo to b b "to b b co <t © ffr. botobbcobb'co 



Cubic feet 
per minute, 



CC{OtOtOlOtOtOMH- l J-ih-'j--'H*j--ih-i 

bo bo <r b co "to 'o'<o '•<! b "^ co to h b co bo m b bi '^ co 

I— i^J©CO-J)-»Cn©astO<0-<ICnCOtO)— 'I— 'tOCO^ascO 



Loss of head 
in feet. 



oo^TasosasascncncncnhU^i^tCk^cocococotototo 

tsJOCOCHWMOOp^MtOMj^tOp^JCnWOGOOiW 

I&. <i co b b "to b b *H-i bo b ?-* ^ j^ b <r co b b "to b b 



Cubic feet 
per minute, 



co to to to to h 



Cn O tO cn cn 00 ~-3 O 00 h- 4 00 



Loss of head, 
in feet. 



©tO©00qCOtO©^CnCOMtQ<ICnCOI- t tO^ICnCO© 



Cubic feet 
per minute. 



tO tO tO J- 1 ^ £-» M M H-» M J-» J-i 

bo^bbo^osb'^blot-'bbbo^ibbb^^coto 
cncni— 'co^jcn^cototototocoi^qooi— i^nihoko 

KXJ05 00WOJMOCOOH03 



Loss of head, 
in feet. 



OOOOOOOOMI-'(D<tOi^bOOOCO>^(OI-'tg 






Cubic feet 
per minute. 



Doo^jasascna^axcocoto 

_. J COCnOOO^OOtOOil-'as 

<iacocoHOioco^ooiyto^ 



Loss of head, 
in feet. 



OOOOOOOOOOOOO^PQ0Q'(0O0)CCO 



Cubic feet 
| per minute, 



tO J-» h-» H- 1 

to ^r b b 



rfxcoroi-'OOOooo-crciascn^^cocototO 
i- 1 tocoi^en<rotO)^^j»-'^ooco^itoooco 

CncO^I^JtOcO<^OtOeOQOtOOO^J>— t ~J 



Loss of head, 
in feet. 



OOOOOOOOOOOOOOOOtOOsi— 'QiQ^ 



Cubic feet 
per minute. 



■> O CO 00 
3 rf^OsOO 
2OC0 00 



oo<iqsq. cn»£»>*»-cocotototo 



Loss of head, 
in feet. 



to to to to 

^.1 co to to 

<I^3 CO tO 



to to i- 1 *- 

rf^OiOOC 



OOOOOOOOOO O.O OOOOOOOOl- 



Cubic feet 
per minute. 



* I— i I— ' h- 1 I— » h- ' t- 



SCO 00 0C 
hfi* CO Cn 



-JOsOsCnCnos-^COCOtOtOM 
>£>. 00 tO — ' ii O 05 M <I W O 
tO Cn g a. O ~j ^3 O Cn CO ^- 00 



Loss of head. 
in feet. 



co to to ro 

COOC^-TOs 

pcocojt^ 
bbbbooo 



to to tc 

en 4- Zjj 
^ Cn Cn 



r c to i o 
tO'-'O 
Os <I^J 

bob 



oooooooooooto 



Cubic feet 
per minute. 



oro. 13* 



^crq 

a* 

p DO 

go tr* 
3 o 

&* 

° — 
So 

Bo- 



5" 



U ?. » 2 

O 

o 



P 0> 



SB -o 



3 q 



£ r^ 



O 

3 






P 

P 

P^ 



534 



Flow of Water. 



ojuutui jad 
;ooj oiqno 



COCOOOOQOOOt>M>Mt>McOHCOH(OH500iC» 

^COHOOOt»iO^WHa5COOiOCC(NOOil^tOTl<!D 

QC 05 O H H M CO T< iO O C ^ 00 O) O H O) M « 1< lO X 

HHHHHHHHHHHt-lNM(M(NfNiMM(M 



"j)90J UI 

pisoq jo ssoi 



CDCOHTf*05iC(MOSOOI>OOOiH^XMX'^H00550 
«0 1> OS O ri CC iC '^ X O ri t N O) H -r X Oi M iO t^ M 
OOOHHHHHH(MC^W(N(MCOWCOCC'*t)<t}<0 







ajimira jod 
^ooj oxqiio 



OSXl^-'Xl^mMHOaiXr-OiO^^lHOOiXl^H 

iO<Dl>I>XXO>OOHH(MO)W*^iOiO<£:L^NO 
i-liHi-»i-lr-li-lr-li-lr-li-lT-lr-ii-li-HCN 



HCOtOO^M 



•o)aoj ui 
pisoq jo ssoi 



rfi as c 

ODOSr 
O Or 



JXNI>MOXX050Ml^C>lXiOI> 
1Ht)*C005!N^NO'#1>0'^1>HH 
H<M(MOaC^COCOCCTTiTriTtliOlOiCtOGO 



ojnuiuuod 
jooj oiqno 



CN cOr 

-r X c 



iocs 

t^i-H 



COCO 
CO O 



lOOSrJHXiMl^HCiOiOOCCX'MTt* 
l>t>XX050>OOHHHM(MMiO 



*J09J HI 

p-eoq jo ssoi 



HHHHrflN 



iCHOOiCiCiCt>-OTf<OCOCO(NiMO> 
CCCOXHTft^O-^t^H^XINCir^ 
C^<M(MCCCOCOTTlTrlTnLOiOiOCOCOOO 



ojnuuu jod 
*99J oiqno 



^TXNOXiCMHXtD^Hast^iCMOXiCCOHOl 
l^HiOOilNOOTOHiOOKN^O^XHiOOlCOH 
M-^^t)*lOiC?DCD01>1>I>XX05010)OOOHCO 



'^99J UI 

P'eoq jo ssoi 



X50(DNOTfiOkOM(NM'*t^H!OM01H(Nrfl>»M 
OHCOiCXOCMiCXH^t^O^l^HiCOCONHiC 
OrHiHrHi-ICNCNCNCNCOCOCO^Ttl^iOiOiOCOCOl>as 



oo)nuiui J9d 
;99j oiqno 



5DXON«iONXOHCOinNXO(MCOiCI>XOa> 



cococoTFT*rfiiOioioco<©coco£^i>i>aoaoco0scsr-i 



'J99J UI 

: pi3oq jo ssot; 



XN05HiO(N05XXOM«^P)HH(NiOOiO(NO 
OMTt<Na3(M^I^O^t^O-*X(M^O'^05COX'<J< 
i— ItHiHi— lrHCNCNcNCOCOCOTTiTriTtHLOtOCOCOC0 1>l>0 



9jimixn jgd 
^99j oiqno 



MX^O^OMOSLOHt^WOOMX-^Ot^MOiiC^ 
OXH^«005HTft^05(MtOt^OMLOXOMiCXH 

(M<Mcocococoxj<TriTtiTj<iOiOLO«r>^o^c>?r>t^i>i>-t^a5 



"o)OOJ UI 

pisoq jo ssoq; 



a)OTf*OitDiOiOcOOiTj(003XOi(NI>C^OXaiHCO 
H^«OXHTt<NOCONHfXC x H>HCDHiOOCO'* 

HHHHCiri:i:c:::o^TjHTt<iCiCcDOt^^xXH 



ojnuiui J9d 
*99j oiqno 



1—1 U'J 11.^ I.'- *— H 1 1 U'J ^i-' ^*^ *— ' .N '"T ^k^ ^,' ^ — J V»< «-'.*' 1"" W 1 ' UW ^J 1 



*^99J III 

pi3oq jo ssoq; 



(NtONOOHiOO)NffitDOi(NX<OiO!DMMO>> 

coiocx)th^i^ocoi^i— iioas^cocococoaoTfiasiot^ 



o^nniin jod I 
jooj oiqno 



I^TfiiHCO^i-ICO^i-ICOiOeMCCiOCNaSiOCNascOCNcO 
tOXOHCOiO(CXOHMiO«OXOHO:iO©XOX 
rHiHCNCO CN<^COCNCOCOCOCOCOCO^^T*Ttl^TfiOiO 



•J89J UI 

p^oq jo ssoi 



NiCiO«DO«DmcOiOXCOHHCMiOOXXO>HCi 

TfNOCOt^O-fXM'OHCDHCOHl^C-lX-^Hl^M 

HH(NM(NC0C000Tt<T}<i0iO<£i«:i>l>XX05OOTf 



o^nuira jed 
%^l oiqno 



N(M^OHtDHiOOiOOTfHa>Tf*aiCOXCON(NN(NiC 



•;ooj UI 
'p'eoq jo ssoi 



Xt^XMXiOcDX'MOSXXH^fOMCO^HXX 

iOXHiCXCJ©OiOO>'f05iOO©(MXt<Ht^'*(N 

HH(NN(MWM'^rf«T)<iOiC!OI>l>XX0500HiO 



omnium aod 
i)ooj oiqno 



XH*fN05Mi(5XHCOC0 05MiOXH«tOOiMinO> 

(M -f irr x> n Oi o ^ co -f io -^ x J3 o w m i* ic n x ^ 

HHHHHH(NCN(N(N(NCM^OlCOCOCOC0MMM1* 



•^99j at 

'prcoq jo ssot; 



OlO'fOXa)0)XiCiONHXNXHN'*COi005 
CO O CO 1-^ O -M OS CO' X CO <X -f OS iO i— I CO -f i— < CO iO CN CO 

HONlNiMCOCOCOt<T)iiOiO(O^Ot^XXaiOOHMtO 



oiniuiujod 
ojooj oiqno 



HCNCNCNCNCNCNCNC 



>ioj ui 
'pi3oq jo ssoi 



CO CO CO O ON iC 00 i-l Ol CO CN i-i i-H iO t-< as O CI l^ \o 

x i—i io as co i- oi i^ oi i- co oi io h x t c-i 05 (O i* 

HCNatMOOCO^TfiOiOcOCDl^XXOSOOHlN 



•puooos jod 

'o)09J UI 'iCjIOOlOA 



q^T(jccQOO(NTfcocoo(N^coQoq(NTij?ooqqo 

Ol CN ON CN CN CO* CO* CO CO CO Tj5 Tin t* ■<*' «*' tO tO tO \0 UC CO r^ 



Flow of Water. 



535 



Flow of Water in Open Channels. 

In computing the flow of water in channels, canals, rivers, ditches, etc., 
the form of the Chezy formula is retained, the various working formulas 
* being arranged to permit the value of the coefficient, C, in the formula, 

v = C\/rs, 

to be determined for the various classes of channels. 

In France the formula of Bazin is generally used, as follows : 



157.6 



-]/ rs, for English measures ; 



V~ 



87 



1 + 



j/ rs, for metric measures. 



|/ r 



In these formulas 7 is a coefficient dependent upon the character of the 
wetted surface ; r is the hydraulic radius, or cross-section, divided by the 
wetted perimeter ; and s is the slope, or sine of the angle of inclination. 

Bazin divides channels into six classes, with a value of 7 for each. 



Class. 



Character of wetted surface. 



Feet. Metres. 



I. 

II. 
III. 
IV. 

V. 
VI. 



Smooth cement, planed wood 

Planks, bricks, cut masonry, etc 

Rubble masonry 

Earth, dry rubble, etc 

Earthen channels in ordinary condition 

Earthen channels or rivers, with stony beds and 
grassy banks 



.109 
.290 



1.540 
2.355 



3.170 



.06 
.16 
.46 
.85 
1.30 

1.75 



Although the formula appears complicated, it is not difficult of applica- 
tion, and its use may be simplified by the use of the diagram on page 536, 
which is for the metric system, and is due to M. Soreau. 



Diagram for Flow of Water. 

Bazin's Formula. 
Metric System. 



87 



1 + 



j/rs. 



v* 



Join 7 to r. Then draw a line parallel to this through s, and it will 
intersect v at the velocity value. In the diagram, r = 4, 7 = 1.30, s = 0.004, 
and v is found to be 6.63 metres per second. 

In Switzerland, Germany, and to some extent in the United States and 
in England, Kutter's formula is used. This is also in the Chezy form, and 
consists of a rather complicated expression for the value of the coefficient, 
C, in the formula, 

v = CV rs, 



536 



Plow of Water. 



s being the slope of the stream, and r the hydraulic radius, or cross-section, 
divided by the wetted perimeter. In the English measure r is taken in 




*.5 



Diagram for Bazin Formula. (See page 535.) 



feet,— i.e., the cross-section in square feet is divided by the wetted perimeter 
in feet. In the metric system the cross-section is taken in square metres 
and the wetted perimeter in metres. The Kutter formula, then, is 



41. 6 + -^L + i^L 



c= 



/., . , .00281 \ 
(41.6 + — — )n 



ICnglish system; 



1 + 



Yr 



C= — — „„, Fg — , metric system. 



1 + 



(« + -sa-.). 



/. 



Plow of Watek. 537 



In this formula the quantity, n, is a factor, the value of which depends 
upon the character of the channel. The value of n to be used in the 
formula may be taken from the following list : 

Artificial Channels, Uniform Section. 

Surface. n 

Planed boards 009 

Cement, neat 010 

Plaster, 3 cement, 1 sand Oil 

Rough boards 012 

Ashlar, or brickwork 013 

Rubble 017 

Natural Channels. 

Canals in very firm gravel 020 

Canals and rivers in fairly good order, free from stones 

and weeds 025 

Canals and rivers with occasional stones and weeds 030 

Streams in bad condition, with many stones and weeds . . .035 

The whole subject of the derivation and use of Kutter's formula, with 
many examples, are given in the book entitled, "A General Formula for 
the Uniform Flow of Water," by Ganguillet and Kutter, translated by 
Hering and Trautwine. In order to avoid the tedious computations with 
the formula, values of C are computed and tabulated ; but sufficient pre- 
cision may be attained by the use of a diagram, which is appended. This 
is a modification, by M. Soreau, of the original diagram by Kutter, the 
change being only to put it in more convenient form for the page. The 
diagram is for use in the metric system. 

The use of the diagram will be best understood by an example. Taking 
the same data as were used with the Bazin formula, page 535, let r = 4 and 
8 = .004. For a canal in fairly good condition take n = .025. We then join 
4, on the horizontal line, r, with .025, on the curve. The intersection of 
the dotted line with the inclined scale gives the value, C= 49. We then 

have 

v = 49|/4 X .004 = 49-j/ .016 = 6.17 metres per second. 

Diagram for Flow of Water. 

(See page 538.) 

Kutter's Formula. 

Metric System. 

Join point on line, r, corresponding to given hydraulic radius, with 
point on the curves, corresponding to given values of s and n. The inter- 
section with the inclined line gives the value of C in the formula, 



cVr 



The formula of Tutton, as used for pipes, may be modified for open 
channels, as follows : 

1.54 a i 

v = r 3 s 2 . 

n 

in which n is the same as in Kutter's formula, English measures being 
used. This has the advantage of greater simplicity, and gives equally 
reliable results. 

The difficulty with all these formulas lies in the fact that the flow 
depends to a great extent upon the condition of the channel, and there- 
fore upon the selection of the coefficient of roughness, n. 

Whenever possible, the actual velocity of the stream should be meas- 
ured, computations based upon assumptions as to slope and condition of 
roughness being made only for canals and ditches prior to construction. 



538 



Flow of Water. 



0.000025 
0.000030- 

0.00005< 




O.00O05 

000\20^,\ 

.0002^^ 
0.00030°- 

O.C0050' 
«g01 



Diagram for Kutter's Formula. (See page 537.) 

The following details of measurements represent the practice of the 
Pelton Water-wheel Company, and are based on large experience : 

Select a stretch on the stream or ditch which will afford as straight and 
uniform a course as possible. If the water is at any point carried in a 
flume, it is better to measure at this point. Lay off a distance of, say, 300 
feet; measure the width of flowing water at about six different places in 
this distance, and obtain the average width; likewise, at these same points, 
measure the depth of water at three or four places across the stream, and 
obtain the average depth. Next, drop a float in the water, noting the 
number of seconds it takes to travel the given distance. From this can be 
calculated the velocity of the water, in feet, per second. The quantity is 
the product obtained by multiplying the average width, in feet, by the 
average depth, in feet, by the velocity, which (if in feet per second) will 
give the flow of the stream, in cubic feet, per second. From the figures so 
obtained it is advisable to deduct about 20 per cent., as surface velocity of 
the water is in excess of the actual average velocity. 

When the stream is of sufficient depth— say 3 feet or over— the average 



Plow of Water. 



539 



velocity can be more closely obtained by using a pole, to one end of which 
is attached a stone or piece of lead of necessary weight to allow the pole 
to sink nearly to the bottom. In this way the velocities at the surface and 
bottom of the stream counteract one another, and a closer approximation 
»of the average velocity is obtained. 

The most accurate method of measuring the volume of water flowing 
in a stream is by the use of a weir. 

The principle of the weir is that for a notch of given dimensions and 
determinate head of water the flow through it is constant and uniform. 
If, therefore, the flow of a stream can all be caused to pass through a notch 
of a certain shape, the volume can be determined from the size of the 
notch and the depth of the water. 

The general arrangement of a weir will be seen in the illustration, the 
dimensions being determined by the volume of water flowing in the 
6tream. The width of the notch can be carefully measured before it is set 
in place, and the depth of water measured afterwards. 




General arrangement of Weir. 

The instructions of the Pelton Water-wheel Company are as follows : 
Place a board or plank in the stream, as shown in the drawing, at some 
point where a pond will form above. The length of the notch in the dam 
should be from two to four times its depth for small quantities, and longer 
for large quantities. The edges of the notch should be bevelled towards 
the intake side, as shown. The overfall below the notch should not be 
less than twice its depth,— that is, 12 inches, if the notch is 6 inches deep, 
and so on. 

In the pond, about 6 feet above the dam, drive a stake, and then 
obstruct the water until it rises precisely to the bottom of the notch, and 
mark the stake at this level. Then complete the dam so as to cause all the 
water to flow through the notch, and, after allowing time for the water to 



540 



Flow of Water. 



settle, mark the stake again for this new level. If preferred, the stake can 
be driven with its top precisely level with the bottom of the notch, and the 
depth of the water be measured with a rule after the water is flowing free ; 
but the marks are preferable, in most cases. 

The theoretical quantity of water passing over a weir is given by the* J 
formula, 

Q = %Vlg . bHi, 

in which b is the width of the notch, or the length of the weir ; H, the 
depth of water; g, the acceleration of gravity, -■= 32.2. 

The actual quantity of water has been determined by numerous experi- 
ments. According to Francis, we may use 

Q = 3.33&tf§ = 3.S3bHi/H, 

H and b both being taken in feet. 

The following table also may be used. 



Table for Weir Measurement. 

Pelton Water-wheel Company. 

Giving Cubic Feet of Water per Minute that will Flow over a Weir 1 inch 
wide and from % to 20% inches deep. 



Inches. 


Vs 


M 


% 


V* 


% 


% 


Vs 





.00 


.01 


.05 


.09 


.14 


.19 


.26 


.32 


1 


.40 


.47 


.55 


.64 


.73 


.82 


.92 


1.02 


2 


1.13 


1.23 


1.35 


1.46 


1.58 


1.70 


1.82 


1.95 


3 


2.07 


2.21 


2.34 


2.48 


2.61 


2.76 


2.90 


3.05 


4 


3.20 


3.35 


3.50 


3.66 


3.81 


3.97 


4.14 


4.30 


5 


4.47 


4.64 


4.81 


4.98 


5.15 


5.33 


5.51 


5.69 


6 


5.87 


6.06 


6.25 


6.44 


6.62 


6.82 


7.01 


7.21 


7 


7.40 


7.60 


7.80 


8.01 


8.21 


8.42 


8.63 


8.83 


8 


9.05 


9.26 


9.47 


9.69 


9.91 


10.13 


10.35 


10.57 


9 


10.80 


11.02 


11.25 


11.48 


11.71 


11.94 


12.17 


12.41 


10 


12.64 


12.88 


13.12 


13.36 


13.60 


13.85 


14.09 


14.34 


11 


14.59 


14.84 


15.09 


15.34 


15.59 


15.85 


16.11 


16.36 


12 


16.62 


16.88 


17.15 


17.41 


17.67 


17.94 


18.21 


18.47 


13 


18.74 


19.01 


19.29 


19.56 


19.84 


20.11 


20.39 


20.67 


14 


20.95 


21.23 


21.51 


21.80 


22.08 


22.37 


22.65 


22.94 


15 


23.23 


23.52 


23.82 


24.11 


24.40 


24.70 


25.00 


25.30 


16 


25.60 


25.90 


26.20 


26.50 


26.80 


27.11 


27.42 


27.72 


17 


28.03 


28.34 


28.65 


28.97 


29.28 


29.59 


29.91 


30.22 


18 


30.54 


30.86 


31.18 


31.50 


31.82 


32.15 


32.47 


32.80 


19 


33.12 


33.45 


33.78 


34.11 


34.44 


34.77 


35.10 


35.44 


20 


35.77 


36.11 


36.45 


36.78 


37.12 


37.46 


37.80 


38.15 



Example. Suppose the weir to be 66 inches long, and the depth of water 
on it to be 11% inches. Follow down the left-hand column of the figures . 
in the table until you come to 11 inches. Then run across the table on a 
line with the 11 until under %, on top line, and you will rind 15.85. This, 
multiplied by 66, the length of weir, gives 1046.10, the number of cubic 
feet of water passing per minute. 



Flow of Water. 



541 



The Miner's Inch. 

The term Miner's Inch is used in a number of Western States, being 
used in the measurement of water for mining and irrigation. The term is 
more or less indefinite, for the reason that the water companies do not all 
use the same head above the centre of the aperture, and the inch varies 
from 1.36 to 1.73 cubic feet per minute each ; but the most common measure- 




ment is through an aperture 2 inches high and whatever length is re- 
quired, and through a plank 1% inches thick, as shown in the illustration. 
The lower edge of the aperture should be 2 inches above the bottom of the 
measuring box, and the plank 5 inches high above the aperture, thus 
making a 6-inch head above the centre of the stream. Each square inch 
of this opening represents a miner's inch, which is equal to a flow of 1% 
cubic feet per minute. 

The use of the miner's inch is to be discouraged, because of its indefi- 
nite value. In some States its legal value has been made 1.5 cubic feet per 
minute. 



542 



Water-power. 



Tables for Calculating the Horse=power of Water. 



Miner's Inch Table. 




Cubic Feet Tabic 




The 


following 


table gives the 


The following: table 


gives the 


horse-power ot l 


miner's 


men 01 


horse- 


power of 


1 cubic foot of 


water under heads from 


1 up to 


water 


per minute under heads 


1100 feet. This 


men equals 1>£ 


from ] 


up to 1100 feet. 




cubic feet per minute. 












Heads, 


Horse- 


Heads, 


Horse- 


Heads, 


Horse- 


Heads, 


Horse- 


in feet. 


power. 


in feet. 


power. 


in feet. 


power. 


in feet. 


power. 


1 


.002 4147 


320 


.772 704 


1 


.001 6098 


320 


.515 136 


20 


.048 2294 


330 


.796 851 


20 


.032 196 


330 


.531 234 


30 


.072 441 


340 


.820 998 


30 


.048 294 


340 


.547 332 


40 


.096 588 


350 


.845 145 


40 


.064 392 


350 


.563 430 


50 


.120 735 


360 


.869 292 


50 


.080 490 


360 


.579 528 


60 


.144 882 


370 


.893 439 


60 


.096 588 


370 


.595 626 


70 


.169 029 


380 


.917 586 


70 


.112 686 


380 


.611 724 


80 


.193 176 


390 


.941 733 


80 


.128 784 


390 


.627 822 


90 


.217 323 


400 


.965 880 


90 


.144 892 


400 


.643 920 


100 


.241 470 


410 


.990 027 


100 


.160 980 


410 


.660 018 


110 


.265 617 


420 


1.014 174 


110 


.177 078 


420 


.676 116 


120 


.289 764 


430 


1.038 321 


120 


.193 176 


430 


.692 214 


130 


.313 911 


440 


1.062 468 


130 


.209 274 


440 


.708 312 


140 


.338 058 


450 


1.086 615 


140 


.225 372 


450 


.724 410 


150 


.362 205 


460 


1.110 762 


150 


.241 470 


460 


.740 508 


160 


.386 352 


470 


1.134 909 


160 


.257 568 


470 


.756 606 


170 


.410 499 


480 


1.159 056 


170 


.273 666 


480 


.772 704 


180 


.434 646 


490 


1.183 206 


180 


.289 764 


490 


.788 802 


190 


.458 793 


500 


1.207 350 


190 


.305 862 


500 


.804 900 


200 


.482 940 


520 


1.255 644 


200 


.321 960 


520 


.837 096 


210 


.507 087 


540 


1.303 938 


210 


.338 058 


540 


.869 292 


220 


.531 234 


560 


1.352 232 


220 


.354 156 


560 


.901 488 


230 


.555 381 


580 


1.400 526 


230 


.370 254 


580 


.933 684 


240 


.579 528 


600 


1.448 820 


240 


.386 352 


600 


.965 880 


250 


.603 675 


650 


1.569 555 


250 


.402 450 


650 


1.046 370 


260 


.627 822 


700 


1.690 290 


260 


.418 548 


700 


1.126 860 


270 


.651 969 


750 


1.811 025 


270 


.434 646 


750 


1.207 350 


280 


.676 116 


800 


1.931 760 


280 


.450 744 


800 


1.287 840 


290 


.700 263 


900 


2.173 230 


290 


.466 842 


900 


1.448 820 


300 


.724 410 


1000 


2.414 700 


300 


.482 940 


1000 


1.609 800 


310 


.748 557 


1100 


2.656 170 


310 


.499 038 


1100 


1.770 780 



When the Exact Head is Found in Above Table: 

Example. Have 100-foot head and 50 inches of water. How many 
horse-power? 

By reference to above table the horse-power of 1 inch under 100-foot 
head is .241470. This amount, multiplied by the number of inches, 50, will 
give 12.07 horse-power. 

When Exact Head is Not Found in Table: 

Take the horse-power of 1 inch under 1-foot head and multiply by the 
number of inches, and then by number of feet head. The product will 
be the required horse-power. 

The above formula will answer for the cubic feet table by substituting 
the equivalents therein for those of miner's inches. 

Note.— The above tables are based upon an efficiency of 85 per cent. 



Contents of Pipes. 



543 



Contents, in Cubic Feet and United States Gallons, of 

Pipes and Cylinders of Various Diameters and 

1 Foot in Length. 

1 gallon = 231 cubic inches. 1 cubic foot = 7.4805 gallons. 





For 1 foot 


in length. 


, *RO 


a 


For 1 foot 


in length. 




a 










*£ © 




Cubic 


United 


- «M © O 


"1 


Cubic 


United 


<D .3 OH O 


® DC 

e3 G 


feet ; 
also, area 


States 
gallons 


0> aJ 


feet ; 
also, area 


States 
gallons 


in square 


231 cubic 


in square 


231 cubic 


ft" 


feet. 


inches. 


H 


ft" 


feet. 


inches. 




.0003 


.0025 




6 


.1963 


1.469 


61.13 


.0005 


.0040 




6% 
6% 
6% 
6% 
6% 
6% 
6% 
7 


.2046 


1.531 


58.65 


.0008 


.0057 




.2131 


1.594 


56.31 




.0010 


.0078 




.2217 


1.662 


54.01 


.0014 


.0102 




.2304 


1.724 


52.08 


.0017 


.0129 




.2394 


1.791 


50.13 


P 
i 

15 


.0021 


.0159 




.2485 


1.859 


48.29 


.0026 


.0193 




.2578 


1.928 


46.55 


.0031 


.0230 




.2673 


1.999 


44.89 


.0036 


.0269 




7% 

7% 


.2769 


2.071 


43.34 


.0042 


.0312 




.2867 


2.145 


41.86 


.0048 


.0359 




.2967 


2.219 


40.45 


l" 


.0055 


.0408 


"2i8i.8l" 


.3068 


2.295 


39.11 


1% 


.0069 


.0516 


1739.13 


7% 


.3171 


2.372 


37.84 


IK 
1% 


.0085 


.0638 


1411.76 


7% 


.3276 


2.450 


36.63 


.0103 


.0770 


1165.04 


m 


.3382 


2.530 


35.48 


.0123 


.0918 


975.69 


8 


.3491 


2.611 


34.37 


.0144 


.1077 


833.33 


8% 


.3601 


2.694 


33.32 


I'M 


.0167 


.1249 


718.56 


8% 
3% 


.3712 


2.777 


32.33 


.0192 


.1436 


625.00 


.3826 


2.862 


31.36 


2 


.0218 


.1632 


550.44 


8$ 


.3941 


2.948 


30.45 


2% 


.0246 


.1840 


487.80 


m 


.4057 


3.035 


29.58 


1 


.0276 


.2066 


434.76 


&A 
m 


.4176 


3.125 


28.74 


.0308 


.2304 


389.52 


.4296 


3.214 


27.93 


2% 


.0341 


.2550 


351.84 


9 


.4418 


3.305 


27.16 


2% 


.0376 


.2813 


319.14 


9% 


.4541 


3.397 


26.43 


1 


.0412 


.3085 


291.26 


9M 
«% 


.4667 


3.491 


25.71 


.0451 


.3374 


266.07 


.4794 


3.586 


25.03 


8 


.0491 


.3672 


244.39 


9 <^ 


.4922 


3.682 


24.38 


3% 


.0533 


.3987 


225.14 


9/8 


.5053 


3.780 


23.75 


3% 


.0576 


.4309 


208.33 


l 


.5185 


3.879 


23.14 


3% 


.0621 


.4645 


193.23 


.5319 


3.979 


22.56 


i 


.0668 


.4998 


178.14 


10 


.5454 


4.080 


22.00 


.0717 


.5361 


167.36 


10% 


.5591 


4.182 


21.46 


3% 

3% 


.0767 


.5738 


156.45 


iom 
1051 


.5730 


4.286 


20.94 


.0819 


.6127 


146.52 


.5871 


4.392 


20.44 


4 


.0873 


.6528 


137.43 


10% 


.6013 


4.498 


19.96 


4% 


.0928 


.6942 


129.31 


io-% 


.6157 


4.606 


19.49 


4% 
4% 

4% 

4% 

1 


.0985 


.7369 


121.82 


10% 

ioj| 


.6303 


4.715 


19.04 


.1044 


.7810 


114.94 


.6450 


4.825 


18.60 


.1104 


.8263 


108.69 


11 


.6600 


4.937 


18.18 


.1167 


.8727 


102.82 


11% 


.6751 


5.050 


17.78 


.1231 


.9206 


97.50 


11% 

11% 


.6903 


5.164 


17.38 


.1296 


.9695 


92.59 


.7057 


5.279 


17.00 


5 


.1364 


1.020 


87.98 


11% 
11% 

Hi 


.7213 


5.396 


16.63 


5% 


.1433 


1.072 


83.74 


.7370 


5.513 


16.28 


5% 

5% 


.1503 


1.125 


79.84 


.7530 


5.633 


15.94 


.1576 


1.179 


76.14 


.7691 


5.753 


15.60 


1 


.1650 


1.234 


72.73 


12 


.7854 


5.875 


15.28 


.1726 


1.291 


69.52 


12% 


.8018 


5.998 


14.94 


m 


.1803 


1.349 


66.56 


12i| 
12% 


.8184 


6.122 


14.66 


w 


.1883 


1.409 


63.72 


.8352 


6.248 


14.37 



















544 



Contents of Pipes. 



Contents, in Cubic Feet and United States Gallons, of 
Pipes and Cylinders of Various Diameters and 

1 Foot in Length. —Continued. 
1 gallon = 231 cubic inches. 1 cubic foot = 7.4805 gallons. 





For 1 foot 


in length. 


?th, in 
ties, of 
inder of 
ibic foot 
acity. 


«3 C 


For 1 foot 


in length. 




a-S 


Cubic 

feet ; 

also, area 


United 

States 
gallons 


Cubic 

feet ; 

also, area 


United 
States 
gallons 




in square 


231 cubic 


h3 


in square 


231 cubic 


?1k § £ 


Q*~ 


feet. 


inches. 


5" 


feet. 


inches. 


1> .m OH O 


12% 


.8522 


6.375 


14.080 


21% 
21% 
21% 


2.463 


18.42 


4.872 


12^1 


.8693 


6.503 


13.800 


2.521 


18.86 


4.760 


1% 


.8866 


6.632 


13.530 


2.580 


19.30 


4.651 


12% 


.9041 


6.763 


13.270 


22 


2.640 


19.75 


4.545 


13 


.9218 


6.895 


13.020 


22% 


2.700 


20.20 


4.445 


13% 


.9395 


7.028 


12.780 


22% 
22% 


2.761 


20.66 


4.347 


13% 
13% 


.9575 


7.163 


12.530 


2.823 


21.12 


4.251 


.9757 


7.299 


12.300 


23 


2.885 


21.58 


4.160 


13% 

13% 


.994 


7.436 


12.070 


23% 


2.948 


22.05 


4.070 


1.013 


7.578 


11.850 


23% 


3.012 


22.53 


3.990 


% 


1.031 


7.712 


11.640 


23% 


3.076 


23.01 


3.901 


1.051 


7.855 


11.420 


24 


3.142 


23.50 


3.819 


14 


1.069 


7.997 


11.230 


25 


3.409 


25.50 


3.520 


14% 


1.088 


8.139 


11.030 


26 


3.678 


27.58 


3.263 


14% 


1.107 


8.281 


10.840 


27 


3.976 


29.74 


3.018 


14% 


1.127 


8.431 


10.650 


28 


4.276 


31.99 


2.806 


1.147 


8.578 


10.460 


29 


4.587 


34.31 


2.616 


1.167 


8.730 


10.280 


30 


4.909 


36.72 


2.444 


1.187 


8.879 


10.110 


31 


5.241 


39.21 


2.290 


1.207 


9.029 


9.940 


32 


5.585 


41.78 


2.149 


15 


1.227 


9.180 


9.780 


33 


5.940 


44.43 


2.020 


15% 


1.248 


9.336 


9.620 


34 


6.305 


47.16 


1.903 


m 


1.268 


9.485 


9.460 


35 


6.681 


49.98 


1.796 


1.289 


9.642 


9.310 


36 


7.069 


52.88 


1.698 


15% 

15% 


1.310 


9.801 


9.160 


37 


7.467 


55.86 


1.607 


1.332 


9.964 


9.010 


38 


7.876 


58.92 


1.527 


15 % 


1.353 


10.121 


8.870 


39 


8.296 


62.06 


1.446 


15% 


1.374 


10.278 


8.730 


40 


8.727 


65.28 


1.375 


16 


1.396 


10.440 


8.600 


41 


9.168 


68.58 


1.309 


16% 


1.440 


10.772 


8.330 


42 


9.621 


71.91 


1.247 


16% 


1.485 


11.11 


8.081 


43 


10.085 


75.44 


1.190 


16^| 


1.530 


11.45 


7.843 


44 


10.559 


78.99 


1.136 


17 


1.576 


11.79 


7.511 


45 


11.045 


82.62 


1.087 


1| 

17% 


1.623 


12.14 


7.394 


46 


11.541 


86.33 


1.040 


1.670 


12.49 


7.186 


47 


12.048 


90.13 


.996 


1.718 


12.85 


6.985 


48 


12.566 


94.00 


.955 


18 


1.768 


13.22 


6.787 


49 


13.095 


97.96 


.916 


18% 


1.817 


13.59 


6.604 


50 


13.635 


102.00 


.880 


1.867 


13.96 


6.427 


51 


14.186 


106.12 


.846 


1.917 


14.34 


6.259 


52 


14.748 


110.32 


.814 


19 


1.969 


14.73 


6.094 


53 


15.320 


114.60 


.783 


19% 


2.021 


15.12 


5.938 


54 


15.904 


118.97 


.755 


11 


2.074 


15.51 


5.786 


55 


16.499 


122.82 


.727 


2.128 


15.92 


5.639 


56 


17.104 


127.95 


.702 


20 


2.182 


16.32 


5.500 


57 


17.720 


132.55 


.677 


20% 
20% 


2.237 


16.73 


5.365 


58 


18.347 


137.24 


.654 


2.292 


17.15 


5.236 


59 


18.985 


142.02 


.632 


20% 


2.348 


17.56 


5.110 


60 


19.637 


146.89 


.611 


21 


2.405 


17.99 


4.989 











To find the capacity of pipes greater than the largest given in the table, 
look in the table for a pipe one-half the given size and multiply its capacity 
by 4, or one of one-third its size, and multiply its capacity by 9, etc. 



Water-wheels. 



545 



Table for Tank Measurement. 

Giving the Number of Cubic Feet of Water Discharged per Minute through 
an Orifice 1 inch square under any Head of Water from 3 to 72 inches. 



a 
M 


Cubic feet 
discharged 
per minute. 


a 
"1 °° 


is -2 

— ■- c 

3 T3 ft 

O 


I -2 • 

! s§ 


Cubic feet 
discharged 
per minute. 


a 

«f ° 

a; a 


Cubic feet 
discharged 
per minute. 


a 

■73 -a 
i a 


Cubic feet 
discharged 
per minute. 


3 


1.12 ! 


17 


2.51 


31 


3.36 


45 


4.05 


59 


4.63 


4 


1.27 


18 


2.58 


32 


3.41 


46 


4.09 


60 


4.65 


5 


1.40 


19 


2.64 


33 


3.47 


47 


4.12 


61 


4.72 


6 


1.52 


20 


2.71 


34 


3.52 


48 


4.18 


62 


4.74 


7 


1.64 


21 


2.78 


35 


3.57 


49 


4.21 


63 


4.78 


8 


1.75 


22 


2.84 


36 


3.62 


50 


4.27 


64 


4.81 


9 


1.84 


23 


2.90 


37 


3.67 


51 


4.30 


65 


4.85 


10 


1.94 


24 


2.97 


38 


3.72 


52 


4.34 


66 


4.89 


11 


2.03 


25 


3.03 


39 


3.77 


53 


4.39 


67 


4.92 


12 


2.12 


26 


3.08 


40 


3.81 


54 


4.42 


68 


4.97 


13 


2.20 


27 


3.14 


41 


3.86 


55 


4.46 


69 


5.00 


14 


2.28 


28 


3.20 


42 


3.91 


56 


4.52 


70 


5.03 


15 


2.36 


29 


3.25 


43 


3.95 


57 


4.55 


71 


5.07 


16 


2.43 


30 


3.31 


44 


4.00 


58 


4.58 


72 


5.09 



Example. Suppose the opening to be 36 inches long and 2 inches high, 
and the head of water above the opening 25 inches. Multiply the length, 
36, by 2, the height of the opening, and it gives 72. Referring to the above 
table, opposite 25-inch head will be found 3.03. This, multiplied by 72, 
gives 218.16, the number of cubic feet of water passing through the 
opening per minute. 

Water=wheels. 



Water-wheels may be divided into two classes, vertical and horizontal, 
according to the position of the plane in which the revolving wheel is 
placed. 

Vertical wheels include Overshot-wheels, Undershot-wheels, Breast- 
wheels, and Impact-wheels of the Pelton type. Horizontal wheels 
include practically all forms of turbines, although in some cases turbine 
wheels are placed on horizontal axes and revolve in vertical planes, 
without, however, suffering any material change in construction or action. 
In the pages immediately following the data for the various kinds of 
vertical wheels are : 

Q = quantity of water, in cubic feet, per second ; 

h = head, in feet ; 

V = velocity of water, in feet, per second ; 

v = velocity of wheel buckets, in feet, per second ; 

u = angle of entrance ; 

a = area of float, in square feet. 

Example. The vertical section of the immersed floats of an undershot- 
wheel in a mid-stream is a = 27 square feet; velocity of the stream, V= 
8.6 ; and v = 4 feet per second. Required the horse-power of the wheel ? 



EP = 



200 K } 200 

35 



8.6 — 4)2 = 11.4 IP. 



546 



Water-wheels. 



Example. On a breast-wheel is acting Q = 88 cubic feet of water per 
second ; the head, h = 8 feet ; velocity of the wheel at the centre of the 
buckets, v = 5 feet per second. The water strikes the buckets at an angle, 
u = 8°, and velocity, V= 7 feet per second. Required the horse-power of 
the wheel ? « 

IP = ^ ( 8 + |(7 X cos 8° - 5) ) = 65 IP. 

Example. Required the effect of a Poncelet wheel : the head, h — 4 feet ; 
and the orifice, a = 5 square feet ; the velocity of the wheel at the centre 
of the pressure of the floats is v — 6.78 feet per second ? 

V= 6.91 ;/ 4 = 13.82 feet per second ; 
Q = 6.5 X 5 X V 4 =65 cubic feet per second ; 
65 X 6.78 , 



IP = 



197 



-(13.82 — 6.78) = 15.8 IP. 



Example. A saw-mill wheel is to be built under a fall of h = 18 feet, 
and to make n = 110 revolutions per minute. Required the proper diam- 
eter of the wheel ? 

D = ^-l/is - 3.857 feet 

at the centre of pressure of the buckets. 
Velocity, . _ 

F= 8j/l8 = 33.94 feet per second. 
Velocity, 

3.14 X 3.857 X 110 00 n . . 

v = „„ — == 22.2 feet per second. 

60 

The fall discharged 30 cubic feet of water per second. Required the 
horse-power of the wheel ? 

30 V 99 1 
IP = 200 (88,94 _ 22,2) = 39 IR 

In general, the maximum efficiency of such wheels is obtained when 



Undershot Stream-wheel. 




IP-- 



200 



{v-vy. 



When V = 2v, about, the effect 
will be 

aV z 
IP= t^m* a = area ot float. 
1500 



Undershot-wheel. 




When V = 2v, about, /P = 



ah}/ h 
3.9 ' 



Water-wheels. 



547 



Poncelet Wheel. 



Low Breast-wheel. 




IP = JgL( V—v), when h > 5 feet ; 



iP = Jji_( v— v) , when ft < 5 feet ; 



Q = 8ma j/ /i ; 
F — 6.91 i/T. 



Breast-wheel with Parabolic 
Drain. 




* = &[* + &*-*)] 

Q=6.ba\/~h' . 




&=^\_ h +U v ™* u -^> 



■- kb ; 



a ' 



See table for weirs. 



Breast-wheel. 




IP= *m[ h + ^ Vco3U -*]- 



548 



Water-wheels. 



Overshot=wheel. 




Proper velocity about 
352) + 100 

n = D 

revolutions per minute. 



Saw-mill Wheel. 




Proper diameter of the wheel : 

_ 100 /-- . , . 
D = — y h , m feet ; 
n 

n = revolutions per minute. 



For high heads of water the most effective form of wheel is the tan- 
gential impact type, of which the well-known Pelton wheel is a good 
example. 




Pelton Water-wheel. 



Pelton Bucket. 



The buckets of the Pelton wheel are made double, with centre fin, 
producing a side discharge, and permitting the maximum transfer of 
energy from the jet to the wheel, an efficiency of 85 per cent, being 
obtained. 



Water-wheels. 



549 



Pelton Water=wheel Table. 

The calculations for power in this table are based upon the application 
of one stream to the wheel, as also upon an 85 per cent, efficiency and effec- 
tive heads, no allowance being made for loss of head, in pipe, by friction. 
The smaller figures under those denoting the various heads give the equiv- 
alent pressure, in pounds, and spouting velocity of the water, in feet, per 
minute. The cubic feet measurement is also based on the flow per minute. 



Size 
of wheels. 



2 
inch. 



5 

inch. 



men. 



24 

inch. 



3 

foot. 



4 

foot. 



5 

foot. 



Horse-power .60 
Cubic feet... ' 3. 74 
Miner's inc's (2.33 
Revolutions. 1530 



Horse-power 
Cubic feet. . . 
Miner's inc's 
Revolutions . 
Horse-power 
Cubic feet. . . 
Miner's inc's 
Revolutions. 
Horse-power 
Cubic feet. . . 
Miner's inc's 
Revolutions. 
Horse-power 
Cubic feet. . . 
Miner's inc's 
Revolutions. 
Horse-power 
Cubic feet. . . 
Miner's inc's 
Revolutions. 
Horse-power 
Cubic feet.. . 
Miner's inc's 
Revolutions . 
Horse-power 
Cubic feet.. . 
Miner's inc's 
Revolutions . 
Horse-power 
Cubic feet. . . 
Miner's inc's 
Revolutions . 
Horse-power 
Cubic feet.. . 
Miner's inc's 
Revolutions. 
Horse-power 
Cubic feet.. . 
Miner's inc's 
Revolutions. 
Horse-power 
Cubic feet.. . 
Miner's inc's 
Revolutions . 
Horse-power 
Cubic feet. . . 
Miner's inc's 
Revolutions. 



.79 
4.10 
2.56 
1677 

99 
4.43 
2.76 
1812 
1.22 
4.73 
2.95 
1938 
1.45 
5.02 
3.13 
2049 
1.70 
5.29 
3.30 
2160 
1.96 
5.55 
3.46 
2268 
2.24 
5.80 
3.62 
2370 
2.52 
6.04 
3.77 
2466 
2.82 
6.26 
3.91 
2562 



1.40 
8.74 
5.46 
765 
1.84 
9.57 
5.98 
838 
2.33 

10.34 
6.46 
906 
2.84 

11.05 
6.90 
969 
3.39 

11.75 
7.32 
1024 
3.97 

12.36 
7.72 
1080 
4.59 

12.96 
8.10 
1134 
5.23 

13.54 
8.46 
1185 
5 

14.09 
8.8C 
123^ 



2.32 

14.81 

9.25 

612 

3.12 

16.21 

10.13 

671 

3.94 

17.53 

10.95 

725 

4.82 

18.74 

11.71 

775 

5.75 

19.87 

12.41 

820 

6.74 

20.94 

13.08 

864 

7.77 
21.96 
13.72 

906 
8.86 
22.93 
14.33 

948 
10.05 
23.88 
14.92 

986 



4.21 

26.22 
17.48 

510 
5.54 
28.72 
19.15 

559 
6.99 
31.03 
20.68 

604 
8.54 
33.17 
22.11 

646 
10.19 
35.18 
23.45 

683 
11.93 
37.08 
24.72 

720 
13.77 
38.89 
25.93 

756 
15.69 
40.62 
27.08 

790 
17.69 
42.28 
28.19 

822 



6.59 11.16 19.7' 

14.62 24.79 43.8! 

9.13 15.49 29.25 

1281 1025 854 



3.13 7.31 
6.4815.13 



4.05 
2652 
3.45 
6.70 
4.18 
2739 
3.78 



9.45 
1326 
8.05 
15.63 
9.76 
1369 
82 



12.38 
25.66 
16.03 
1060 
13.64 
26.50 



21.93 
45.42 
30.28 
884 
24.16 
46.91 



6.90 16.12 
4.3110.07 
28231 1411 1 



16.56131.27 
1095! 913 
14.94126.46 



7.31 

17.06 
1130 



48.35 

32.24 

941 



7.49 
46.58 
31.05 

382 

9.85 
51.02 
34.01 

419 
12.41 
55.11 
36.74 

453 
15.17 
58.92 
39.28 

484 
18.10 
62.49 
41.66 

513 
21.20 
65.87 
43.91 

540 
24.46 
69.08 
46.05 

567 
27.87 
72.16 
48.10 

592 

31.43 
75.10 
50.07 

617 
35.12 
77.94 
51.29 

639 
38.95 
80.67 
53.78 

663 
42.91 
83.32 
55.55 

685 
47.00 
85, 
57.26 

706 



16.84 

104.88 

69.93 

255 

22.18 

114.91 

76.60 

279 

27.96 

124.12 

82.72 

302 

34.16 

132.68 

88.46 

323 

40.77 

140.74 

93.82 

342 

47.75 

148.35 

98.90 

360 

55.09 

155.59 

103.72 

378 

62.77 

162.50 

108.34 

395 

70.78 
169.14 
112.76 

411 
79.11 
175.53 
117.02 

427 
87.73 
181 
121.12 

442 

96.65 

187.65 

125.10 

456 

105.86 188.02 
193.42 343.55 
128. 98! 229. 04 

470! 353 



29.93 
186.32 
124.21 

191 

39.41 

204.10 

136.06 

209 
49.64 
220.44 
146.96 

226 

60.68 

235.68 

157.12 

242 

72.41 

249.97 

166.64 

256 
84.81 
263.49 
175.66 

270 

97.85 

276.35 

184.23 

283 
111.50 
288.64 
192.42 

296 
125.72 
300.43 
200.28 

308 
140.51 
311.77 
205.18 

319 
155.83 
322.71 
215.14 

331 
171.68 
333.29 
222.19 

342 



46.85 
291.51 
194.34 

152 

61.66 

319.33 

212.89 

167 

77.71 

344.92 

229.94 

181 

94.94 

368.73 

245.82 

193 
113.30 
391.10 
260.73 

206 
132.70 
412.25 
274.83 

216 
153.10 
432.38 
288.25 

226 
174.45 
451.60 
301.07 

237 
196.71 
470.04 
313.36 

247 
219.84 
487.79 
325.19 

255 
243.82 
504.91 
336.60 

265 
268.60 
521.46 
347.64 

274 
294.18 
537.51 
358.34 

282 



550 



Water-wheels. 



Pelton Water=wheel Table.— continued. 

The calculations for power in this table are based upon the application 
of one stream to the wheel, as also upon an 85 per cent, efficiency and effec- i 
tive heads, no allowance being made for loss of head, in pipe, by friction. 
The smaller figures under those denoting the various heads give the equiv- 
alent pressure, in pounds, and spouting velocity of the water, in feet, per 
minute. The cubic feet measurement is also based on the flow per minute. 



Head, 
in feet. 


Size 
of wheels. 


6 

in. 


12 

inch. 


15 

inch. 


18 
inch. 


24 

inch. 


3 

foot. 


4 

foot. 


5 

foot. 


6 

foot. 


360 

156 lb. 
9130.14 

380 

165 lb. 
9380.32 

400 

173 lb. 
9624.00 

420 

182 lb. 
9861.66 

440 

191 lb. 
10093.74 

460 

200 lb. 
10320.58 

480 

208 lb. 
10542.56 

500 

217 lb. 
10759.96 

600 


Horse-power 
Cubic feet.. . 
Miner's inc's 
Revolutions . 
Horse- power 
Cubic feet. . . 
Miner's inc's 
Revolutions. 
Horse-power 
Cubic feet.. . 
Miner's inc's 
Revolutions. 
Horse-power 
Cubic feet.. . 
Miner's inc's 
Revolutions. 
Horse-power 
Cubic feet.. . 
Miner's inc's 
Revolutions. 
Horse-power 
Cubic feet. . . 
Miner's inc's 
Revolutions. 
Horse-power 
Cubic feet.. . 
Miner'sinc's 
Revolutions. 
Horse-power 
Cubic feet. . . 
Miner'sinc's 
Revolutions. 
Horse-power 
Cubic feet. . . 


4.10 
7.10 
4.43 
2907 
4.46 
7.30 
4.56 
2985 
4.82 
7.49 
4.68 
3063 
5.19 
7.67 
4.79 
3141 

5.56 
7.85 
4.90 
3213 
5.95 
8.03 
5.01 
3285 
6.34 
8.20 
5.12 
3357 
6.74 
8.37 
5.23 
3426 


9.61 
16.58 
10.36 

1453 
10.42 
17 04 
10.65 

1492 
11.25 
17.48 
10.92 

1531 
12.11 
17.91 
11.19 

1570 
12.98 
18.33 
11.45 

1606 
13.88 
18.74 
11.71 

1642 

14.79 
19.15 
11.96 

1678 
15.73 
19.54 
12.21 

1713 


16.28 
28.10 
17.56 

1161 
17.66 
28.88 
18.03 

1194 
19.07 
29.63 
18.51 

1225 
20.52 
30.36 
18.93 

1255 
22.01 
31.07 
19.41 

1285 
23.53 
31.77 
19.79 

1315 
25.07 
32.45 
20.28 

1343 

20.66 
33.12 
20.72 
1370 


28.83 
49.75 
33.17 
969 
31.27 
51.12 
34.08 
995 
33.77 
52.45 
34.96 

1021 
36.33 
53.74 
35.83 

1047 
38.96 
55.01 
36.66 

1071 
41.65 
56.24 
37.50 

1095 
44.39 
57.45 
38.30 

1119 

47.20 
58.64 
39.09 
1142 


51.21 
88.37 
58.91 

726 
55.54 
90.80 
60.53 

746 
59.98 
93.16 
62.10 

765 
64.54 
95.46 
63.64 

785 
69.20 
97.70 
65.13 

803 
73.97 
99.90 
66.60 

821 

78.85 

102.05 

68.00 

839 

83.83 

104.15 

69.41 

856 


115.34 

199.03 

132.68 

484 

125.08 
204.48 
136.32 

497 
135.08 
209.80 
139.84 

510 
145.34 
214.98 
143.32 

523 
155.85 
220.04 
146.64 

535 
166.60 
224.98 
150.00 

547 
177.58 
229.82 
153.20 

559 
188.80 
234.56 
156.36 

571 
248.16 
256.95 
171.30 

625 
312.73 
277.54 
185.02 

675 
382.09 


204.86 
353.51 
235.64 

363 
222.16 
363.20 
242.13 

373 
239.94 
372.64 
248.40 

382 

258.16 
381.84 
254.56 

392 
276.82 
390.82 
260.53 

401 
295.91 
399.61 
266.40 

410 
315.42 
408.20 
272.12 

419 
335.34 
416.62 
277.64 

428 
440.77 
456.38 
304.24 

469 
555.46 
492.95 
328.63 

506 
678.66 
526.99 
351.32 

542 
809.82 
558.96 
372.64 

574 
948.48 
589.19 
392.79 

605 


320.52 
553.10 
368.73 

290 
347.60 
568.25 
378.83 

298 
375.40 
583.02 
388.68 

306 
403.91 
597.41 
398.28 

313 
433.11 
611.47 
407.65 

320 
462.97 
625.22 
416.80 

327 
493.49 
638.66 
425.78 

335 
524.66 
651.83 
434.56 

342 
689.63 
714.05 
476.03 

375 
869.06 
771.26 
514.18 

405 
1061.81 
824.51 
549.68 

433 
1267.02 
874.53 
583.02 

459 
1483.97 
921.83 
614.56 

484 


461.36 
796.14 
530.75 

242 
500.33 
817.95 
545.29 

248 
540.35 
839.20 
559.35 

255 
581.39 
859.93 
573.28 

261 
623.40 
880.16 
586.56 

267 
666.40 
899.95 
600.00 

273 
710.33 
919.29 
612.80 

279 
755.20 
938.25 
625.44 

285 
992.65 


260 lb. 












1027.80 


11786.94 


Miner'sinc's 
Revolutions. 
Horse-power 
Cubic feet. . . 












685.20 






i 






312 


700 




..... "... 






1250.92 


3041b. 








1110.16 


12731.34 


Miner'sinc's 
Revolutions . 










740.09 












337 


800 


Horse-power 
Cubic feet. . . 




i 







1528.36 


348 lb. 




1 






296.70 
197.80 

722 
455.94 
314.70 
209.80 

766 
534.01 


1186.81 


13610.40 


Miner'sinc's 
Revolutions. 
Horse-power 
( Jubic feet . . 












791.21 














361 


900 













1823.76 


391 lb. 












1258.81 


14436.00 


Miner's inc's 
Revolutions. 
Horse-power 











839.20 














383 


1000 












2136.04 


434 lb 












331.72 


1326.91 


15! '16.su 


Miner's inc's 
Revolutions. 












221.15 
807 


884.61 














403 















Water-wheels. 



551 



Turbines. 

For moderate and low heads and large volumes of water turbines 
are now generally used. Various forms oi turbines are in use, but they 
may all be considered as variants of 
the two original types, the Fourney- 
ron and the Jonval. 

In the Fourneyron turbine the 
water flows down and out through 
the guides, LL, and is delivered into 
the curved buckets, AA, of the 
wheel, this latter being connected 
to the vertical shaft by the plate, 
BB. In the illustration the power is 
transmitted by the gear-wheels, DE, 
but the rotor of a dynamo may be 
mounted directly on the shaft. 

The proportions of the Fourney- 
ron turbine may be determined as 
follows, according to Weisbach : * 
The data are given in the accom- 
panying diagram, the dimensions 
being in feet, and Q being the quan- 
tity of water delivered, in cubic feet, 
per second under a head, h, in feet. 
The inner radius, r : = CE = 0.326 j/Q. 
The outer radius, r = CM = vr\ = 

5 - - A 
2 



at the entrance, E, may then be 
made a = 15° to 30°, and the bucket 
angle at the same point = = 2a + 
20° to 2a + 30°. The inner velocity 
of the wheel will then be determined 
by the formula, 




Fourneyron Turbine. 



Vi 



■4 



2gh 



2 sin cos a 
sin (0 — a) 



+o.i 17 . g y ,y+^\ 

T |_ V SHI (0 — a) / ^ J 




Curves for the Fourneyron Turbine. 



The outer velocity will then be 

given v= w 1 = —v lt v being the 

ratio of the inner and outer 
radii of the wheel. From this 
we obtain the number of revo- 
lutions : 



30v 

7JT 



: 9.55- 



:9.55 



*>i 



The velocity, c, with which the 
water issues through the guides 
will be 

_ v 1 sin /3 



sin (a — /3)' 

and the cross-section, F, of the 
sum of all the openings will be 



Q _ Qsin(a — 0) 



square 



c i\ sin 

feet. If e be the height of a 



' Der Ingenieur," Seventh Edition, 1896, pages 596-602. 



552 



Water-wheels. 



bucket, and d = the width, as at AB\, we have for their ratio A. = — == 2 to 

5, according to the head of water, the larger value being for the lower 
head. The thickness of metal in the floats may be made 

s = 0.015r. 

We then have for the height, e, of the wheel, 

F / 271-r sin a . \s \ 
a\ 1+ ~F )' 

\F 



2ttvi sin 
The number of guide buckets, n\ 



and the number of wheel buckets, 



sin 

n = —. n\ 



Ai^sin £ 
e 2 sin a 



The exit angle, 8, at the middle point, M, of a bucket is found from 

. „ F 2 + nse 

sin 8 = —4-^ , 

2nre 

F 2 being the sum of all the discharge openings of the wheel buckets. 
" The curvature of the floats is determined as follows : 
From CM = r lay off the angle, CME = 8, and drop the perpendicular, 
CE, to ME. From M and from E lay off MA = MB X = EO = E0 X = r sin 8 

tan ~, <f> being = -. ABi will then be the width of a bucket mouth, 

neglecting the thickness of the metal of the floats, and and 0\ will be 
the centres for the arcs, AB and A\B\, of the outer portions of the floats. 

Lay off the line, AF = CE = r 1? making the angle, EAF, = 180° — /3, 
join CF, and at the middle point, H, of the latter line erect the perpen- 
dicular, HK. The intersection, K, with EA will then be the centre from 
which the remainder of the bucket curve is struck. 

Lay off the angle, a, from the ends, C and E, of the inner radius, CE, 
making the isosceles triangle, CEG, and G will be the centre from which 
to strike the curve, DE. of the guide. Having determined the centres for 
one pair of guides and bucket floats, the rest of the buckets can be readily 

drawn. 

The inward discharge turbines, such 
as that of Francis, may be designed in the 
same manner, simply by changing r to r*i 
and v to vi, and vice versa. 

The Jonval turbine is constructed as 
shown in the illustration, the guide 
blades, LL, being arranged in a ring 
above the turbine buckets, AA, the flow 
of water being parallel to the axis. This 
form of turbine is especially adapted for 
use with a draft tube, surrounding the 
wheel and adding the suction head of the 
discharge to the head above the wheel. 
When such a draft tube is used, the effec- 
tive head, h, used in the computations is 
the sum of the heads above and below 
the wheel. 

In this form of turbine a = 15° to 25° 
and ^ = 100° to 120°, and the most efficient velocity for the wheel is given 
by the formula, 




Jonval Turbine. 



■Vi 



sin £ cos a 
sin (/3 — a) 



0.1 



2qh 



1 + 



/ sin/3 \8-| 
V sin (0 - a) ) J 



Water-wheels. 553 



The velocity of entrance of the water is 

vsin /3 



sin (j8 — a) 



The total cross-section of the entrance spaces between the guides will 
then be 

c 
and the total discharge section of the wheel buckets, 

V 

Q being the quantity of water, in cubic feet, per second. 

If ri and r 2 be the inner and outer radii of the wheel, the mean radius 
will be 

r= n_±j± 

2 ' 

and the width of the annular operative portion of the wheel, measured 
radially, will be 

e = To — ?'!. 

Usually, e is made equal to pr = 0.4r, whence 

n = r(l — %p) =0.8r, 
and 

r* = r{l + y 2P ) = 1.2r. 
The ratio, 

*-* 

of the length, e, of the floats to their width, d, may be made from 2 to 4. 
The radius may be determined from the formula, 



V^5t( 1 + W^l^)- 



Approximately, we may take 



-^ 



2irp sin a ' 
and the thickness of the floats may be made 

s = 0.02r. 
The length of a float will then be 

e — pr. 

The number of guides, 

F \F 
de <% 

while the number of floats in the wheel, 

sin /3 

71 = — : . TOi. 

sin a 
The angle of discharge, 8, is obtained from 

F 2 + nse 



sin 6 : 



2irre 
and the number of revolutions, 

u = = 9.55 — . 

nr r 



•i 



554 



Pumps. 



The height, a, of the wheel, as well as of the guides, may he made 0.5r 
to 0.6r. 

Both guides and floats are formed of warped surfaces, whose generating 

line is at right angles with the 
axis and follows the curves, HE' ~ 
and EA, of the illustration. The 
lower portions of the guides and 
buckets are straight lines, inclined 
at angles, a and 8, with the hori- 
zontal lines, as at BE and BA. 
The upper portions are arcs of 
circles, DH and BE. The centre, 
K, for DH, is at the intersection of 
a normal, DK, to the straight part 
of the guide with the upper line 
of the guides. To find the centre, 
C, of the curve, BE, of the wheel 
float s draw BC perpendicular to 
AB, make the angles, CBE and 

BEC, equal to ^—x — , when the in- 
tersection, C, of jEOand i? (7 will be the desired centre. 




Let 



PUMPS. 

Dimensions. 

D = diameter of plunger, in inches ; 

Q = quantity of water, in cubic feet, per minute; 

v = plunger speed, in feet, per minute. 



D = \ 0.00545?'' 

Q = 0.00545-yD 2 . 

Approximately, the number of United States gallons delivered per minute, 
with a plunger speed of 100 feet per minute, will be 

G = 4D2. 

The loss by leakage and slip varies from 10 to 40 per cent. For a new 
pump, well packed, the delivery should be 90 per cent, of the theoretical. 

Ordinarily, the speed of pump plungers should not exceed 100 feet per 
minute. At higher speeds there are apt to be concussions and water- 
hammer produced, due mainly to the sadden stoppages in the movement 
of the column of water. The study of the movement of water in pumps 
has been greatly facilitated by the use of the indicator, the instrument 
being attached, not only with the recording drum operated from the pump 
plunger, but also by the life movement of the valves. The result of the 
latter method of investigation has shown the necessity for proper timing 
of the valves, especially when pumping against high heads at high speeds. 
In the designs of Professor Riedler a combination movement is used, the 
valve being closed mechanically, and opened by the action of the water. 

Smooth running may also be improved by judicious use of air chambers 
to receive the impact of the water. Air chambers on the suction side of 
the plunger are especially important. The proper arrangement is to have 
the air chamber in the direct line of the suction; and recent designs of 
high-speed pumps provide a large air chamber directly beneath the cylin- 
der, each suction valve having its own suction tube extending down 
nearly to the bottom of the air chamber. This construction is both simple 
and effective, and should be followed, when practicable. 

General practice indicates that the air vessel on the delivery side should 
be from 3 to 6 times the capacity of the pump, while on the suction side it 
may be made from 2 to 3 times the capacity of the pump. 



Pumps. 



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Ol 


Ol 


Ol 


Ol 


CO- 


co 


CO 


CO 


Tfl 


Tfl 


TTl 


-t< 


LO 


LO' 


LO 


© 


i T* 


H 


OS 


CO 


CO 


o 


CO 




X 


CO 


OS 


-o 


!> 


© 


X 


CO 


00 


OS 


lO 


© 


CO 


CO 


X 


© 


§ 


© 


CO 


CO 


tH 


SS 


o 


CO 


~H 


CO 


co 


CO 


lO 


t- 




lO 




X 


-^ 


:-. 


1^ 


lO 


CO 


01 




lO 


01 


CO 


© 


© 


r^ 


: 


CO 


r^ 


o 


~H 


X 


Ol 


CO 


o 


TH 


a. 


:: 


X 


01 


Ol CM 


01 


- 


3 


CO 


X 


3 

01 


co 


- 


© 


CO 


X 


Ol 


r^ 


CO 


© 


* 


t> 


l> 


t> 


X 


00 


as 


Oi 


a. 


o 


© 




Ol 


CO 


-r 


lO 


t> 


zr - 


Ol 


Ol 

Ol 


ol 


LO 

Ol 


Ol 


85 


© 

Ol 


CO 


t> 


to 


X 




kC 


rH 


as 


X 


q 


Ol 


CO 


01 


^ 


© 


CO 


q 


X 


Tt] 


L^ 


q 


Ol 


•H 


CO 


X 


© 


co 


CO 


TH 


CM 


lo 


iH 


IS 




d 


»> 


CO* 


d 


r^ 


lO 


oi 


d 


CO 


"H 


© 


t> 


lO 


-J^ 


CO 


CO* 


Tfl 


LO 


l> 


© 


■co 


d 




cd 


CM* 


© 


rH 




N 


CO 


CO 


-h 


iO 


iO 


CO 


i- 


X 


X 


© 


Ol 


CO 


lO 


r^ 


— 




CO 


uO 


l> 


© 


01 


TH 


£^ 


© 


Ol 




rH 












rH 








H 


rH 


Ol 


01 


Ol 


O^ 


Ol 


01 


CO 


CO 


CO 


co 


CO' 


TH 


TH 


Tf 


-H 


in 


o 


o 


q 


q 


o 


q 


q 


q 


o 


q 


o 


© 


© 


© 


© 


© 


© 


© 


© 


q 


© 


© 


© 


© 


© 


© 


© 


© 


© 


»> 


— ■ 




lO* 


^ 


X 


i> 


co 


co 


T-^ 


CO 


oi 


d 


© 


o4 


£• 


-H 


LO 


X 


CO 


H 


oi 


LO 


oi 


© 


oi 


cS 


oi 


oi 


co 


X 


© 


© 


i-O 


OS 


Tfl 


a. 


■o 


CO 




o 


o 


CO 


a- 


a. 


01 


X 


X 




I> 


o 


© 


LO 


H 


01 




I'M 


lO 


X 


oi 


lO 


g 




m 


X 


01 


•o 


a 




Ol 


© 


X 


i> 


I> 


CO 


o 


r^ 


I> 




© 




CO 


lO 


X 




r° 


lit 


lO 


© 


CO 


t> 


t> 


l> 


X 


X 


OS 


© 




rH 


01 


co 


-r 


lO 


CO- 


t> 


X- 


© 




Ol 


CO 


-H 


© 






























rH 


rH 


H 


rH 


1-1 


H 


rH 


H 


rH 


rH 


Ol 


Ol 


Ol 


Ol 


CM 


h 


© 


OS 


tH 


X 


CO 


a- 


t> 


q 


q 


co 


Ol 


q 


© 


CO 


(N 


c : 


r^ 


co 


co 


lO 


Ol 


TH 


01 


lO 


CO 


l> 


t> 


CM 


CO 


oo 


X 


CO 


X 


th 


OS 


ol 




r^ 


CO 


© 


cd 


© 


CO* 


CO 


CO 


X* 


-H 




x3 


CO 


Tfl 


CO 


oi 


oi 


oi 


CO 


in 


X 


04 


OS 


o 


o 






CO 


co 


~H 


m m 


k> 


X 


as 


rH 


Ol 


^ 


r ^ 


1-; 


© 




co- 


% 


l> 


© 


rH 


CO 








T— 


iH 


^ 


lH 


^ 


^ 


1-1 


rH 


^ 


rH 


T-H 


rH 


rH 


Ol 


Ol 


Ol 


Ol 


0-1 


Ol 


CO 


co 


CO 


oo 


** 


TH 


! 


^ 


H^ 


^ 


55 


r^ 


r^ 


































X 


X 


OS 


OS 


o 


o 


rH 




01 


CM 


CO 


CO 


-H 


iQ 


CO 


t> 


X 


'Os 


© 




Ol 


CO 


-H 


lO 


CO 


l> 


X 


© 


© 


^ 






H 


Ol 


01 


Ol 


ol 


Ol 


Ol 


Ol 


Ol 


Ol 


Ol 


01 


01 


Ol 


01 


CO 


CO 


CO' 


CO 


co 


CO 


CO 


co 


CO' 


CO 


TH 



) 



558 



Pumps. 



Standard Sizes of Deane Steam Pumps. 

For Ordinary Service. 



I 


©~ a? 
u >> © 

S «,d 
g 1 a 

p H 


M 
| 

t_ CO 

6 2 

ti)P 

^ .5 
P 


GO 

o u 

j§ ft 


u 

4> 
ft . 

- o 

la 

GQ 


Capacity, per 
minute, at 
given speed. 


"So 

a 

® CO* 

B ° 
p 


.a 

is 

k. m 

^ Oi 

2^ 

a 2 
p 


2 ® 

£ ft 

o ft 

J 8 

m 


g ft 

00 to 

QQ 


d 

.2 

o 

o 

© 

33 
2 


CD 

o3 

.a 
o 
. DD 

-3 

o 

<D 

IS! 

33 


« ,2 00 


Strokes. 


United 

States 
gallons. 


4 


3% 


5 


.14 


1 to 300 


130 


18 


33 


9% 


V* 


% 


IX 


4 


4 


5 


.27 


1 to 300 


130 


35 


33 


9% 


y* 


% 


2 


1^ 


5 


4 


7 


.39 


1 to 300 


125 


49 


45% 


15 


% 


1 


3 


2% 


5% 


5 


7 


.51 


1 to 275 


125 


64 


45% 


15 


% 


1 


3 


2% 


5% 


5% 


7 


.72 


1 to 275 


125 


90 


45% 


15 


% 


1 


3 


2% 


7 


7 


10 


1.64 


1 to 250 


110 


180 


58 


17 


i 


V/2 


5 


4 


•% 


7^ 


10 


1.91 


1 co 250 


110 


210 


58 


17 


i 


IK 


5 


4 


7K 


8 


10 


2.17 


1 to 250 


110 


239 


58 


17 


i 


V/* 


5 


4 


8 


6 


12 


1.47 


1 to 250 


100 


147 


67 


20% 


i 


v/* 


4 


4 


8 


7 


12 


2.00 


1 to 250 


100 


200 


67 


20% 


i 


Vi 


5 


4 


8 


8 


12 


2.61 


1 to 250 


100 


261 


68 


30 


i 


V/* 


5 


5 


8 


10 


12 


4.08 


1 to 250 


100 


408 


68 


20% 


i 


V& 


8 


8 


10 


8 


12 


2.61 


1 to 250 


100 


261 


68% 


30 


V4 


2 


5 


5 


10 


10 


12 


4.08 


1 to 250 


100 


408 


68% 


30 


i% 


2 


8 


8 


10 


12 


12 


5.87 


1 to 250 


100 


587 


68% 


30 


iK 


2 


8 


8 


12 


10 


12 


4.08 


1 to 250 


100 


408 


64 


24 


2 


2% 


8 


8 


12 


10 


18 


6.12 


1 to 200 


70 


428 


68% 


30 


2 


2% 


8 


8 


12 


12 


12 


5.87 


1 to 250 


100 


587 


64 


28% 


2 


2% 


8 


8 


12 


12 


18 


8.80 


1 to 175 


70 


616 


88 


28% 


2 


2% 


8 


8 


12 


11 


18 


12.00 


1 to 175 


70 


840 


88 


28% 


2 


2% 


8 


8 


14 


10 


12 


4.08 


1 to 250 


100 


408 


69 


30 


2 


2% 


8 


8 


14 


10 


18 


6.12 


1 to 175 


70 


428 


93 


25 


2 


2% 


8 


8 


14 


10 


24 


8.16 


1 to 150 


50 


408 


112 


26 


2 


2% 


8 


8 


14 


12 


12 


5.87 


1 to 250 


100 


587 


69 


30 


2 


2% 


8 


8 


14 


12 


18 


8.80 


1 to 175 


70 


616 


88 


28% 


2 


2% 


8 


8 


14 


12 


24 


11.75 


1 to 150 


50 


587 


112 


26 


2 


2% 


10 


8 


14 


14 


J I 


15.99 


1 to 150 


50 


800 


112 


34 


2 


2% 


12 


10 


11 


L6 


16 


13.92 


1 to 175 


80 


1114 


84 


34 


2 


2% 


12 


10 


11 


1G 


24 


20.88 


1 to 150 


50 


1044 


112 


38 


2 


2% 


12 


10 


10 


14 


18 


12.00 


1 to 175 


70 


840 


89 


27 


2 


2% 


8 


8 


16 


11 


24 


15.99 


1 to 150 


50 


800 


109 


34 


2 


2% 


12 


10 


L6 


L6 


L6 


13.92 


1 to 175 


80 


1114 


85 


34 


2 


2% 


12 


10 


16 


16 


21 


20.88 


1 to 150 


50 


1044 


115 


34 


2 


2% 


12 


10 


16 


is 


21 


26.48 


l to 125 


50 


1322 


115 


40 


2 


2% 


11 


12 


L8 


16 


21 


20.88 


l to 125 


50 


1044 


118 


38 


3 


3% 


12 


10 


18 


L8 


21 


26, 13 


1 to 125 


50 


1322 


118 


40 


3 


3% 


11 


12 


is 


20 


21 


32.64 


1 to 125 


50 


1632 


118 


40 


3 


3% 


16 


14 


'JO 


Is 


24 


26.43 


1 to L25 


50 


1322 


118 


40 


3 


3% 


11 


12 ' 


20 


20 


21 


32.64 


1 to 125 


50 


1632 


118 


40 


3 


3% 


16 


14 


•Jo 


22 


21 




1 to 125 


50 


1975 


120 


40 


3 


3% 


18 


14 



Pumps. 



559 



Sizes of Worthington Standard Duplex Feed Pumps. 



Size of pump. 


la 






Size of 


pipes. 




«•-. 






ll 












O OD 
© Jl 

©a d 


O m 

© £ 


o 
+3 -^ 




fa 


a 

o3 
© 

OQ 


CO 


d 


£> 


Diam 

stea 
cyli 


III 

ft 


&c o 

G £ 
©. 






o3 
X 




© 

ft 


Inch. 


Inch. 


Inch. 




Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


2 


iy s 


2% 


100 


21 X 6 


3/ 


X 


1 


% 


3 


1% 


3 


200 


26X10 


78 


% 




1 


3 


3 


300 


26 X 10 


I 


i/ 


1 


3 


2 


3 


400 


26 X 10 




1 


4% 


2% 


4 


1000 


33 X 13 


% 


£ 


2 


2 


33^ 


5 


1800 


38 X 15 


% 


2% 


2% 


6 


4 


6 


2500 


44X16 


1 


i% 


3 


3 


7% 
7K 


4% 


6 


3300 


48X24 


% 


2 


4 


3 


5 


6 


4000 


48X24 


2 


4 


3 


7 >l 


4^ 


10 


4000 


72 X 29 


2 


4 


3 


5 


10 


5000 


72X30 


2 


5 


5 


534 


10 


5500 


72X30 


2 


5 


5 


9 


10 


5500 


72X30 


2 


2% 


5 


5 


9 


6 


10 


7200 


72X31 


2 


2% 


6 


5 


10 


6 


10 


7200 


72X31 


2 


2% 


6 


5 


10 


7 


10 


10000 


72 X 33 


2 


2% 


6 


6 


12 


83^ 
9M 


10 


15000 


80 X 42 


23^ 
2% 
2>1 


3 


6 


6 


12 


10 


18000 


80X42 


3 


8 


7 


14 


10 


18000 


80X42 


3 


8 


7 



Sizes of Knowles Standard Duplex Feed Pumps. 



a 


h 




u S 


*H«H A 


«m a © fe 






© 

ft 




© 




^ 






©«M -jJ 

a 0.2 

CD © ft 


of -2 ft o3 . 


03 ftp, 
>> ^--2 - © 


ft 
ft 




ft 


>> 


ft 5-1 


a «2 

Si 


1 £ 


© 

i 




© S _- O T3 

^.S^ ©■ 

E fl * in a 

-S a © Sc 


ft © £ *S 


a 

o3 
© 


3 
o3 

,d 
X 


.2 

3 


> © 

•-^ a 

©ft 


O © 


OQ 


OQ 


O 


OQ 


O 


OQ 


H 


OQ 


ft 


Eh 


In. 


In. 


In. 






Gallons. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


3 


2 


3 


.046 


75 to 200 


9 to 18 


S 


% 


i x 4 


1 


36X12 


4i% 


2M 
3i? 


4 


.10 


75 to 150 


15 to 30 


% 


2 


1% 


38X13 


514 


6 


.24 


75 to 150 


36 to 72 


3 4 


l x 4 


2% 


2 


52X20 


6 


4 


7 


.39 


75 to 125 


58 to 97 


1 


1% 


3 


2% 


54 X 24 


VA 


4% 


10 


.69 


60 to 120 


82 to 164 


1% 


1% 


4 


3 


60X24 


■8 


5 


1-2 


1.02 


50 to 100 


102 to 204 


1% 


1% 


5 


4 


80X31 


8 


6 


12 


1.47 


50 to 100 


147 to 294 


1% 


1% 


5 


4 


80 X 31 


10 


6 


12 


1.47 


50 to 100 


147 to 294 


2 


2% 


5 


4 


80X31 


12 


7 


12 


2.00 


50 to 100 


200 to 400 


2 


2% 


6 


5 


83 X 35 


12 


8% 


12 


3.00 


50 to 100 


300 to 600 


2 


2% 


6 


5 


83 X 37 


14 


8% 


12 


3.00 


50 to 100 


300 to 600 


2% 


3 


6 


5 


83X37 


14 


10% 


12 


4.50 


50 to 100 


450 to 900 


2% 


3 


8 


7 


83X44 


16 


10% 


12 


4.50 


50 to 100 


450 to 900 


3 


3 


8 


7 


83X44 


16 


12 


12 


5.87 


50 to 100 


587 to 1174 


3 


3 


10 


8 


83X50 


18% 


10% 


12 


4.50 


50 to 100 


450 to 900 


3 


3 


8 


7 


83X44 


18% 


12 


12 


5.87 


50 to 100 


587 to 1174 


3 


3 


10 


8 


83X50 


18% 


14 


12 


8.00 


50 to 100 


800 to 1600 


3 


3 


12 


10 


83X57 


20 


14 


18 


12.00 


40 to 70 


960 to 1680 


4 


6 















560 Pumping Engines. 



Standard Method of Conducting Duty Trials of 
Pumping Engines. 

Report of Committee of the American Society of Mechanical 
Engineers, 1890. 

Abstract. 

The basis upon which the duty of a pumping engine is to be determined 
is 1,000,000 British thermal units. This is the equivalent of 100 pounds of 
coal when each pound of coal imparts 10,000 heat units to the water in the 
boiler, this corresponding to an evaporation of 10.355 pounds of water from 
and at 212° F. per pound of fuel. 

The duty should be computed from the quantity of heat supplied to the 
complete plant, including all auxiliaries. The work done by the pump is 
to be determined by the plunger displacement, the loss by leakage to be 
subsequently determined. 

The necessary data having been obtained, the duty of an engine may 
be computed by the use of the following formulas : 

-, t^ i. Foot-pounds of work done w1AftftAnA 

1. Duty = =— ~, ^ ^r — i 77 -r X 1 000 000, 

Total number of heat units consumed 

. A(P±p + 8)XL.XX x j 000 000 Qntrvounto). 



H 

CX 144 

2. Percentage of leakage = X 100 (per cent.). 

A X J-' X Jy 

3. Capacity = number of gallons of water discharged in 24 hours, 

AX LX NX 7.4805X24 



DX 144 
A X L X N X 1.24675 



D 
4. Percentage of total frictions 



(gallons). 



tup A{P±p + s)XLXN 

' ' * DX60X 33000 

I.H.P. 

i A(P±p + 8)XLX N ' 



xioo, 



or, in the usual case, where the length of the stroke and number of strokes 
of the plunger are the same as that of the steam piston, this last formula 
becomes 

Percentage of total frictions » |l a ^MEP 1 X 10 ° (p€r cent ')* 

In these formulas the letters refer to the following quantities: 

A = Area, in square inches, of pump plunger or piston, corrected for 
area of piston-rod. (When one rod is used at one end only, 
the correction is one-half the area of the rod. If there is more 
than one rod, the correction is multiplied accordingly.) 



Pumping Engines. 561 



P == Pressure, in pounds, per square inch, indicated by the gauge on 
the force main. 

p = Pressure, in pounds, per square inch, corresponding to indication 
of the vacuum gauge on suction main (or pressure gauge, if 
the suction pipe is under a head). The indication of the 
vacuum gauge, in inches of mercury, may be converted into 
pounds by dividing it by 2.035. 

s = Pressure, in pounds, per square inch, corresponding to distance 
between the centres of the two gauges. The computation for 
this pressure is made by multiplying the distance, expressed in 
feet, by the weight of one cubic foot of water at the tempera- 
ture of the pump well, and dividing the product by 144. 

L = Average length of stroke of pump plunger, in feet. 

jV = Total number of single strokes of pump plunger made during 
the trial. 

A s = Area of steam cylinder, in square inches, corrected for area of 
piston-rod. The quantity, A s X M.E.P., in an engine having 
more than one cylinder is the sum of the various quantities 
relating to the respective cylinders. 

L s = Average length of stroke of steam piston, in feet. 

2T S = Total number of single strokes of steam piston during trial. 

M.KP. = Average mean effective pressure, in pounds, per square inch, 
measured from the indicator diagrams taken from the steam 
cylinder. 

I.H.P. = indicated horse-power developed by the steam cylinder. 

C = Total number of cubic feet of water which leaked by the pump 
plunger during the trial, estimated from the results of the 
leakage test. 

D = Duration of trial, in hours. 

H = Total number of heat units (B.T.U.) consumed by engine = 
weight of water supplied to boiler by main feed pump X total 
heat of steam of boiler pressure reckoned from temperature of 
main feed water + weight of water supplied by jacket pump X 
total heat of steam of boiler pressure reckoned from tempera- 
ture of jacket water + weight of any other water supplied X 
total heat of steam reckoned from its temperature of supply. 
The total heat of the steam is corrected for the moisture or 
superheat which the steam may contain. For moisture the 
correction is subtracted, and is found by multiplying the latent 
heat of the steam by the percentage of moisture, and dividing 
the nroduct by 100. For superheat the correction is added, 
and is found by multiplying the number of degrees of super- 
heating— i.e., the excess of the temperature of the steam above 
the normal temperature of saturated steam — by 0.48. No allow- 
ance is made for heat added to the feed water, which is derived 
from any source, except the engine or some accessory of the 
engine. Heat added to the water by the use of a flue heater at 
the boiler is not to be deducted. Should heat be abstracted 
from the flue by means of a steam reheater connected with the 
intermediate receiver of the engine, this heat must be included 
in the total quantity supplied by the boiler. 

The leakage test of the pump plunger should be made as soon as possi- 
ble after the completion of the main trial. 

The leakage of an inside plunger (the only type which requires testing) 
is most satisfactorily determined by making the test with the cylinder 
head removed. A wide board or plank may be temporarily bolted to the 
lower part of the end of the cylinder, so as to hold back the water in the 
manner of a dam, and an opening made in the temporary head thus pro- 
vided for the reception of an overflow pipe. The plunger is blocked at 

36 



562 Pumping Engines. 



some intermediate point in the stroke (or, if this position is not practica- 
ble, at the end of the stroke) and the water from the force main is admitted 
at full pressure behind it. The leakage escapes through the overflow pipe, 
and is collected in barrels and measured. 

Should the escape of the water into the engine-room be objectionable, ■_. 
a spout may be constructed to carry it out of the building. Where the 
leakage is too great to be readily measured in barrels, or where other ob- 
jections arise, resort may be had to weir or Orifice measurement, the weir 
or orifice taking the place of the overflow pipe in the wooden head. The 
apparatus may be constructed, if desired, in a somewhat rude manner, 
and yet be sufficiently accurate for practical requirements. The test 
should be made, if possible, with the plunger in various positions. 

In the case of a pump so planned that it is difficult to remove the 
cylinder head, it may be desirable to take the leakage from one of the 
openings which are provided for the inspection of the suction valves, 
the head being allowed to remain in place. 

It is here assumed that there is a practical absence of valve leakage, a 
condition of things which ought to be attained in all well-constructed 
pumps. Examination for such leakage should be made first of all, and if 
it occurs, and it is found to be due to disordered valves, it should be 
remedied before making the plunger test. Leakage of the discharge 
valves will be shown by water passing down into the empty cylinder at 
either end when they are under pressure. Leakage of the suction valves 
will be shown by the disappearance of water which covers them. 

If valve leakage is found which cannot be remedied, the quantity of 
water thus lost should also be tested. The determination of the quantity 
which leaks through the suction valves, where there is no gate in the 
suction pipe, must be made by indirect means. One method is to measure 
the amount of water required to maintain a certain pressure in the pump 
cylinder when this is introduced through a pipe temporarily erected, no 
water being allowed to enter through the discharge valves of the pump. 

The exact methods to be followed in any particular case, in determining 
leakage, must be left to the judgment and ingenuity of the person con- 
ducting the test. 



Table of Data and Results. 

In order that uniformity may be secured, it is suggested that the data 
and results, worked out in accordance with the standard method, be 
tabulated in the manner indicated in the following scheme : 

DUTY TRIAL OF ENGINE. 
Dimensions. 

1. Number of steam cylinders 

2. Diameter of steam cylinders • . ins. 

3. I )iameter of piston-rods of steam cylinders ins'. 

4. Nominal stroke of steam pistons ft. 

5. Number of water plungers 

6. Diameter of plungers ins. 

7. I Ma meter of piston-rods of water cvlinders ins.' 

8. Nominal stroke of plungers ft. 

9. Net area of plungers sq. ins. 

10. Net area of steam pistons sq.* ins! 

11. Average length of stroke of steam pistons during trial ....... ft.' 

12. Average length of stroke of plungers during trial ft. 

(Give also complete description of plant.) 

Temperatures. 

L8. Temperature of water in pump well degs. 

n. Temperature of water supplied to boiler by main feed pump.' degs'. 
l .. Temperature of water supplied to boiler from various other 

Bources degs. 



Pumping Engines. 563 



Feed Water. 

16. Weight of water supplied to boiler by main feed pump lbs. 

17. Weight of water supplied to boiler from various other sources, lbs. 
k»18. Total weight of feed water supplied from all sources lbs. 

Pressures. 

19. Boiler pressure indicated by gauge lbs. 

20. Pressure indicated by gauge on force main lbs. 

21. Vacuum indicated by gauge on suction main ins. 

22. Pressure corresponding to vacuum given in preceding line. . . lbs. 

23. Vertical distance between the centres of the two gauges ins. 

24. Pressure equivalent to distance between the two gauges lbs. 

Miscellaneous Data. 

25. Duration of trial hrs. 

26. Total number of single strokes during trial 

27. Percentage of moisture in steam supplied to engine, or ( per cent. 

number of degrees of superheating \ or deg. 

28. Total leakage of pump during trial, determined from results 

of leakage test lbs. 

29. Mean effective pressure, measured from diagrams taken from 

steam cylinders M. E. P. 

Principal Results. 

30. Duty ft.-lbs. 

31 . Percentage of leakage per cent. 

32. Capacity gals. 

33. Percentage of total frictions , per cent. 

Additional Results.* 

34. Number of double strokes of steam piston per minute 

35. Indicated horse-power developed by the various steam cylin- 

ders I. H. P. 

36. Feed water consumed by the plant per hour lbs. 

37. Feed water consumed by the plant, per indicated horse-power, 

per hour, corrected for moisture in steam lbs. 

38. Number of heat units consumed, per indicated horse-power, 

per hour B. T. U. 

39. Number of heat units consumed, per indicated horse-power, 

per minute B. T. U. 

1 40. Steam accounted for by indicator at cut-off and release in the 

various steam cylinders lbs. 

I 41. Proportion which steam accounted for by indicator bears to 

the feed water consumption .* 

Sample Diagrams taken from Steam Cylinders. 

(Also, if possible, full measurements of the diagrams, embracing press- 
ures at the initial point, cut-off, release, and compression; also, back- 
pressure and the proportions of the stroke completed at the various points 
noted.) 

42. Number of double strokes of pump per minute 

43. Mean effective pressure, measured from pump diagrams M. E. P. 

44. Indicated horse-power exerted in pump cylinders I. H. P. 

45. Work done (or duty) per 100 pounds of coal ft.-lbs. 

(Sample diagrams taken from pump cylinders.) 

* These are not necessary to the main object, but it is desirable to give them. 



564 



Hydraulic Ram. 



The Hydraulic Ram. 

The hydraulic ram is a device by means of which a large volume of 
water under a low head may be used to force a smaller quantity of water 
up against a higher head. Its chief advantages are its simplicity, moder-*" 

ate cost, and freedom from 
care or attendance. 

The operation is as fol- 
lows : 

The water working the 
ram is supplied through 
the pipe, S, and escapes 
through an opening at o 
until it has gained a veloc- 
ity sufficient to raise the 
valve or ball, B, which 
suddenly stops the current 
and causes an excessive 
pressure in the ram, E, 
which opens the valve or 
ball, C; the water is forced 
into the vessel and air chamber, A, and finally through the delivery pipe, 
d, to its destination. When equilibrium of pressure is restored between 
S and R the valve, B, falls and the operation is repeated. The ram can 
make as many as 200 strokes per minute, depending upon its size. 

The length of the supply pipe, S, should not be less than 5 times the 
height of the fall, F, because it is the dynamic action of the water in the 
pipe which works the ram. The delivery pipe may be made 10 or more 
times the height of the fall. 

The useful effect of the ram, like that of water-wheels and turbines, 
depends much upon its construction. In ordinary cases it returns about 
50 per cent, of the natural effect,— that is, the quantity of water, q, multi- 
plied by the height, h, of the delivery above the ram will be about 50 per 
cent, of the quantity of water. Q, working the ram, multiplied by the 
head of fall, F, in the same unit of time. 




qh = 0.bQF. 



Q = 



0.5QF 



2qh 



Q and q can be expressed in any unit of volume or weight, 
i^and h can be expressed in any unit of length. 
But let us assume Q and g to be cubic feet per minute ; 
i^and h = fall and height, in feet; 

L = length, in feet, and D — diameter, in inches, of the 

supply pipe, S\ 
I — length, and d = diameter of the delivery pipe, d ; 



then 



z>=y 



2Q-'(L + 5Z>) 



_. >l4qHl + 5d) 



Hydrometer. 



A body wholly immersed in a liquid will lose as 
much of its weight as the weight of the liquid it 
displaces. 

A floating body will displace its own weight of the 
liquid in which it floats. 

A cylindrical rod of wood or some light materials, 
being set down in two liquids, A and B, of different 
specific gravities, when in equilibrium will sink to 
the mark a in the liquid .1, and to b in the liquid B ; 
then the specific gravity of A : B = 6, c : a, c, or in- 
versely as the immersed parts of the rod. This is the principle upon which 
a hydrometer is constructed. 





i 

a 


m 


c 


' l® 


k* 


M 


^\ 


If 


J(L-~ 


izl. 


Jc£ 


!=l£ 



Hydrostatics. 



565 



Table Showing the Comparative Scales of Gay Lussac and Baume, 

with the Specific Gravity and Proof, at the Temperature of 

60° Fahr. 





£ 100 

:9o 

-80 
- 70 
-GO 
3 50 

: 40 








:*fei 


'-■^ 


tl~ 



i 

1 Gay Lussac's. 


I 


f 100 




95 




90 






o 


85 


o 
o 


80 


08 


75 


f-H 


70 


£ 




£. 


65 


60 


o 


55 






08 


50 






s 


45 


o 






40 


Ph 


35 




30 




I 25 



Baume 's. 



46 
40 
36 
33 
31 
28 
26 
24 
23 
21 
19 
18 
17 
16 
15 
14 



Specific gravity. 



Proof. 



.796 

.815 
.833 
.848 
.863 



.901 
.912 
.923 
.933 
.942 
.951 
.958 
.964 
.970 



100 1 
92 
82 
72 
62 
52 
42 
32 
22 
12 J 
Proof. 



Ph 



18 
29 
35 
48 



^2 



Hydrostatics. 

Notation. 

A and a = areas of the pressed surfaces, in square feet ; 
Zand p = hydrostatic pressure, in pounds ; 

d = depth of the centre of gravity of A or a under the surface 
of the liquids, in feet ; 

5 = specific gravity of the liquid. 

Example. Case I. — The plane A = 3.3 square feet at a depth of d = 6 
feet under the surface of fresh water. Required the pressure, P =? 
Specific gravity of fresh water, S = 1. 

P = 62.3Ad = 62.3 X 3.3 X 6 = 1237.5 pounds. 



Example. Case IV. — The area of the pistons, A = 8.5 square feet, a == 
0.02 square feet, I = 4 feet, e = 9 inches, and F= 18 pounds. Required the 
pressure, P = 



FIX 

ea, 



18 X 4 X 8.5 
0.75 X 0.02 



= 40800 pounds. 



It must be distinguished that the centre of pressure and centre of 
gravity of the planes are two different points ; the centre of pressure is 
below the centre of gravity when the plane is inclined or vertical. 



566 



Hydrostatics. 



A = 




The Hydrostatic Paradox. 

The pressure, P, is independent 
of the width of column, C. 

P = G2.3Sah. (Same as above. ) 

III. 






p = a (~ — 62.SSh\ 




Centre of Pressure of a Rec- 
tangle, 

the upper edge at the surface of the 
liquid, d = %h. 



VI. 




Centre of Pressure of a Triangle, 

the base being at the surface of the 
liquid, d = %h. 



VII. 




h = 



Pa — pA 




Centre of Pressure of a Triangle, 

the vertex being at the surface of 
the liquid, d = %h. 



VIII. 



P = 



Bramah's Hydraulic Press. 

FIA . Pea 

FAl 



ea 

Pea 

A 



Pe 




d = % 






3^ 2 — %hh x + hi* 



2h — hi 



Pipe Bends. 



567 



Resistance Caused by Obstructions. 

In nearly all hydraulic work numerous bends, valves, etc., must be 
inserted in the connections, and these produce frictional resistance to the 
flow. Such resistances are conveniently computed in terms of additional 
head, representing the number of feet-head to be added to that for a 
smooth pipe in order that the final discharge or pressure may be realized. 

Resistance in Angles and Bends.— The resistance due to an angle is 
important, and is dependent upon what Weisbach calls the semi-angle of 
deviation, j3, according to the following formula : 



h 2 = & 

from which we get : 



2g 



v 2 
= (0.9457 sin2 /3 + 2.047 sin^ p)— — , 
^9 



0= 10 

£ 2 = 0.046 



20 
0.139 



30 
0.364 



40 
0.74 



45 
0.985 



50 
1.26 



60 
1.861 



70 
2.431 



Example, 
equal to 



In a right-angle bend £ = 45°, and the loss is practically 




In the case of bends the resistance is not so great, but is too large to 
be neglected, since we have 

The ratio of the radius of the tube to the radius of the curvature of the 
bend affects the coefficient, as below : 

°-" D 0.2 

0.138 

0.7 
r 

£ 2 = 0.440 0.661 

Example. For a right-angle bend in which r = D, we have 

45 v 2 v 2 

ho = 0.294^ . i- = 0.147^-, 

90 2g 2g 

or only about } the resistance of a sharp bend with any curvature. 

Resistances due to Sudden Changes of Cross=section.— When water 

which is moving at a velocity, v lt suddenly changes to another velocity, v, 
as at a, it experiences a loss of pressure which, according to Weisbach, is 
equivalent to a height : 




0.3 


0.4 


0.5 


0.158 


0.206 


0.294 


0.8 


0.9 


1.0 


0.977 


1.408 


1.978 



u 



2g ~\F 1 ) 2g~ **2g' 



.Fand Fi being the respective cross-sections ; also, Fv = J^. Doubling the 

v 2 
cross-section causes a loss of head equal to — . 

2g 



508 



Centrifugal Pumps. 



For gate valves, as at 6, or cocks, as at c, there is a loss due to the 
amount of contraction. For gate valves we have from Weishach : 





Openings = % 


% 


Vs 




% 




% 


% 


Va 


^ = 0.159 


0.315 


0.466 




0.609 


0.74 


0.856 


0.948 


& = 97.8 


17.0 


5.52 




2.06 




0.81 


0.26 


0.07 


and for cocks : 


















Angle = 10° 


20° 


30° 


40° 




50° 


60° 


65° 


82%° 


Q? = 0.850 
F 


0.692 


0.535 


0.385 




0.25 


0.137 


0.091 





£= 0.29 


1.56 


5.47 


17.3 




52.6 


206 


486 


CO 



From the above tables it will be seen how important an influence is 
exerted by valve chests, mud traps, and the like upon the flow of water. 
In all such cases it is important to modify the suddenness of the change of 
velocity by rounding and curving all angles in the passages, and in this 
way a large part of the loss may be obviated. For gaseous fluids the re- 
sistance is less, but is at the same time sufficiently important to be carefully 
considered. For a fuller discussion of the resistances offered to water in 
canals and streams the reader must be referred to special treatises on the 
subject. 



Let 



Then 



Centrifugal Pumps. 

v = velocity of rim of wheel, in feet, per second ; 

h = height of delivery, in feet, including suction ; 

D = diameter of wheel, in feet ; 

Q = cubic feet of water per minute ; 

d = diameter of discharge pipe, in feet. 

v = 10 + 8]/^ 



d = 0.36- 






Q 



29/i 



D 



■4 



\/h 



X 0.18. 



The inlet opening in the side of the wheel is made equal to 0.52). The 
blades are sometimes made in the form of an Archimedean spiral, but a 
better efficiency is obtained with the reversed curve, designed according 
to the method of Rittinger, as follows : 
Let 

r = the radius of the propeller wheel ; 
n = the radius of the inlet opening ; 

a ss the angle between radius and initial line of blade ; I 

I = radius of curvature of blade ; 
n — number of revolutions per minute; 
c a velocity of inflowing water per minute ; 
Q 

irri* ' 



Hydraulic Transmission. 



569 



We have 



tan a = 0.1047 



nri 



I - 



c 

r 2 — n 2 



2ri sin a 
The case is made in the form of an Archimedean spiral. 




Centrifugal Pump. 



Hydraulic Transmission of Power. 

For many purposes where power has to be distributed over a limited 
area for working machinery, such as presses, lifts, cranes, riveting and 
flanging machinery, and the like, it has been found advantageous to use 
water under high pressure, the system being piped to accumulators and 
pumps, so that a supply of stored energy is available for the varied and 
irregular demands. 

The accumulator is simply a vertical cylinder fitted with a weighted 
plunger, the area of the plunger and its load being proportioned to equal 
the pressure, in pounds, per square inch to be maintained in the system. 
The pumps deliver water under the plunger, and the pipe system is also 
connected to the cylinder. Unless the demand upon the pumps is equal 
to their full capacity, the plunger of the accumulator will be forced 
upward, the excess energy being thus stored in the lifted weights. When 
the plunger reaches its upper limit it shuts off the steam to the pumps 
and checks their action ; as it falls, the steam is turned on, and the pumps 
are again started. When any machine connected with the system, as a 
riveter or a press, is put in motion the accumulator plunger falls as the 
water is drawn from the pipes, but the pressure is maintained by the 
weights upon the plunger. The pumps promptly respond to the fall of the 
plunger, so that the latter is kept oscillating up and down in response to 
the demand from the machines and the supply from the pumps. 

The amount of energy stored in an accumulator will be 



in which 



ird 2 

Foot-pounds = 2240 Ws = -j~sp, 

W = weight on plunger, in tons ; 
d = diameter of plunger, in inches ; 
p = pressure, in pounds, per square inch ; 
s = vertical travel of plunger. 



570 



Hydraulic Transmission. 



The efficiency of an accumulator may be as high as 98 per cent., 1 per 
cent, being lost in charging and as much in discharging. While the total 
amount of energy which can be stored is not great, it can be discharged at 
a high rate for a short time, and by care in proportioning the capacity of 

the pumps to the probable *~ -" 
demand a very satisfactory 
service may be maintained. 
The pressures in such 
systems range from 600 to 
800 pounds per square inch. 
At Hull, England, a press- 
ure of 610 pounds per 
square inch is maintained. 
In London the pressure is 
800 pounds, and at Bir- 
mingham it is 730 pounds. 
For lifts the water is used 
in cylinders, usually hav- 
ing a stroke equal to but 
a fraction of the entire 
hoist, the travel of the cage 
being multiplied by a re- 
duplication of the hoisting 
cable over a system of 
sheaves. Plunger eleva- 
tors are now coming into 
use, however, the valves 
used permitting speeds of 
600 feet per minute, when 
required. 

The high-pressure water is used in various forms of motors. On the 
Continent the Schmid oscillating engine is much used, while in England 
the 3-cylinder engine of Brotherhood and the 4-cylinder engine of Rigg 
are found. 

The principal feature of such engines is the regulation. Throttling is 
unsatisfactory, and the rigidity of the water column must be contended 
with. The most satisfactory principle is that of a variable stroke, the 
pressure of the water being left unchanged. 

An example of such a regulator is that of Helfenberger. This is made 
with a hydraulic ratchet mechanism arranged in the crank disk in such 
a manner as to move the crank pin to or from the centre, the ratchet being 
operated by tappets, which strike each time the crank passes the dead 
centres. The throw of the crank is thus varied to correct for variations of 
speed, the mechanism being controlled by a governor. 

The Pelton water-wheel has also been employed with success as a small 
motor for use with high-pressure water, and is both simple and convenient. 




Schmid Hydraulic Motor. 




Professor Reuleaux has suggested a very effective method for power 
distribution by hydraulic pressure, using the water in a circuit or "ring," 
not unlike methods of electrical distribution. 

Taking into consideration high-pressure hydraulic systems, we find two 
distinct kinds of " ring" systems which may be used. 

In the first method, shown above, the flow of water under pressure 
starts from the power station, T , with a pressure, p , and proceeds to the 



Hydraulic Transmission. 



571 



first station, Ti, where it operates a water-pressure engine, and passes on 
with a reduced pressure, p x . It has, therefore, operated at the station, T if 
with a pressure, p — p x . With the pressure, p u it passes on to the second, 

third, fourth, nth station, T n , each time losing pressure until it returns 

f to the power station with a final pressure, p n , where it is again raised to 
the initial pressure of p . It is apparent that the water-pressure engines 

(escapements) at T lf T 2 , T s , T n , should all "be of equal size, in order 

to utilize the entire flow without excessive resistance. Automatic regula- 
tion, such as Helfenberger's, is also desirable. 




The second system is shown above. It will be seen that at each station 
there is a branch or shunt tube leading through the motor (or escape- 
ment), T 2 , and then reuniting with the main pipe. The main pipe, A, 
forks at the station into the two branches, B and C, of which the first 
diverts any required fraction of the power of the main flow, as T V, i, %, as 
the case may be. At the fork is a swing valve, C", operated by a speed 
governor, R, driven by the motor. This governor requires the assistance 
of some form of power reinforcement. The discharge pipe, D, of the 
motor unites with the by-pass, C, to form again the main conductor, E. 
At the entrance in the main pipe, A, we have the pressure, p±, of the origi- 
nal flow. The motor, T 2 , is now supposed to be stationary, the stop valve 
at B' having been closed by hand. The flap valve, C" ', which has been 
disconnected from the regulator before stopping the motor, is also closed. 
The flow of water then passes through (7 to E with the pressure, p x . 

When the motor, To, is to be started, the valve, B f , is opened and the 
flap valve, C", gradually opened until the motor begins to move, when it 
is connected to the governor, which regulates it thereafter so as to keep 
the motor at its normal speed. When a heavy load is thrown on the valve 
is opened so that the pressure, p 2> in B becomes a greater fraction of p lf 
and when the work is less it is reduced. The pressure of discharge, p 3 , acts 
as a back pressure, so that the motor works with an effective pressure, 
Vi — P3- The flow of water in the by-pass pipe, C, also passes the valve, 
C, with a pressure, p 3 , and unites with the discharge at E, to be further 
utilized at subsequent stations until it returns to the power station, where, 
if it has reached the minimum pressure, it is permitted to flow into a tank, 
from which it is again drawn by the pressure pumps. If the return water 
is delivered under pressure it may be allowed to enter the suction pipe of 
the pressure pumps direct, and so form a closed ring system to start anew 
on the circuit. 

The ring system of hydraulic power transmission is to be recommended 
when the various stations are distributed over a wide area and are readily 
connected by a continuous line of pipe. The pipe can be kept from freez- 
ing in winter by occasional gas flames, as has already been demonstrated 
by experience with Armstrong's hydraulic cranes. The ring system should 
be carefully distinguished from those forms in which the flow of water 
passes through the motor and is allowed to flow off at lowest pressure of 
discharge. 

Full detailed descriptions of a variety of hydraulic machinery will be 
found in Professor Henry Robinson's treatise on "Hydraulic Power and 
Hydraulic Machinery," and reference should also be made to Reuleaux's 
"Constructor." 



572 Fuel. 



FUEL. 



The fuels used in engineering consist of compounds of carbon and of 
hydrogen, which when uniting with oxygen produce heat. v 

Fuels may be classified as solid, liquid, and gaseous. 

The solid fuels are coal in its various grades from anthracite, bitumi- 
nous, lignite, and peat ; charcoal, coke, and wood. Liquid fuels include 
the mineral, vegetable, and animal oils. The gaseous fuels are natural 
gas and the various artificial gases, as coal gas produces gas, etc. 

The calorific power of solid and liquid fuels, or, as it is sometimes 
called, the thermal value, is measured by the number of thermal units 
or calories developed during the combustion of a unit weight. Usually, 
the calorific power is expressed in the number of British thermal units 
evolved by the combustion of a pound of the fuel, or the number of 
calories produced by the combustion of a kilogramme of fuel. To convert 
B. T. U. per pound to calories per kilogramme multiply by 0.555, to convert 
calories per kilogramme to B.T.U. per pound multiply by 1.8, this being 
the ratio of the Centigrade to the Fahrenheit degree. The calorie involves 
the raising of the temperature of a kilogramme of water, and the B.T.U. 
involves the raising of the temperature of only a pound of water, but 
this corresponds exactly to the different weights of fuels respectively con- 
sumed, so that the ratio is simply that due to the thermometer scales. 

The calorific power of gaseous fuels is generally determined in B. T. U. 
per cubic foot or in calories per cubic metre. To convert calories per 
cubic metre to B. T. U. per cubic foot multiply by 0.11235, to convert B. T. U. 
per cubic foot to calories per cubic metre multiply by 8.9. 

When the calorific powers of solids and gases are compared, the gas 
should be taken by weight, in order to have the data in comparable form, 
otherwise it is more convenient to consider gases separately by volume, as 
they are measured. 

Since the fuels used in engineering are carbon, hydrogen, and their 
compounds, the calorific value of these elements form the foundations 
upon which other values are computed. 

One pound of pure carbon, completely burned to carbonic acid, C0 2 , 
evolves 14,500 B. T. U. One kilogramme of carbon, burned in like manner, 
evolves 8080 calories. 

One pound of pure hydrogen, burned to water, evolves 62,100 B. T. U., 
and one kilogramme of hydrogen evolves 34,500 calories. 

Having these facts we may determine the calorific value of any combi- 
nation of carbon and hydrogen when we know the ultimate chemical 
composition, — i.e., the percentage of carbon and hydrogen contained. 
When there is only carbon and hydrogen present, the calorific value of 
the combination is expressed by the sum of the calorific value of the con- 
stituents. Thus, if h be the number of heat units evolved by the complete 
combustion of a combination of carbon and hydrogen, we have 

h = 8080 C + 3450077" calories, 
or 

h = 14500 C + 6210077 B. T. U. 

When, however, as is usually the case, there is oxygen present in the 
fuel it will unite with a portion of the hydrogen, and in such case a de- 
duction should be made. We therefore have, for the computation of the 
calorific value of a fuel from its chemical composition, 

h = 8080 C + 34500 (lI—-\, for calories per kilogramme, 
h = 14500C + 62100^77 — ~ V for B. T. U. per pound. 



and 



Whenever practicable, it is desirable that the calorific power of a fuel be 
determined directly by experiment. Various devices have been made for 



Fuel. 



573 



this purpose, depending for their action upon the complete combustion of 
a determinate weight of the fuel in a closed chamber immersed in a known 
weight of water. The rise in the temperature of the water then gives the 
information from which the heat evolved may be determined. The most 
f reliable apparatus of this kind is the so-called calorimetric "bomb" of 
Bethelot, Vielle, and Mahler, in which the fuel is enclosed in a steel vessel, 
lined with platinum or enamel, together with sufficient compressed oxygen 
to complete the combustion. The ignition is effected by means of an elec- 
tric current, and the heat evolved is measured by the rise in temperature 
of the bath of water in which the bomb is immersed. 

For full details of calorimetric apparatus and work reference may be 
made to Poole's work on the "Calorific Power of Fuels." 

In important investigations the fuel used should be carefully sampled, 
and its calorific power determined, either by computation— using Dulong's 
formula — from a chemical analysis or by the use of the bomb, such work 
being performed in the testing laboratory. For general purposes, however, 
the calorific value of a fuel may well be taken by selecting from existing 
tests that of a fuel corresponding most nearly with the one under con- 
sideration. 



Calorific Values of Fuels. 



Substance. 



Approximate 

total heat of 

combustion of 1 

pound of fuel. 



Equivalent evapo- 
ration from and at 
212° Fahr. per 
pound of fuel. 



Hydrogen 

Petroleum oils (benzine, etc.) 

Petroleum, crude 

Petroleum refuse 

Coal gas 

Coal gas, per cubic foot, at 62° Fahr. 

Coal, good average quality 

Carbon, pure — 

Coke 

Wood charcoal, dessicated 

Wood, dessicated 

Peat, dessicated 

Wood, air dried 

Straw 

Peat, 25 per cent, moisture 

Sulphur 



Thermal units. 
62000 
27500 
20400 
20000 
17800 
630 
14700 
14500 
13500 
13000 
11000 
10000 

8000 

8000 

7000 

4000 



Lb. water. 
64.20 

28.56 
21.13 
20.70 
18.43 

0.70 
15.22 
15.07 
13.87 
13.46 
11.39 
10.35 

8.28 

8.40 

7.25 

7.14 



574 



Fuel. 



Theoretical Heating Value of Coals. 

(Babcock and Wilcox.) 

Heating Power of Coals of Great Britain, United States, Germany, France,^ i 
Belgium, and Austria- Hungary. 



Coals. 
Locality of beds. 



B.T.U. 


Calories. 


16214 


89981 


15715 


8710 


14998 


8318 y 


14964 


8305| 


14689 


8152 J 


15000 


8325 


15123 


8402 


14820 


8225) 


13860 


7692 V 


13918 


7724] 


14164 


7861 


14221 


7892 


13143 


7293 


13155 


7301) 

7987/ 


14391 


15198 


8434 


9326 


5175 


13123 


7283) 
7851) 


14146 


13097 


7268 


13100 


7270 


9215 


5114 


14518 


8066) 


15125 


8403 V 


13514 


7508 j 


13212 


7340 


14985 


8325) 


11511 


6395 


11964 


6647 [ 


11343 


6302 


10674 


5930 j 


5769 


32051 


6444 


3580 


6093 


3385 


::s5^ 


2140 y 


4165 


2314 


3830 


2128 


4563 


2535 J 



Nature. 



Great Britain. 

Welsh Coal: 

Ebbw Vale, 1848.. 

Powell Duffryn, 1848 . . . 

Llangennech, 1848 

Llangennech, 1871 

Graigole, 1848 

Nixon's Navigation 

Gwaun Cae Gurwen 

Newcastle 

Derbyshire and Yorkshire. . 

Lancashire 

Scotch 

United States. 

Pennsylvania 

Pennsylvania 

Pennsylvania 

Kentucky 

Kentucky 

Kentucky , 

Illinois 

Indiana 

Indiana 

Virginia 

Arkansas 

Germany. 

Rhenish Prussia. 
Ruhr Coal: 

Dortmund 

Witten 

Bochum 

Bommern 

Essen 

Saar coal 

Saxony. 

Zwickau 

Hohndorf 

Oelsnitz . 

Lower Saxony, Anhalt, and 
Brunswig. 

Unseburg 

Atzendorf 

Neudorf 

Gorzig 

Halle a S 

Bitterfeld 

Naumburg 



Almost pure anthracites, 
having 84 to 89 per cent, of 
carbon. 

Pure hard anthracite. 
Called smokeless. 

Bituminous coal, having 77 
to 82 per cent, of carbon. 

Bituminous coal, having 78 
per cent, of carbon. 

Anthracite, having 88 per 

cent, of carbon. 
Cannel coal. 

Bituminous coking. 

Cannel coke. 
Lignite, good. 

Bituminous coking. 

Cannel coal. 
Bituminous coking. 
Lignite, good. 



Cannel coal. 

Short-flame coal, semi-an- 
thracite. 



Cannel coal. 



Brown coal or lignite, low J 
grade. ^ 



Fuel. 



575 



Theoretical Heating Value of Coals. 



-Continued. 



Coals. 
Locality of beds. 



Germany. — Continued. 

Hanover. 

Osnabmck 

Obernkirchen ... 

Silesia (Prussia). 

Carlssegen 

Myslowitz 

Waterloa ... 

Konigshiille 

Paulusgrube 

Waldenburg . . — 

Brandenburg 

Neurode , 

Freienstein 

Maxgrube 

Bavaria. 

Hanshamer coal 

Peipenberg 

Penzberg .............. — 

France. 

Anthracite de la Mayenne . . 
Anthracite de Lamure (Isere) 

Bassin du Bas-de-Calais. 

Maries. 

Bully 

Hessin 

Lens 

Naux. ,. 

l'Escarpelle 

Les Courrieres 

Bassin de la Sadne. 
Blanzy ... 

Epinac 

Bassin de la Loire. 

Rive-de-Gier, puits Henry 

Rive-de-Gier, No. 1 

Rive-de-Gier, Cimetierel. 
Rive-de-Gier, Cimetiere2. 
Rive-de-Gier, Couson 



Bassin de PAveyron. 

Lavaysse 

Ceral 

Bassin d' Alais Rochbelle. . 



B. T. U. 



10789 
12718 



10422 
10758 
11412 
12247 
12425 
12637 
12193 
13393 
9651 
10087 



9821 

8186 
8921 



15566 
13782 



14175 
15120 
15352 
15258 
15256 
15400 
14265 



13127 
14086 



15481 
15472 
14493 
15309 
14770 



14S30 
13203 
15643 



Calories. 



5994 
7066 



5790 
5977 
6340 
6804 
6903 
7021 
6774 
7441 
5362 
5604 



5456 

4548 
4956 



8646) 
7657/ 



7875) 

8400] 

8529 

8477 

8476) 

8556 ] 

7925 



7293 

7826 



8601) 
8596] 
8052 ~ 
8505 
8206 



8128 
7335 
8691 



Nature. 



Semi-anthracite, low grade. 
Bituminous. 



Long-flaming, semi-bitumi- 
nous. 



Lignite or brown, low grade. 



Anthracite. 



Bituminous hard coal. 

Bituminous coking. 
Bituminous hard coal. 

Bituminous coking. 

Semi-bituminous coal. 



Semi-bituminous coal, long 

flame. 
Bituminous coal, long flame. 



Bituminous hard coal. 



Bituminous hard coal, long 
flame. 



Semi-bituminous coal. 
Bituminous coking. 



576 



Fuel. 



Theoretical Heating Value of Coals— continued. 



Coals. 
Locality of beds. 



Prance. — Continued. 

Bassin de Valenciennes. 

Denain, Fosse Renard 

Denain, Fosse Lelvet 1 

Denain, Fosse Lelvet 2 

St. Wast, Fosse de la Reussite 

St. Wast, Grande Fosse 

St. Wast, Fosse Tinchon 

Anzin, Fosse Chauffour 

Anzin, Fosse la Cave 

Anzin, Fosse St. Louis 

Fresne, Fosse Bonnepart ..... 
Vieux-Conde\ Fosse Sarteau. 

Belgium. 

Bassin de Mons. 

Haut-flenu 

Belle et Bonne, Fosse No. 21. 

Levant de flenu 

Couchant 

Midi 

Grand-Hornu 

Nord du bois de Bossu 

Grand-Buisson 

Escouffiaux 

St. Hortense, bonne veine. . . 

Bassin du Centre. 

Haine St. Pierre 

Bois du Luc 

La Louviere 

Bracquegnies 

Mariemont 

Bascoup 

Sars-Longchamps 

Houssu 

Bassin de Charleroi. 

St. Martin, Fosse No. 3 

Trieukaisin 

Poirier, Fosse St. Louie 

Bayemont, Fosse St. Charles. 

Sacr6-Madame 

Sars-les-Moulins, Fosse No. 7 
Carabinier-francaise No. 7. . . 

Roton, veine Greffier 

Pont-du-Loup 

Austria-Hungary. 

Lower Austria. 

Griinbach 

Thallern 

Upper Austria. 
Wolfsegg-Trannthal 



B.T.U. 


Calories. 


15244 


8469) 


15100 


8389 V 


15316 


8509) 


15105 


8392) 


15188 


8438 V 


15082 


8379 j 


14353 


7974) 


14549 


8083 V 


15397 


8554) 


15228 


8460 \ 


15409 


8561/ 


14576 


8098) 


14326 


7959 


14508 


8060 


14446 


8037 


14553 


8085 


14943 


8302 " 


14407 


8004 


14877 


8265 


15217 


8454 


15107 


8393 J 


14702 


8168) 


14358 


7977 


15127 


8404 [ 


15363 


8535 | 


15168 


8427 J 


14911 


8284) 


14895 


8275 V 


14945 


8303 j 


14954 


8308) 


15069 


8372 V 


14421 


8012 


13806 


7670) 


15204 


8447 


15125 


8403 


14911 


8284 <* 


14311 


7951 


14947 


8304 J 


11458 


6366 


7057 
6006 


3921>| 
3337 J 



Nature. 



Bituminous coal, long flame. 

Bituminous coal, short 
flame. 

Bituminous coking. 

Semi-bituminous coal. 



Semi-bituminous hard coal. 



Semi-bituminous coking. 



Bituminous hard coal. 



Semi-bituminous coking. 



Semi-bituminous hard coal. 



Semi-bituminous coal. 



Lignite or brown coal. 



« 



Fuel. 



577 



Theoretical Heating Value of Coals — continued. 



Coals. 
Locality of beds. 



B.T.U. 


Calories 




9666 


53701 




9187 


5104 




6222 


3457 \ 




7997 


4443 




7556 


4198 J 




10675 


59311 




8865 


4925 




9900 


5500 




7979 


4433 \ 




7257 


4032 




9318 


5177 




6552 


5307 




6408 


35601 




7808 


4338 , 




8182 


4546 f 




8274 


4597 J 




12553 


6974 




12623 


7013 




4858 


2699\ 
2809] 




5056 




12564 


69801 




12389 


6883 




11057 


6143 [ 




13021 


7234 1 
6632 J 




11932 




10276 


5709) 
6309/ 




11356 




5200 


28891 




8325 


4625 




6913 


3841 




7966 


4426 




7709 


4283 




8069 


4483 




8087 


4493 




10182 


5657 




11286 


6270 




8692 


4829 




7911 


4359. 





Nature. 



Austria=Hungary.— Con- 
tinued. 
Styria. 

Leoben 

Fohnsdorf 

Goriagh 

Wies 

Trifail 

Bohemia. 

Kladno 

Buschtehrad 

Libuschin 

Schlan 

Rakonitz-Lubna 

Pilsen 

Schatzlar 

Aussig 

Dux 

Bilin 

Briix 

Moravia. 

Rossitz , 

M. Ostran , 

Gaya 

Goding 

Silesia. 

P. Ostran 

Orlan-Lazy 

Poremba 

Karwin 

Taklowetz 

Hungary. 

Fiinfkirchen 

Anina 

Neufeld 

Brennberg 

Aika 

Salgo-Tarjan 

Dorog-Annathal 

Tokod 

Dalmatia. 
Siveric 

Istria. 
Arsa 

Transylvania. 

Petrozseny 

Egeres 

Bosnia. 
Zenica 



Lignite or brown coal. 



Semi-bituminous coal. 



Lignite or brown coal. 



Lignite or brown coal. 



Bituminous coal. 



Cannel coal. 



Lignite or brown coal. 



578 



Fuel. 



American Coals. 



State. 
Kind of coal. 



Per cent, 
of ash. 



Theoretical value. 



In heat 
units. 



Pounds of 

water 
evaporated. 



Pennsylvania anthracite , 

Pennsylvania cannel , 

Pennsylvania, Connellsville . . , 
Pennsylvania semi-bituminous 

Pennsylvania, Stone's gas , 

Pennsylvania, Youghiogheny . . 

Pennsylvania brown 

Kentucky caking 

Kentucky cannel , 

Kentucky lignite , 

Illinois, Bureau County 

Illinois, Mercer County 

Illinois, Montauk , 

Indiana block 

Indiana caking , 

Indiana cannel , 

Maryland, Cumberland , 

Arkansas lignite 

Colorado lignite 

Texas lignite 

Washington Territory lignite . . 



14199 
13535 
14221 
13143 
13368 
13155 
14021 
14265 
12324 
14391 
15198 
13360 

9326 
13025 
13123 
12659 
13588 
14146 
13097 
12226 

9215 
13562 
13866 
12962 
11551 



14.70 
14.01 
14.72 
13.60 
13.84 
13.62 
14.51 
14.76 
12.75 
14.89 
16.76 
13.84 

9.65 
13.48 
13.58 
13.10 
14.38 
14.64 
13.56 
12.65 

9.54 
14.04 
14.35 
13.41 
11.96 






Wood 

as fuel is estimated to have about 0.4 times the calorific value as the same 
weight of coal. The relative calorific values of various woods are there- 
fore proportional to their weights. The following table gives the weight, 
in pounds, per cord. 



Kind of wood. 


Weight. 


Kind of wood. 


Weight. 


Hickory, shell-bark 

Hickory, red heart 

White oak 


4469 
3705 
3821 
3254 
2325 
2137 


Beech 


3126 


Hard maple 


2878 


Southern pine 


3375 


Red oak 


Virginia pine 


2680 


Spruce 


Yellow pine 


1904 


New Jersey pine 


White pine 


1868 









Fuels. 



579 



Total Heat of Combustion of Fuels. 

w (Rankine.) 

The following table shows the total heat of combustion with oxygen of 
1 pound of each of the substances named in it, in British thermal units, 
and also in pounds of water evaporated from 212°. It also shows the 
weight of oxygen required to combine with each pound of the combusti- 
ble and the weight of air necessary in order to supply that oxygen. The 
quantities of heat are given on the authority of MM. Favre and Silber- 
mann. 



Combustible. 


Pounds of 
oxygen per 

pound of 
combustible. 


Pounds 

of air 

(about). 


Total 

British 

heat 

units. 


Evaporative 
power from 
212° Fahr. 


Hydrogen gas 


8 

2% 
3? 


36 

6 

12 

15f 


62032 

4400 

14500 
21344 
( from 
J 21700 
1 to 
1 19000 

10000 


Lb. 
64.20 


Carbon, imperfectly burned, 
so as to make carbonic oxide 

Carbon, perfectly burned, so 
as to make carbonic acid . . . 

Olefiant gas 1 pound 


4.55 

15.0 
22.1 


Various liquid hydrocarbons, 


from 
22^ 


Carbonic oxide, as much as is 
made by the imperfect com- 
bustion of 1 pound of car- 
bon,— viz., 2% pounds 


VA 


6 


to 
20 

10.45 



Evaporative Power and Composition of Liquid Fuels. 



Fuel. 


Specific 
gravity 
at 32° 
Fahr. 
Water =1. 


Chemical 
composition. 


Heating 
power, 

in 
B. T. U. 


Theoretical 
evaporation, in 
pounds, of water 


C. 


H. 


0. 


per pound of 
fuel, from and 
at 212° Fahr. 


Pennsylvania heavy crude 
oil 


.886 
.884 
.938 
.928 

1.380 


P.c. 

84.9 
86.3 
86.6 

87.1 

80.0 


P.c. 

13.7 
13.6 

12.3 
11.7 

5.0 


P.c. 

1.4 

.1 

1.1 

1.2 

8.0 


20736 
22027 
20138 
19832 

14112 


Lb. 
21 48 


Caucasian light crude oil . 
Caucasian heavy crude oil 
Petroleum refuse 


22.79 
20.85 
20 53 


Good English coal, mean 
of 98 samples 


14 61 







580 



Fuels. 



Comparative Evaporation of Coal and Oil. 

Taken from the United States Geological Report on Petroleum for 1900. 



1 pound of combustible. 



Pounds of 
water evapo- 
rated at 212° 
per pound of 
combustible. 



Barrels of 

petroleum 

required to do 

same amount of 

evaporation as 

1 ton of coal. 



Petroleum, 18° to 40° Baume 

Pittsburg lump and nut, Pennsylvania . 
Pittsburg nut and slack, Pennsylvania . 

Anthracite, Pennsylvania 

Indiana block 

Georges Creek lump, Maryland 

New River, West Virginia 

Pocahontas lump, West Virginia 

Cardiff lump, Wales 

Cape Breton, Canada 

Nanaimo, British Columbia 

Co-operative, British Columbia 

Greta, Washington 

Carbon Hill, Washington 



10.0 
8.0 
9.8 
9.5 

10.0 
9.7 

10.5 

10.0 
9.2 
7.3 
8.9 
7.6 
7.6 



4.0 
3.2 
3.9 
3.8 
4.0 
3.8 
4.2 
4.0 
3.7 
2.9 
3.6 
3.0 
3.0 



Under favorable conditions 1 pound of oil will evaporate from 14 to 16 
pounds of water from and at 212° ; 1 pound of coal will evaporate from 7 
to 10 pounds of water from and at 212° ; 1 pound of natural gas will evapo- 
rate from 18 to 20 pounds of water from and at 212°. 



Relative Values in Coal and Oil. 

Petroleum residuum 19500 

Beaumont crude 18500 

Anthracite coal— East Middle coal-field 13400 

Semi-bituminous coal— Cumberland, Maryland 14400 

Be mi-bituminous coal— Pocahontas, Virginia 15070 

Bituminous coal— Jackson County, Ohio 13090 

Bituminous coal— Hocking Valley, Ohio 12130 

Bituminous coal— Missouri 12230 

Bituminous coal— Alabama 13500 

Bituminous coal — McAllester, Indian Territory 12789 

Bituminous coal— New Mexico 12000 

Bituminous coal— Texas lignite 10000 



Gaseous Fuels. 

The most valuable gaseous fuel is the natural gas of Pennsylvania and 
Ohio, the calorific power being about 1100 B. T. U. per cubic foot, or 10,000 
calories per cubic metre. In comparison, 57.25 pounds of coal or 63 pounds 
of coke are about equal to 1000 cubic feet of natural gas. 

Producer gas, made by the partial combustion of coal to carbonic oxide, 
is a lean gas composed of about 25 per cent, of CO and about 60 per cent. 
of nitrogen, with small quantities of C0 2 and hydrogen. The calorific 
value is about 150 B. T. U. per cubic foot. 



Steam. 



581 



Blast-furnace gas is almost identical with producer gas in composition, 
except that there is usually more C0 2 present, the calorific power falling 
to about 120 B. T. U. per cubic foot. 

These lean gases can be used to advantage in properly designed gas 
engines with a high thermal efficiency, and engines of 1000 horse-power 
and more are in successful operation, using the waste gases from blast 
furnaces. 

Calorific Power of Gas Fuels. 



Authority. 


Gas. 


B. T. U. 


A. G. Glasgow, M.E.. 


Plain water gas. 


327,268 per 1000 cubic feet. 


A. C. Humphreys . . . 


Plain water gas (theoreti- 
cal). 
Plain water gas. 


323,003 per 1000 cubic feet. 


F. E. Taylor, M.E.... 


8,335 per pound. 


Dr. Greene 


Plain water gas. 
Plain water gas. 


6,223 per pound. 
290,000 per 1000 cubic feet. 


Newbigging 1 


Plain water gas. 


6,649 per pound. 


r 


Carburetted water gas, 22 


650,000 per 1000 cubic feet. 


Dr. Gideon Moore < 


candle-power. 




I 


Coal gas, 18 candle-power. 


642,000 per 1000 cubic feet. 


Newbigging \ 


Coal gas, 17 candle-power. 


673,224 per 1000 cubic feet. 


Coal gas, 17 candle-power. 


21,696 per pound. 


' 


Coal gas. 


735,000 per 1000 cubic feet. 




Water gas. 


322,000 per 1000 cubic feet. 


R. D. Wood & Co. . < 


Producer gas (anthracite). 


137,000 per 1000 cubic feet. 




Producer gas (bitumi- 


156,000 per 1000 cubic feet. 




nous). 





STEAM. 

Steam is the common name for water which has been converted into 
the gaseous state by heat. When heat is applied to water in an open vessel 
at or near the level of the sea the temperature of the water will rise until 
it reaches 212° F. or 100° C, after which it will remain constant until all 
the water is vaporized. 

If we consider one pound of water at atmospheric pressure, it will 
require the expenditure of 180.9 B. T. U. to raise the temperature from the 
freezing-point, 32° F., to the boiling-point, 212° F. If heat is further sup- 
plied until the pound of water at 212° is converted into a pound of steam 
at 212°, it will be found to require 965.7 additional thermal units, so that a 
pound of steam at amospheric pressure will have a sensible temperature of 
212° F., and will contain energy equal to 180.9 + 965.7 = 1146.6 B. T. U. 

Furthermore, its volume will have increased to 1641.5 times that of the 
original pound of water at its greatest density (39° F.). 

If the steam is confined, and heat further applied, its temperature will 
rise,— the temperature, pressure, volume, and heat absorbed bearing certain 
relations to each other. These relations, of continual importance in steam 
engineering, have been the subject of much study and investigation, and 
have been tabulated in various ways by numerous authors. 

The data upon which all steam tables at present in use are founded are 
the result of experiments made by the French physicist, Begnault, in 1847. 
Regnault's observations covered only temperatures from 40° C. (104° F.) to 
230° C. (446° F.), advancing by 10° C, there being thus 20 observations in 
all ; and upon these 20 observations all the existing steam tables have been 
built by various computers who have devised formulas representing more 



582 Steam. 

or less accurately the results of the experiments, and therefore available 
for interpolating the intermediate values. 

Since modern steam engineering is beginning to demand the use of 
pressures higher than the maximum examined by Regnault, some of the 
tables have been extended to higher pressures ; but it must be understood < 
that such figures are based upon the assumption that the relations devel- 
oped within the range of Regnault's experiments continue beyond the 
limit of his work. A study of this feature of the subject will be found in 
the important paper of Macfarlane Gray, presented before the British Insti- 
tution of Mechanical Engineers in 1889. 

The tables here given in British units are those computed from the 
experiments of Regnault by the late John W. Nystrom, and may be ac- 
cepted as being as reliable as any. The figures for temperatures above 
446° F. agree fairly well with those deduced by Macfarlane Gray, and, 
until experimental researches at these higher pressures and temperatures 
are made, they may be used. 

The metric steam tables given have been compiled from those of 
Zeuner and Fliegner. 



Introduction to Steam Tables. 

(Nystrom.) 
Properties of Steam. 

Column P contains the total steam pressure, in pounds, per square inch, 
including the pressure of the atmosphere. 

Column I is the same pressure, in inches, of mercury. The specific 
gravity of mercury at 32° F. is 13.5959, compared with water of maximum 
density at 39°. 1 cubic inch of mercury weighs 0.49086 pound, of which a 
column of 29.9218 inches is a mean balance of the atmosphere, or 14.68757 
pounds per square inch. 

Column T contains the temperature of the steam on Fahrenheit's scale, 
deduced from Regnault's experiments. 

Column V con tains the volume of steam of the corresponding tempera- 
ture, T, compared with that of water of maximum density at 39° F. This 
column is calculated from the formula of Fairbairn and Tate, namely,— 

F= 25.62 + - 49513 



J + 0.72* 

Column W contains the weight per cubic foot in fractions of a pound ; 
and 

Column Cthe cubic feet per pound of saturated steam under the press- 
ure, P, and temperature, T. 

Column JL contains the heat units per pound of steam from 32° to 
temperature. T, and pressure, P, calculated from the formula, 

II =1081.91 + 0.305 T. 

Column IV contains the heat units per cubic foot of steam from 32° to 
temperature, T. 

The columns //and ZFgive the heat units required to heat the water 
from 32° to the boiling-point, and evaporate the same to steam under the 
pressure, P, and of temperature, T. 

Column L contains the latent, units of heat per pound in steam of tem- 
perature, T t and pressure, P. The latent heat expresses the work done in 
the evaporation, or the difference between the number of heat units per 
pound in the -tram and in the water of t lie same temperature. 

Column V contains the latent heat per cubic foot of steam. 

Latent heat, /, // - A, the heat units required to evaporate each 
pound oi water trom the boiling-poin! intosteam. 

In the nut ne tables the pressures are given in kilogrammes per square 
centimetre, or so-called metric atmospheres (1 kilogramme per square 
centimetre - l4.22pounds per square inch), the temperatures in degrees 
Centigrade, and the total and latent heats in calories per kilogramme and 
calories per cubic metre. 



Properties of Steam. 



583 



Steam Table. British System. 



Absolute press. 


Temp. 
Fahr. 
scale. 


Volume 
water = 
1 at 39°. 


Wt. 

lb. per 
cubic 
foot. 


Bulk, 

cubic 

feet per 

pound. 


Units 


of heat, 


from 32° to T°. 


Press. 


Lb. 

per 

sq. in. 


Inch, 
of mer. 


Total 
per 
lb. 


Total 

per 

cubic ft. 


Lat'nt 
per 
lb. 


Latent 

per 
cubic ft. 


Ab. at, 
lb. per 
sq. in. 


P 


I 


T 


V 


W 


G 


R 


B! 


L 


U 


V 


1 


2.037 


101.36 


17983.00 


.00347 


288.2400 


1112.8 


3.8614 


1043.40 


3.6337 


—14 


2 


4.074 


126.21 


10353.00 


.00602 


165.9400 


1120.4 


6.7449 


1026.00 


6.1165 


—13 


3 


6.111 


141.67 


7283.80 


.00856 


116.7500 


1125.1 


9.6308 


1015.20 


8.6901 


—12 


4 


8.149 


153.27 


5608.40 


.01112 


89.8950 


1128.7 


12.551 


1007.10 


11.199 


—11 


5 


10.18 


162.51 


4565.60 


.01366 


73.1800 


1131.5 


15.456 


1000.60 


13.714 


—10 


6 


12.22 


170.25 


3851.00 


.01619 


61.7420 


1133.8 


18.156 


995.17 


16.113 


— 9 


7 


14.26 


176.97 


3330.80 


.01872 


53.3880 


1135.9 


20.846 


990.44 


18.194 


- 8 


8 


16.29 


182.96 


2935.10 


.02125 


47.0460 


1137.7 


24.176 


986.22 


20.957 


— 7 


9 


18.33 


188.36 


2624.00 


.02377 


42.0590 


1139.4 


27.083 


982.41 


23.352 


— 6 


10 


20.37 


193.20 


2373.00 


.02628 


38.0370 


1140.8 


29.980 


978.99 


25.728 


— 5 


11 


22.41 


197.60 


2166.30 


.02880 


34.7230 


1142.2 


32.895 


975.88 


28.099 


— 4 


12 


24.44 


201.90 


1993.00 


.03130 


31.9450 


1143.5 


35.791 


972.84 


30.450 


— 3 


13 


26.48 


205.77 


1845.70 


.03380 


29.5840 


1144.7 


38.691 


970.11 


32.789 


— 2 


14 


28.52 


209.55 


1718.90 


.03629 


27.5510 


1145.8 


41.581 


967.43 


35.435 


— 1 


14.7 


29.92 


212.00 


1641.50 


.03800 


26.3110 


1146.6 


43.571 


965.70 


36.706 





15 


30.55 


213.04 


1608.60 


.03878 


25.7840 


1146.9 


44.476 


964.96 


37.421 


.3125 


16 


32.59 


216.33 


1511.70 


.04123 


24.2300 


1147.9 


47.328 


962.63 


39.690 


1 


17 


34.63 


219.45 


1426.20 


.04374 


22.8590 


1148.8 


50.248 


960.49 


42.012 


2 


18 


36.67 


222.40 


1349.80 


.04622 


21.6360 


1149.7 


53.138 


958.32 


44.393 


3 


19 


38.71 


225.25 


1281.10 


.04868 


20.5390 


1150.6 


56.011 


958.30 


46.698 


4 


20 


40.74 


227.95 


1219.70 


.05119 


19.5500 


1151.4 


58.894 


954.38 


48.655 


5 


21 


42.78 


230.60 


1163.80 


.05360 


18.6540 


1152.2 


61.758 


952.50 


51.924 


6 


22 


44.82 


233.10 


1112.90 


.05605 


17.8380 


1153.0 


64.637 


950.62 


53.282 


7 


23 


46.85 


235.49 


1066.30 


.05851 


17.0920 


1153.7 


67.503 


949.03 


55.529 


8 


24 


48.89 


237.81 


1023.60 


.06095 


16.4070 


1154.5 


70.367 


947.37 


57.743 


9 


25 


50.93 


240.07 


984.23 


.06338 


15.7760 


1155.1 


73.410 


945.76 


59.942 


10 


26 


52.97 


242.24 


947.86 


.06582 


15.1930 


1155.8 


76.074 


944.25 


62.161 


11 


27 


55.00 


244.32 


914.14 


.06824 


14.6520 


1156.4 


78.913 


942.74 


64.423 


12 


28 


57.04 


246.35 


882.80 


.07067 


14.1500 


1157.1 


81.772 


941.29 


66.521 


13 


29 


59.08 


248.33 


853.60 


.07308 


13.6820 


1157.7 


84.604 


939.88 


68.686 


14 


30 


61.11 


250.26 


826.32 


.07550 


13.2450 


1158.2 


87.444 


938.50 


70.857 


15 


31 


63.15 


252.13 


800.79 


.07791 


12.8350 


1158.8 


90.166 


937.17 


73.015 


16 


32 


65.19 


253.98 


766.83 


.08031 


12.4510 


1159.4 


93.121 


935.45 


75.126 


17 


33 


67.23 


255.77 


754.31 


.08271 


12.0900 


1159.9 


95.861 


934.57 


77.298 


18 


34 


69.26 


257.52 


733.09 


.08510 


11.7500 


1160.5 


98.782 


933.32 


79.425 


19 


35 


71.30 


259.22 


713.08 


.08749 


11.4290 


1161.0 


101.48 


932.10 


81.549 


20 


36 


73.34 


260.88 


694.17 


.08987 


11.1270 


1161.5 


104.38 


930.92 


83.662 


21 


37 


75.38 


262.50 


676.27 


.09225 


10.8400 


1162.0 


107.19 


929.76 


85.770 


22 


38 


77.41 


264.09 


659.31 


.09462 


10.5680 


1162.5 


109.98 


928.62 


87.866 


23 


39 


79.45 


265.65 


643.21 


.09700 


10.3100 


1162.9 


112.79 


927.51 


89.968 


24 


40 


81.49 


267.17 


627.91 


.09936 


10.0640 


1163.4 


115.59 


926.42 


92.059 


25 


41 


83.52 


268.66 


613.34 


.10172 


9.8310 


1163.9 


118.39 


925.35 


94.126 


26 


42 


85.56 


270.12 


599.46 


.10407 


9.6086 


1164.3 


121.17 


924.30 


96.192 


27 


43 


87.60 


271.55 


586.23 


.10642 


9.3963 


1164.7 


123.95 


923.28 


98.255 


28 


44 


89.64 


272.96 


573.58 


.10877 


9.1938 


1165.2 


126.74 


922.27 


100.32 


29 


45 


91.67 


274.33 


561.50 


.11111 


9.0002 


1165.6 


129.51 


921.29 


102.36 


30 


46 


93.71 


275.68 


549.94 


.11344 


8.8149 


1166.0 


132.29 


920.32 


104.40 


31 


47 


95.75 


277.01 


538.87 


.11577 


8.6374 


1166.4 


135.07 


919.36 


106.43 


32 


48 


97.78 


278.32 


528.25 


.11810 


8.4673 


1166.8 


137.83 


918.43 


108.46 


33 


49 


99.82 


279.62 


518.07 


.12042 


8.3040 


1167.2 


140.69 


917.49 


110.48 


34 


50 


101.86 


280.89 


508.29 


.12273 


8.1472 


1167.6 


143.30 


916.58 


112.49 


35 


51 


103.90 


282.14 


498.89 


.12505 


7.9966 


1167.9 


146.08 


915.68 


114.50 


36 


52 


105.93 


283.39 


489.85 


.12736 


7.8517 


1168.4 


148.85 


914.79 


116.51 


37 


53 


107.97 


284.58 


481.15 


.12966 


7.7122 


1168.7 


151.63 


913.93 


118.50 


38 


54 


110.01 


285.76 


472.77 


.13196 


7.5779 


1169.0 


154.48 


913.08 


120.49 


39 


55 


112.04 


286.96 


464.69 


.13428 


7.4468 


1169.4 


157.02 


912.22 


122.47 


40 


56 


114.08 


288.09 


456.90 


.13652 


7.3236 


1169.8 


159.74 


911.42 


124.43 


41 


57 


116.12 


289.24 


449.38 


.13883 


7.2030 


1170.1 


162.45 


910.48 


126.40 


42 



584 



Peopekties of Steam. 







Steam Table 


. British 


System. 






Absolute press. 


Temp. 
Fahr. 
scale. 


Volume 
water = 
1 at 39°. 


Wt. 

lb. per 
cubic 
foot. 


Bulk, 

cubic 

feet per 

pound. 


Units of heat, from 32° to T°. 


Press. 


Lb. 

per 

sq. in. 


Inch, 
of mer. 


Total 
per 
lb. 


Total 

per 

cubic ft. 


Lat'nt 
per 
lb. 


Latent 

per 
cubic ft. 


Ab. at. 
lb. per 
sq. in. 


P 


I 


T 


V 


W 


C 


H 


B! 


L 


V 


P 


58 


118.16 


290.37 


442.12 


.14111 


7.0866 


1170.5 


165.15 


909.78 


128.38 


43 


59 


120.19 


291.48 


435.10 


.14338 


6.9741 


1170.8 


167.84 


908.97 


130.33 


44 


60 


122.23 


292.58 


428.32 


.14566 


6.8654 


1171.2 


170.58 


908.18 


132.28 


45 


61 


124.27 


293.66 


421.75 


.14792 


6.7601 


1171.5 


173.27 


907.40 


134.22 


46 


62 


126.30 


294.73 


415.40 


.15018 


6.6583 


1171.8 


175.96 


906.63 


136.16 


47 


63 


128.34 


295.78 


409.25 


.15244 


6.5597 


1172.1 


178.65 


905.87 


138.09 


48 


64 


130.38 


296.82 


403.29 


.15469 


6.4642 


1172.5 


181.34 


905.13 


140.01 


49 


65 


132.42 


297.84 


397.51 


.15694 


6.3715 


1172.8 


184.03 


904.39 


141.93 


50 


66 


134.45 


298.85 


391.90 


.15919 


6.2817 


1173.1 


186.72 


903.66 


143.85 


51 


67 


136.49 


299.85 


386.47 


.16130 


6.1994 


1173.4 


189.40 


902.94 


145.64 


52 


68 


138.53 


300.84 


381.18 


.16366 


6.1099 


1173.7 


192.07 


902.23 


147.66 


53 


69 


140.56 


301.81 


376.06 


.16590 


6.0277 


1174.0 


194.74 


901.53 


149.56 


54 


70 


142.60 


302.77 


371.07 


.16812 


5.9478 


1174.3 


197.42 


900.84 


151.45 


55 


71 


144.64 


303.72 


366.34 


.17035 


5.8702 


1174.6 


200.08 


900.15 


153.34 


56 


72 


146.68 


304.69 


361.53 


.17256 


5.7948 


1174.9 


202.74 


899.46 


155.21 


57 


73 


148.72 


305.60 


356.95 


.17478 


5.7214 


1175.1 


205.40 


898.79 


157.09 


58 


74 


150.75 


306.52 


352.49 


.17690 


5.6500 


1175.4 


208.04 


898.13 


158.88 


59 


75 


152.79 


307.42 


348.15 


.17919 


5.5805 


1175.8 


210.67 


897.57 


160.83 


60 


76 


154.83 


308.32 


343.93 


.18139 


5.5129 


1176.0 


213.30 


896.83 


162.67 


61 


77 


156.86 


309.22 


339.81 


.18359 


5.4468 


1176.2 


215.93 


896.18 


164.56 


62 


78 


158.90 


310.11 


335.80 


.18578 


5.3825 


1176.5 


218.56 


895.54 


166.37 


63 


79 


160.94 


310.99 


331.89 


.18797 


5.3190 


1176.8 


221.19 


894.92 


168.22 


64 


80 


162.98 


311.86 


328.08 


.19015 


5.2588 


1177.0 


223.82 


894.27 


170.04 


65 


81 


165.01 


312.72 


324.37 


.19233 


5.1992 


1177.3 


226.44 


893.65 


171.87 


66 


82 


167.05 


313.57 


320.74 


.19451 


5.1410 


1177.6 


229.06 


893.03 


173.70 


67 


83 


169.09 


314.42 


317.20 


.19668 


5.0843 


1177.9 


231.68 


892.51 


175.52 


68 


84 


171.12 


315.25 


313.74 


.19885 


5.0289 


1178.1 


234.28 


891.82 


177.33 


69 


85 


173.16 


316.08 


310.36 


.20101 


4.9748 


1178.3 


236.89 


891.22 


179.14 


70 


86 


175.20 


316.90 


307.07 


.20317 


4.9219 


1178.6 


239.50 


890.63 


180.95 


71 


87 


177.24 


317.71 


303.85 


.20532 


4.8703 


1178.8 


242.10 


890.04 


182.75 


72 


88 


179.27 


318.51 


300.70 


.20747 


4.8198 


1179.1 


244.69 


889.46 


184.53 


73 


89 


181.31 


319.31 


297.62 


.20962 


4.7704 


1179.3 


247.29 


888.88 


186.33 


74 


90 


183.35 


320.10 


294.61 


.21185 


4.7222 


1179.6 


249.88 


888.31 


188.12 


75 


91 


185.38 


320.88 


291.66 


.21390 


4.6750 


1179.8 


252.45 


887.74 


189.88 


76 


92 


187.42 


321.66 


288.78 


.21603 


4.6288 


1180.0 


255.02 


887.19 


191.66 


77 


93 


189.46 


322.42 


285.96 


.21816 


4.5836 


1180.3 


257.58 


886.63 


193.43 


78 


94 


191.50 


323.18 


283.21 


.22029 


4.5394 


1180.5 


260.14 


886.08 


195.19 


79 


95 


193.53 


323.94 


280.50 


.22241 


4.4961 


1180.7 


262.69 


885.53 


196.94 


80 


96 


195.57 


324.67 


277.86 


.22453 


4.4537 


1180.9 


265.23 


885.00 


198.71 


81 


97 


197.61 


325.43 


275.27 


.22672 


4.4106 


1181.2 


267.77 


884.45 


200.49 


82 


98 


199.65 


326.17 


272.73 


.22875 


4.3715 


1181.4 


270.30 


883.91 


202.18 


83 


99 


201.68 


326.90 


270.24 


.23085 


4.3316 


1181.6 


273.10 


883.38 


203.92 


84 


100 


203.72 


327.63 


267.80 


.23296 


4.2926 


1181.9 


275.52 


882.85 


205.67 


85 


101 


205.76 


328.35 


265.41 


.23505 


4.2543 


1182.1 


277.85 


882.33 


207.39 


86 


102 


207.79 


329.07 


263.07 


.23715 


4.2167 


1182.3 


280.38 


881.81 


209.12 


87 


103 


209.83 


329.78 


260.77 


.23924 


4.1799 


1182.5 


282.90 


881.29 


210.84 


88 


104 


211.87 


330.48 


2:>s.r>2 


.24132 


4.1438 


1182.7 


285.42 


880.78 


212.55 


89 


105 


213.91 


331.18 


256.31 


.24340 


4.1083 


1182.9 


287.93 


880.27 


214.26 


90 


106 


215.94 


331.87 


254.14 


.24548 


4.0736 


1183.2 


290.45 


879.77 


215.96 


91 


107 


217.9S 


332.56 


252. 01 


.21750 


4.0394 


1183.4 


292.94 


879.27 


217.66 


92 


108 


220.02 


333.24 


219.92 


.2190:' 


4.0058 


1 L83.6 


295.41 


878.79 


219.36 


93 


109 


222.05 


333.92 


247.87 


.25169 


3.9731 


1183.8 


297.91 


878.28 


221.05 


94 


110 


224.10 


334.51 


245.86 


.2:.: 175 


3.9408 


L183.9 


300.44 


877.80 


222.74 


95 


111 


226.13 


335.2( 


243.88 


.25581 


3.9091 


1181.2 


302.93 


877.31 


224.42 


96 


113 


230.20 


336.58 


210.03 


.2599] 


8.8474 


1184.6 


307.90 


876.25 


227.74 


98 


114 


232.24 


337.25 


238.15 


.20201 


3.8100 


1184.8 


310.36 


875.88 


229.51 


99 


115 


234.28 


337.81 


236.31 


.20 KH 


3.7878 


1185.0 


312.86 


875.40 


231.10 


100 


120 


244.4 


311. 


227.56 


.27421 


3.6475 


1185.9 


325.20 


873.09 


239.41 


105 



Properties of Steam. 



585 







Steam Table, 


British 


System. 






Absolute press. 


Temp. 
Fahr. 
scale. 


Volume 
water = 
1 at 39°. 


Wt. 

lb. per 
cubic 
foot. 


Bulk, 

cubic 

feet per 

pound. 


Units of heat, from 32° to T°. 


Press. 


Lb. 

per 
sq. in. 


Inch, 
of mer. 


Total 
per 
lb. 


Total 

per 

cubic ft. 


Lat'nt 
per 
lb. 


Latent 

per 
cubic ft. 


Ab. at. 
lb. per 
sq. in. 


P 


I 


T 


V 


W 


G 


H 


W 


L 


V 


P 


125 


254.6 


344.1 


219.50 


.28422 


3.5184 


1186.9 


337.39 


870.85 


247.51 


110 


130 


264.8 


317.1 


212.07 


.29419 


3.3991 


1187.8 


349.44 


868.68 


255.55 


115 


135 


275.0 


350.0 


205.18 


.30406 


3.2880 


1188.7 


361.42 


866.56 


263.48 


120 


140 


285.2 


352.8 


198.78 


.31385 


3.1862 


1189.5 


373.34 


864.49 


271.32 


125 


145 


295.4 


355.6 


192.83 


.32354 


3.0908 


1190.4 


385.20 


862.48 


278.97 


130 


150 


305.6 


35S.4 


187.26 


.33315 


3.0001 


1191.2 


396.86 


860.45 


286.66 


135 


155 


315.8 


361.6 


180.00 


.3466 


2.8958 


1191.8 


413.20 


858.4 


297.5 


140 


160 


325.9 


364.5 


174.20 


.3601 


2.7916 


1192.5 


429.54 


856.5 


308.3 


145 


165 


336.0 


367.3 


167.90 


.3736 


2.6873 


1193.6 


445.88 


854.0 


319.1 


150 


170 


346.3 


369.8 


161.10 


.3871 


2.5831 


1194.7 


462.22 


852.5 


329.9 


155 


175 


356.5 


372.0 


157.00 


.3973 


2.5171 


1195.4 


475.80 


851.0 


338.7 


160 


180 


366.7 


374.2 


152.80 


.4075 


2.4541 


1196.1 


488.96 


849.4 


347.1 


165 


185 


376.9 


376.4 


148.80 


.4182 


2.3916 


1196.8 


502.10 


847.8 


355.5 


170 


190 


387.1 


378.5 


145.00 


.4292 


2.3299 


1197.4 


515.20 


846.2 


363.9 


175 


195 


397.3 


380.6 


141.50 


.4409 


2.2684 


1198.1 


528.27 


844.8 


372.4 


180 


200 


407.4 


382.6 


138.10 


.4517 


2.2137 


1198.7 


542.07 


843.3 


381.0 


185 


210 


427.8 


386.6 


132.00 


.4719 


2.1192 


1199.8 


568.40 


840.3 


398.0 


195 


220 


448.2 


390.4 


126.30 


.4935 


2.0265 


1201.0 


574.70 


837.5 


414.8 


205 


230 


468.5 


394.0 


120.80 


.5165 


1.9360 


1202.2 


620.96 


835.0 


431.3 


215 


240 


488.9 


397.6 


116.10 


.5364 


1.8646 


1203.2 


647.41 


832.3 


447.9 


225 


250 


509.3 


401.0 


111.70 


.5595 


1.7874 


1204.2 


673.85 


829.8 


464.4 


235 


260 


529.7 


404.3 


107.50 


.5803 


1.7230 


1205.2 


700.28 


827.4 


480.8 


245 


270 


550.0 


407.5 


103.70 


.6016 


1.6621 


1206.2 


726.66 


825.0 


497.1 


255 


280 


570.4 


410.6 


100.20 


.6238 


1.6031 


1207.2 


753.04 


822.8 


513.3 


265 


290 


590 8 


413.5 


97.01 


.6459 


1.5481 


1208.1 


779.40 


820.7 


529.4 


275 


300 


611.1 


416.5 


94.22 


.6681 


1.4967 


1209.0 


805.74 


818.6 


545.4 


285 


310 


631.5 


419.2 


91.13 


.6896 


1.4499 


1209.8 


832.96 


816.5 


561.4 


295 


320 


651.9 


422.1 


88.21 


.7107 


1.4071 


1210.6 


858.36 


814.4 


577.3 


305 


330 


672.3 


424.8 


85.44 


.7302 


1.3695 


1211.5 


884.63 


812.4 


593.2 


315 


340 


692.6 


427.4 


83.19 


.7547 


1.3250 


1212.3 


910.89 


810.5 


608.9 


325 


350 


713.0 


430.0 


80.99 


.7745 


1.2915 


1213.1 


937.13 


808.6 


624.5 


335 


360 


733.4 


432.4 


78.84 


.7943 


1.2590 


1213.9 


963.34 


806.9 


640.2 


345 


370 


753.8 


434.9 


76.74 


.8146 


1.2275 


1214.7 


989.51 


805.1 


655.8 


355 


380 


774.1 


437.3 


74.66 


.8353 


1.1968 


1215.5 


1015.7 


803.4 


671.3 


365 


390 


794.5 


439.6 


72.90 


.8626 


1.1597 


1216.2 


1041.8 


801.7 


686.7 


375 


400 


814.9 


441.9 


71.19 


.8745 


1.1434 


1216.8 


1067.9 


800.0 


702.0 


385 


410 


835.2 


444.1 


69.52 


.8952 


1.1170 


1217.4 


1094.0 


799.4 


717.2 


395 


420 


855.6 


446.4 


67.90 


.9142 


1.0938 


1218.0 


1120.2 


797.7 


732.4 


405 


430 


876.0 


448.5 


66.34 


.9400 


1.0634 


1218.7 


1146.3 


795.0 


747.6 


415 


440 


896.4 


450.6 


64.91 


.9599 


1.0417 


1219.4 


1172.3 


793.5 


762.8 


425 


450 


916.7 


452.6 


63.55 


.9804 


1.0201 


1220.1 


1198.3 


792.0 


777.9 


435 


460 


937.1 


454.6 


62.22 


1.0007 


. .9993 


1220.7 


1224.3 


790.5 


792.9 


445 


470 


957.5 


456.7 


60.94 


1.0211 


.9793 


1221.3 


1250.4 


789.0 


807.8 


455 


480 


977.8 


458.7 


59.72 


1.0446 


.9573 


1221.9 


1276.5 


787.5 


822.7 


465 


490 


998.2 


460.6 


58.54 


1.0652 


.9388 


1222.5 


1302.3 


786.1 


837.4 


475 


500 


1018.6 


462.5 


57.45 


1.0859 


.9209 


1223.0 


1328.1 


784.7 


852.1 


485 


525 


1069.5 


466.1 


54.81 


1.1381 


.8786 


1224.5 


1392.6 


782.3 


881.8 


510 


550 


1120.4 


471.5 


52.47 


1.1890 


.8410 


1225.8 


1456.9 


778.0 


921.3 


535 


575 


1171.4 


475.7 


50.32 


1.2397 


.8066 


1227.2 


1521.0 


775.0 


960.4 


560 


600 


1222.3 


479.8 


48.35 


1.2901 


.7751 


1228.3 


1584.8 


771.8 


1000.0 


585 


650 


1324.2 


487.6 


44.75 


1.3943 


.7172 


1230.6 


1709.5 


766.0 


1082.0 


635 


700 


1426.0 


494.9 


41.70 


1.4961 


.6684 


1232.7 


1933.8 


760.4 


1157.0 


685 


750 


1527.9 


501.8 


39.05 


1.5977 


.6259 


1234.9 


2057.7 


755.4 


1234.0 


735 


800 


1629.8 


508.4 


36.73 


1.6986 


.5887 


1237.0 


2101.2 


750.6 


1307.0 


785 


850 


1731.6 


514.6 


34.68 


1.7989 


.5554 


1238.9 


2228.3 


745.9 


1374.0 


835 


900 


1833.5 


521.4 


32.87 


1.8979 


.5269 


1241.0 


2355.4 


740.0 


1435.0 


885 


950 


1935.5 


526.0 


31.21 


1.9992 


.5002 


1242.4 


2482.5 


737.4 


1490.0 


935 


1000 


2037.2 


531.6 


29.73 


2.0986 


.4765 


1243.5 


2609.6 


732.3 


1538.0 


985 



586 



Properties of Steam. 



Steam Table. Metric System. 



Absolute 










Calories, from 0° C. to T°. 


pressure. 




Volume 


Weight 












Temp. 
Centi- 


cubic 
metres 


kilo- 
grams. 


Volume 
water 




















Kilo- 


Milli- 


grade 


per 


per 


= lat 


Total 


Total 


Latent 


Latent 


grams. 


metres 


scale. 


kilo- 


cubic 


4°C. 


per 


per 


per 


per 


per 


of mer- 




gram. 


metre. 




kilo- 


cubic 


kilo- 


cubic 


sq. cm. 


cury. 










gram. 


metre. 


gram. 


metre. 


.10 


73.6 


45.6 


15.0376 


.0665 


15038 


620.40 


41.25 


574.75 


38.50 


.20 


147.1 


59.8 


7.8064 


.1281 


7806 


624.73 


80.00 


564.84 


72.30 


.30 


220.7 


68.7 


5.3305 


.1876 


5330 


627.46 


116.50 


558.53 


104.40 


.40 


294.2 


75.5 


4.0667 


.2459 


4067 


629.52 


154.70 


553.81 


136.20 


.50 


367.8 


80.9 


3.2971 


.3033 


3297 


631.18 


191.80 


549.99 


167.00 


.60 


441.3 


85.5 


2.7777 


.3600 


2778 


632.58 


228.20 


546.76 


197.00 


.70 


514.9 


89.5 


2.4033 


.4161 


2403 


633.95 


264.10 


544.11 


227.00 


.80 


588.4 


93.0 


2.1191 


.4719 


2119 


634.87 


299.50 


541.44 


255.50 


.90 


662.0 


96.2 


1.8964 


.5273 


1896 


635.64 


335.20 


539.20 


284.70 


1.00 


735.5 


99.1 


1.7173 


.5823 


1717 


636.72 


370.76 


537.15 


312.79 


1.10 


809.1 


101.8 


1.5711 


.6365 


1571 


637.54 


405.5 


535.26 


341.0 


1.20 


882.6 


104.2 


1.4478 


.6907 


1449 


638.29 


447.5 


533.50 


368.2 


1.30 


956.2 


106.5 


1.3430 


.7446 


1343 


639.00 


486.2 


531.86 


396.0 


1.40 


1029.7 


108.7 


1.2527 


.7983 


1253 


639.66 


515.0 


530.32 


423.5 


1.50 


1103.3 


110.8 


1.1740 


.8518 


1174 


640.29 


540.0 


528.27 


450.0 


1.60 


1176.8 


112.7 


1.1050 


.9050 


1105 


640.87 


587.5 


527.49 


477.5 


1.70 


1250.4 


114 5 


1.0438 


.9580 


1044 


641.43 


614.1 


526.18 


504.3 


1.80 


1323.9 


116.3 


.9891 


1.0109 


989 


641.96 


649.0 


524.93 


531.0 


1.90 


1397,5 


118.0 


.9398 


1.0637 


940 


642.48 


683.0 


523.64 


556.0 


2.00 


1471.0 


119.6 


.8960 


1.1161 


896 


642.97 


718.0 


522.60 


583.3 


2.10 


1544.6 


121.1 


.8562 


1.1684 


856 


643.44 


752.0 


521.51 


609.0 


2.20 


1618.1 


122.6 


.8190 


1.2206 


819 


643.88 


785.5 


520.44 


635.0 


2.30 


1691.7 


124.0 


.7855 


1.2726 


785 


644.32 


821.0 


519.42 


661.5 


2.40 


1765.2 


125.4 


.7553 


1.3245 


755 


644.74 


854.1 


518.44 


686.5 


2.50 


1838.8 


126.7 


.7267 


1.3763 


727 


645.15 


888.0 


517.49 


712.0 


2.60 


1912.3 


128.0 


.7003 


1.4280 


700 


645.54 


922.0 


516.57 


738.0 


2.70 


1985.9 


129.3 


.6761 


1.4793 


676 


645.94 


956.0 


515.68 


763.0 


2.80 


2059.4 


130.5 


.6531 


1.5307 


653 


646.29 


990.0 


514.81 


788.0 


2.90 


2133.0 


131.6 


.6321 


1.5820 


632 


646.65 


1024.0 


513.97 


813.0 


3.00 


2206.5 


132.8 


.6124 


1.6332 


612 


647.00 


1057.0 


513.15 


838.0 


3.10 


2280.1 


133.9 


.5938 


1.6843 


594 


647.34 


1091.0 


512.35 


864.0 


3.20 


2353.6 


135.0 


.5763 


1.7352 


576 


647.67 


1123.0 


511.57 


888.0 


3.30 


2427.2 


136.1 


.5599 


1.7864 


560 


648.00 


1158.0 


510.82 


914.0 


3.40 


2500.7 


137.1 


.5444 


1.8369 


544 


648.31 


1195.0 


510.07 


938.0 


3.50 


2574.3 


138.1 


.5296 


1.8879 


530 


648.62 


1227.0 


509.35 


962.0 


3.60 


2647.8 


139.1 


.5160 


1.9384 


516 


648.92 


1258.0 


508.64 


987.0 


3.70 


2721.4 


140.0 


.5027 


1.9889 


503 


649.21 


1292.0 


507.95 


1011.0 


3.80 


2794.9 


141.0 


.4904 


2.0392 


490 


649.50 


1325.0 


507.27 


1034.0 


3.90 


2868.5 


141.9 


.4787 


2 0894 


479 


649.78 


1357.0 


506.61 


1058.0 


4.00 


2942.0 


1 12.8 


.4673 


2.1400 


467 


650.06 


1372.0 


505.96 1088.0 


4.10 


3015.6 


143.7 


.4566 


2.1901 


457 


650.33 


1 125.0 


505.32 1107.0 


4.20 


3089.1 


144.6 


.4464 


2.2401 


446 


650.60 


1457.0 


504.70 


1131.0 


4.30 


3162.7 


1 15.4 


.4367 


2.2904 


437 


650.86 


1492.0 


504.09 


1156.0 


4.40 


3236.2 


1 If..:; 


. 1273 


2.3403 


427 


651.10 


1525.0 


503.47 


1177.0 


4.50 


3309.8 


117.1 


.4184 


2.3901 


418 


651.35 


1558.0 


502.88 


1203.0 


4.60 


3383.3 


1 17. <> 


. 1098 


2.4402 


410 


651.60 


1591.0 


502.30 


1227.0 


4.70 


3459.9 


1 18.7 


.4016 


2.4900 


402 


651.85 


1624.0 


501.73 


1250.0 


4. SO 


3530.4 


1 19.5 


.3938 


2.5394 


394 


652.09 


1658.0 


501.17 


1274.0 


4.90 


3604.0 


150.2 




2.5893 


386 


652.31 


1691.0 


500.61 


1297.0 



Pkoperties of Steam. 



587 



Steam Table. Metric System. 



Absolute 










Calories, from 0° C. to T°. 


pressure. 


Temp. 
Centi- 


Volume 
cubic 
metres 


Weight 
kilo- 
grams. 


Volume 
water 






















Kilo- 


Milli- 


grade 


per 


per 


= 1 at 


Total 


Total 


Latent 


Latent 


grams. 


metres 


scale. 


kilo- 


cubic 


4°C. 


per 


per 


per 


per 


per 


of mer- 




gram. 


metre. 




kilo- 


cubic 


kilo- 


cubic 


sq. cm. 


cury. 










gram. 


metre. 


gram. 


metre. 


5.00 


3677.6 


151.0 


.3786 


2.6412 


379 


652.45 


1723 


500.07 


1321 


5.10 


3751.1 


151.7 


3720 


2.6882 


372 


652.78 


1756 


499.54 


1344 


5.20 


3824.7 


152.5 


.3654 


2.7375 


365 


653.00 


1789 


499.01 


1367 


5.30 


3898.2 


153.2 


.3588 


2.7871 


359 


653.21 


1821 


498.48 


1390 


5.40 


3971.8 


153.9 


.3524 


2.8369 


352 


653.43 


1854 


497.97 


1413 


5.50 


4045.3 


154.6 


.3465 


2.8860 


346 


653.55 


1887 


497.47 


1436 


5.60 


4118.9 


155.3 


.3407 


2.9351 


341 


653.87 


1920 


496.98 


1459 


5.70 


4192.4 


156.0 


.3351 


2.9842 


335 


654.06 


1943 


496.48 


1482 


5.80 


4266.0 


156.6 


.3297 


3.0331 


330 


654.27 


1985 


496.00 


1505 


5.90 


4339.5 


157.3 


.3243 


3.0826 


324 


654.46 


2018 


495.52 


1528 


6.00 


4413.1 


157.9 


.3193 


3.1319 


319 


654.66 


2051 


495.04 


1551 


6.10 


4-186.6 


158.6 


.3144 


3.1807 


314 


654.85 


2083 


494.60 


1573 


6.20 


4560.2 


159.2 


.3096 


3.2300 


310 


655.03 


2116 


494.16 


1596 


6.30 


4633.7 


159.8 


.3049 


3.2787 


305 


655.21 


2148 


493.72 


1618 


6.40 


4707.3 


160 5 


.3004 


3.3278 


300 


655.40 


2181 


493.28 


1641 


6.50 


4780.8 


161.1 


.2962 


3.3761 


296 


655.59 


2213 


492.83 


1663 


6.60 


4854.4 


161.7 


.2919 


3.4247 


292 


655.78 


2246 


492.39 


1686 


6.70 


4927.9 


162.3 


.2879 


3.4734 


288 


655.96 


2278 


491.95 


1708 


6.80 


5001.5 


162.9 


.2839 


3.5224 


284 


656.15 


2311 


491.52 


1731 


6.90 


5075.0 


163.4 


.2800 


3.5714 


280 


656.33 


2343 


491.07 


1753 


7.00 


5148.6 


164.0 


.2763 


3.6193 


276 


656.52 


2376 


490.63 


1776 


7.25 


5332.4 


165.4 


.2673 


3.7411 


267 


656.93 


2443 


489.64 


1822 


7.50 


5516.3 


166.8 


.2590 


3.8610 


259 


657.35 


2540 


488.66 


1890 


7.75 


5700.2 


168.1 


.2511 


3.9825 


251 


657.76 


2620 


487.67 


1942 


8.00 


5884.1 


169.5 


.2437 


4.1034 


244 


658.18 


2700 


486.69 


1998 


8.25 


6068.0 


170.7 


.2368 


4.2230 


237 


658.55 


2782 


485.79 


2052 


8.50 


6251.8 


172.0 


.2302 


4.3440 


230 


658.93 


2867 


484.89 


2109 


8.75 


6435.7 


173.2 


.2241 


4.4623 


224 


659.30 


2942 


483.99 


2161 


9.00 


6619.6 


174.4 


.2182 


4.5830 


218 


659.68 


3022 


483.10 


2216 


9.25 


6803.5 


175.5 


.2127 


4.7015 


213 


660.02 


3105 


482.28 


2270 


9.50 


6987.4 


176.7 


.2074 


4.8216 


207 


660.37 


3185 


481.46 


2321 


9.75 


7171.2 


177.8 


.2024 


4.9407 


202 


660.71 


3265 


480.64 


2375 


10.00 


7355.1 


178.9 


.1975 


5.0607 


197 


661.06 


3345 


479.82 


2432 


10.25 


7539.0 


180.0 


.1931 


5.1787 


193 


661.38 


3425 


479.06 


2483 


10.50 


7722.9 


181.0 


.1888 


5.2966 


189 


661.68 


3505 


478.29 


2535 


10.75 


7906.7 


182.0 


.1847 


5.4142 


185 


662.00 


3585 


477.53 


2586 


11.00 


8090.6 


183.0 


.1807 


5.5340 


181 


662.33 


3665 


476.77 


2638 


11.25 


8274.5 


184.0 


.1769 


5.6497 


177 


662.62 


3745 


476.04 


2690 


11.50 


8458.4 


185.0 


.1733 


5.7703 


173 


662.92 


3825 


475.32 


2742 


11.75 


8642.2 


186.0 


.1698 


5.8858 


170 


663.21 


3905 


474.61 


2794 


12.00 


8826.1 


186.9 


.1665 


6.0060 


166 


663.51 


3985 


473.92 


2846 


12.25 


9010.0 


187.9 


.1634 


6.1200 


163 


663.75 


4064 


473.24 


2897 


12.50 


9193.9 


188.8 


.1603 


6.2383 


160 


664.08 


4143 


472.57 


2948 


12.75 


9377.8 


189.7 


.1573 


6.3573 


157 


664.35 


4222 


471.90 


2998 


13.00 


9561.6 


190.6 


.1545 


6.4725 


154 


664.63 


4301 


471.25 


3049 


13.50 


9929.4 


192.3 


.1491 


6.7069 


149 


665.15 


4467 


469.97 


3155 


14.00 


10297.1 


194.0 


.1441 


6.9396 


144 


665.67 


4620 


468.73 


3254 


14.50 


10664.9 


195.6 


.1394 


7.1737 


139 


666.17 


4780 


467.51 


3353 


15.00 


11032.7 


197.2 


.1351 


7.4019 


135 


666.67 


4935 


466.35 


3452 



588 



Flow of Steam. 



In the preceding tables the temperatures are given which correspond to 
the respective pressures, it being understood that these are the tempera- 
tures at which the steam is formed from the water under those pressures. 
Such steam is said to be saturated ; it contains no moisture ; neither is it 
superheated. If, now, the steam be further supplied with heat, its tern-.** 
perature will rise and it will become superheated. The effect of the 
additional heat upon the steam is similar to that upon a gas, and the more 
highly it is superheated the more nearly it resembles a perfect gas. For 
any given pressure saturated steam can have but one temperature, as 
given in the tables. Superheated steam may have any higher temperature. 

Flow of Steam. 

The flow of steam from one pressure to another increases as the in- 
crease in difference in pressure, until the lower pressure becomes 58 per 
cent, of the higher pressure. If the lower pressure be diminished, or even 
is made a perfect vacuum, the flow will not be affected. Steam will 
expand in a nozzle until it reaches the external pressure, provided the 
latter is not less than 58 per cent, of the internal pressure. The ratio of 
expansion for all external pressures below 58 per cent, of the internal 
pressure is 1 to 1.624. The discharge will then have a constant velocity of 
890 feet per second, and the amount discharged will be proportional to the 
density of the steam, which latter value can be obtained from the steam 
tables. 

The following formulas, by Rankine, may be used in computing the 
discharge of steam : 
Let 

W= weight discharged, in pounds, per minute ; 
a = area of opening, in square inches ; 
p = absolute pressure, in pounds, per square inch ; 
d = difference in pressure, when more than 58 per cent. ; 
k = coefficient = 0.93 for short nozzle = 0.63 for hole in thin plate. 
W=0.$5ap, 

when discharging into atmosphere. 

W=1.9ak\/(p — d)d, 
when the difference between the two pressures is more than 58 per cent. 

The following table, compiled by D. K. Clark from experiments by 
Brownlee, will be useful in this connection. 

Outflow of Steam from a given Initial Pressure into 
Various Lower Pressures. 

Absolute initial pressure in boiler 75 pounds per square inch. 
(D. K. Clark.) 



Absolute 






Velocity of 


Actual 


Discharge 


pressure; in 
boiler iu 


External 


Ratio of 


outflow at 


velocity of 


per square 


pressure, in 


expan- 


constant 


outflow ex- 


inch of 


pounds, per 
square inch. 


pounds, per 


sion in 


density, in 


panded, in 


orifice, in 


square inch. 


nozzle. 


feet, per 


feet, per 


pounds, per 






second. 


second. 


minute. 


75 


74.00 


1.012 


227.5 


230.0 


16.68 


7:> 


72.00 


1.037 


386.7 


401.0 


28.35 


75 


70.00 


1.063 


490.0 


521.0 


35.93 


75 


65.00 


1.136 


600.0 


749.0 


48.38 


75 


61.62 


1.198 


7:56.0 


876.0 


53.97 


75 


60.00 


1.219 


765.0 


933.0 


56.12 


75 


50.00 


l.i::i 


87:5.0 


1252.0 


64 00 


75 


45.00 


1.575 


890.0 


1401.0 


65.24 


75 


48.46 


1.624 


890.6 


1446.5 


65.3 


75 


15 


1.624 


890.6 


1446.5 


65.3 


75 





1.021 


890.6 


1446.5 


65.3 



Flow of Steam. 



589 



Napier's rule, which is a close approximation, is that the absolute 
pressure, in pounds, per square inch, multiplied by the area in square 
inches, divided by 70, equals the discharge, in pounds, per second. 

Brownlee's formula for the discharge of steam of varying pressures of 
a£he atmosphere is _ 

v = 3.5953;/ h, 

in which v = the velocity of outflow, in feet, per second as for steam of 
the initial density, and h — the height, in feet, of a column of steam of the 
given absolute initial pressure of uniform density, the weight of which is 
equal to the pressure on the unit of base. 

Example. Boiler pressure, 80 pounds per square inch above the atmos- 
phere. With what velocity will steam flow out of an orifice in the shell,— 
for example, a safety valve? 

Here the absolute pressure = 80 -f 14.7 = 94.7 pounds per square inch. 
The volume of one pound of steam at this pressure = 4.56 cubic feet; 
consequently, the height of a column of this steam 1 inch square, and 
weighing 94.7 pounds, will be 

4.56 X 144 X 94.7 = 62183.81 feet = h. 

Then by the formula the velocity of outflow will be 



v = 3.5953j//i = 3.5953;/ 62183.81 = 3.5953 X 249.37 - 896 feet per second. 

To find the amount of steam discharged from an orifice of any given 
size in a given time, we have merely to multiply the area of the orifice by 
the above velocity, and this product by the time in seconds, to obtain the 
volume of steam discharged, from which it is easy to calculate its weight 
by reference to a steam table. 



Velocity of Efflux of Steam into the Atmosphere. 



Pressure 

per 
gauge. 


Velocity of 

discharge, 

in feet, per 

second. 


Pounds of steam 
discharged, per 

minute, per 

square inch of 

opening. 


Pressure 

per 
gauge. 


Velocity of 

discharge, 

in feet, per 

second. 


Pounds of steam 
discharged, per 

minute, per 

square inch of 

opening. 


10 


861 


22.2 


70 


894 


73.5 


15 


867 


26.6 


75 


895 


77.6 


20 


871 


30.9 


80 


896 


81.9 


25 


874 


35.3 


85 


898 


86.0 


30 


877 


39.5 


90 


899 


90.3 


35 


880 


43.8 


95 


900 


94.4 


40 


882 


48.0 


100 


902 


98.6 


45 


884 


52.3 


110 


904 


106.9 


50 


886 


56.5 


120 


906 


115.2 


55 


888 


60.7 


130 


908 


123.5 


60 


890 


65.0 


140 


910 


131.9 


65 


892 


69.3 


150 


912 


140.2 



Flow of Steam in Pipes. 

The quantity of steam flowing through a pipe under a given head 
increases directly as the square root of the density of the loss of pressure, 
and inversely as the square ro ot of t he length. A formula used for flow 

of steam in pipes is V = 50a/— D, in which V = velocity, in feet, per 



590 



Plow of Steam. 



second, L = length, and D = diameter of pipe, in feet, H= height, in feet, 
of a column of steam of the pressure of the steam at the entrance, which 
would produce a pressure equal to the difference of pressures at the two 
ends of the pipe. 

If Q = quantity, in cubic feet, per minute, d — diameter, in inches. £,.* 
and H being in feet, formula reduces to 



Q = 4.7233 



V 



JfdP, H = .0448 ^, d 

Li OP 



= .5374^/- 



H ' 



A pipe 1 inch in diameter, 100 feet long, carrying steam of 100 pounds 
gauge-pressure at 6000 feet velocity per minute, would have a loss of press- 
ure of 8.8 pounds per square inch, while steam travelling at the same 
velocity in a pipe 8.8 inches in diameter would lose only 1 pound pressure. 

The following generally-accepted formula gives the weight of steam 
which, with a given vertical pressure, will flow through a given pipe : 

W = weight, in pounds avoirdupois ; 
D = density or weight per cubic foot ; 
d = diameter, in inches ; 
Pi = initial pressure ; 
p 2 = pressure at end of pipe ; 
L = length, in feet. 



W = 87 



D(Pi—p 2 )d^ 



Flow of Steam through Pipes. 



OS «e 


Diameter of pipe, in inches. Length of each = 240 diameters. 


11 


% 


1 


IK 


2 


2V 2 


3 


4 


5 


6 


8 


10 


12 


15 


18 


•43 pu 
3.2 


Weight of steam per minute, in pounds, with 1 pound loss of pressure. 


1 


1.16 


2.07 


5.7 


10.27 


15.45 


25.38 


46.85 


77.3 


115.9 


211.4 


341.1 


502.4 


804 


1177 


10 


1.44 


2.57 


7.1 


12.72 


19.15 


31.45 


58.05 


95.8 


143.6 


262.0 


422.7 


622.5 


996 


1458 


20 


1.70 


3.02 


8.3 


14.94 


22.49 


36.94 


68.20 


112.6 


168.7 


307.8 


496.5 


731.3 


1170 


1713 


30 


1.91 


3.40 


9.4 


16.84 


25.35 


41.63 


76.84 


126.9 


190.1 


346.8 


559.5 


824.1 


1318 


1930 


40 


2.10 


3.74 


10.3 


18.51 


27.87 


45.77 


84.49 


139.5 


209.0 


381.3 


615.3 


906.0 


1450 


2122 


50 


2.27 


4.04 


11.2 


20.01 


30.13 


49.48 


91.34 


150.8 


226.0 


412.2 


665.0 


979.5 


1567 


2294 


60 


2.43 


4.32 


11.9 


21.38 


32.19 


52.87 


97.60 


161.1 


241.5 


440.5 


710.6 


1046.7 


1675 


2451 


70 


2.57 


4.58 


12.6 


22.65 


34.10 


56.00 


103.37 


170.7 


255.8 


466.5 


752.7 


1108.5 


1774 


2596 


80 


2.71 


4.82 


13.3 


23.82 


35.87 


58.91 


108.74 


179.5 


269.0 


490.7 


791.7 


1166.1 


1866 


2731 


90 


2.83 


5.04 


13.9 


24.92 37.52 


61.62 


113.74 


187.8 


281.4 


513.3 


828.1 


1219.8 


1951 


2856 


100 


2.95 


5.25 


14.5 


25.96 


39.07 


64.18 


118.47 


195.6 


293.1 


534.6 


862.6 


1270.1 


2032 


2975 


120 


3.16 


5.63 


L5.5 


27.85 


41.93 


68.87 


127.12 


209.9 


314.5 


573.7 


925.6 


1363.3 


2181 


3193 


150 


345 


6.14 


17.0 


30.37 


15.72 


75.09 


138.61 


228.8 


343.0 


625.5 


1009.2 


1486.5 


2378 


3481 



4 

For any loss of pressure, multiply by the square root of the proposed 

>s. 

For any other length of pipe, divide 240 by the given length expressed 
in diameters, and multiply the table figures by the square root of this 
quotient to get the flow for 1 pound loss of pressure. 



Flow of Steam. 591 



The resistance due to steam entering pipe = 60 diameters additional 
length ; to a globe valve = 60 ; to an elbow = 40, or % of a globe valve. 
All these equivalents must be added in getting out total length of pipe, 
with corresponding losses. 



Moisture in Steam. 

Various methods have been devised for determining the percentage of 
moisture in steam, but the principal difficulty involved in their use lies 
in the impossibility of obtaining an average sample of the steam. 

Professor J. E. Denton has shown that the appearance of an escaping 
jet of steam will reveal to the eye the presence or absence of moisture up 
to about 2 per cent, of moisture. If the jet be transparent close to the 
orifice, the steam may be assumed to be so nearly dry that no portable 
condensing calorimeter will be capable of measuring the small amount of 
moisture present. If the jet be strongly white, the amount of water may 
be roughly judged up to about 2 per cent., but beyond this only a calori- 
meter can determine the amount of moisture present. 

In the appendix to the report of the committee of the American Society 
of Mechanical Engineers on steam boiler trials, Mr. Kent says : " For scien- 
tific research and in all cases in which there is reason to suspect that the 
moisture may exceed 2 per cent., a steam separator should be placed in 
the steam pipe as near to the steam outlet of the boiler as convenient, 
well covered with felting, all the steam made by the boiler passing through 
it, and all the moisture caught by it carefully weighed after being cooled. 
A convenient method of obtaining the weight of the drip from the sepa- 
rator is to discharge it through a trap into a barrel of cold water standing 
on a platform scale. A throttling or a separating calorimeter should be 
placed in the steam pipe, just beyond the steam separator, for the purpose 
of determining, by the sampling method, the small percentage of moisture 
which may still be in the steam after passing through the separator." 

The formula for calculating the percentage of moisture when the throt- 
tling calorimeter is used is the following : 

v , = mx B - h -« T - t \ 

Ju 

in which vj = percentage of moisture in the steam, R — total heat and 
L = latent heat per pound of steam at the pressure in the steam pipe, h = 
total heat per pound of steam at the pressure in the discharge side of the 
calorimeter, k = specific heat of superheated steam, T = temperature of 
the throttled and superheated steam in the calorimeter, and t = tempera- 
ture due to the pressure in the discharge side of the calorimeter, = 212° F. 
at atmospheric pressure. Taking k = 0.48 and t = 212, the formula reduces 

«- = ioox g - 1146 - 6 -°- 48(7 - 212) . 

L 

For descriptions of the throttling calorimeter of Peabody, see " Transac- 
tions of the American Society of Mechanical Engineers," Vol. X., p. 327; 
for the Barrus calorimeter, Vol. XI., p. 790, and Vol. XVII., p. 617; and 
for the Carpenter calorimeter, Vol. XII., p. 640, and Vol. XVII., p. 608. 

In treating of superheated steam it is customary to give the number of 
degrees of superheat, — that is, the excess of temperature over that due to 
the pressure, as shown in the steam tables. It is sometimes desirable to 
give the so-called "quality" of the steam, this being the percentage of 
excess heat. 

The quality of the superheated steam is determined from the number 
of degrees of superheating by using the following formula : 

„_L + 0A$(T—t) 
H ~ L ' 

j in which L is the latent heat, in British thermal units, in 1 pound of steam 
of the observed pressure ; T, the observed temperature ; and t, the normal 



592 Steam Boilers. 



temperature due to the pressure. This normal temperature should be 
determined by obtaining a reading of the thermometer when the fires are 
in a dead condition and the superheat has disappeared, this temperature 
being observed when the pressure as shown by the gauge is the average 
of the readings taken during the trial. « 



STEAM BOILERS. 

A steam boiler is essentially a device for the conversion of water from 
the liquid to the gaseous state by the means of heat. Its performance 
should therefore be based entirely upon thermal considerations : the con- 
version of the energy in the fuel into energy in the steam, regardless of 
the use to which the steam is to be put. To speak of the horse-power of a 
boiler is distinctly unscientific, and is to be as strongly discouraged as the 
expressions horse-power of a feed-water heater, of a condenser, of a chim 
ney, or any similar device. The capacity of a boiler is fully indicated by 
a statement of the quantity of water it is capable of evaporating in a given 
time, and its economy by the proportion of combustible required to the 
quantity of water evaporated. 

The fact that the number of pounds of water evaporated to equal a 
boiler horse-power has varied from time to time shows the unsuitability of 
the application of the term to a steam boiler. At the same time, the com- 
mercial requirements of the business demand some definition of a boiler 
horse-power, and at the present time the evaporation of 30 pounds of water 
from feed water at a temperature of 100° F., as established by the judges 
of the Centennial Exhibition of 1876, may be used. It is always desirable, 
however, that the capacity of a boiler should be stated in terms of the 
number of pounds of water it will evaporate, from and at the boiling- 
point. 

According to the steam tables, it will be seen that 965.7 B. T. U. are 
required to convert a pound of water at 212° to a pound of steam at the 
same temperature. If we assume a pound of combustible in the fuel to be 
capable of supplying 14,500 B. T. U., a perfectly efficient steam boiler 
would be capable of evaporating 

14500 , KIMK , 

9657 = 15.015 pounds 

of water for every pound of combustible burned. The actual efficiency of 
a boiler, therefore, is found by dividing the actual evaporation by 15.015. 
Thus, if a boiler evaporates 10 pounds of water per pound of combustible, 
its efficiency is 

or 66 per cent. 

In order to compute beforehand the proportions which will give the 
best efficiency, the formula of Rankine may be used. 

Let 

E = theoretical evaporative power of fuel used, pounds of water; 
E' — actual evaporative power, pounds of water ; 
S = square feet of heating surface in boiler ; 
F = pounds of fuel burned per square foot of grate per hour ; 

Then we have 

Efficiency = - 



E S+AF' 
'S + AF' 



Steam Boilers. 593 



The value of E varies with the composition of the coal, and may be 
computed by Dulong's formula or determined by a calorimeter. 
The constants A and B may be taken as follows : 

I. Chimney draft, hottest gases meeting hottest water, economizer in 

flue, B = 1,A -0.5. 
II. Ordinary flow of gases, chimney draft, B = 0.916, A = 0.5. 

III. Forced draft, hottest gases meeting hottest water, B = 1, A = 0.3. 

IV. Forced draft, ordinary flow of gases, B = 0.95, A = 0.3. 

From the above it will be seen that a high efficiency may be obtained 
by causing the gases to flow in such a manner as to bring the hottest por- 
tion into contact with that portion of the boiler containing the hottest 
water, the flow of water and gases being in the opposite direction ; also, 
that a moderate rate of combustion is conducive to efficiency. 

When the feed water is supplied to a boiler at a temperature of 212°, the 
only heat required to be supplied is that necessary to furnish the latent 
heat of evaporation and the heat to raise the steam to the working press- 
ure. When, however, the feed water is not at the boiling-point, it is neces- 
sary to supply additional heat to raise it to 212° F. For this reason it is 
necessary to know the temperature of the feed water in order to correct 
the observed evaporation to the equivalent evaporation from and at 212°. 

The factors for making this correction may be computed from the 
formula, 

965.7 ' 

H being the total heat of the steam at the given pressure, and h being the 
total heat of the feed water. 

The table on page 594 gives factors for various temperatures. 

The evaporative performance of steam boilers is such an important 
matter, both from a commercial and technical point of view, that it is 
desirable for all tests to be conducted in such a manner as to be compara- 
ble. The standard method of testing steam boilers, according to the 
report of the Committee of the American Society of Mechanical Engineers, 
enables such uniform methods of testing possible, and an abridgement of 
this code is here given. The complete code will be found in Volume XXI. 
of the "Transactions" of the Society, and may be obtained in pamphlet 
form. 

The Committee recommends that, as far as possible, the capacity of a 
boiler be expressed in terms of the " number of pounds of water evaporated 
per hour from and at 212 degrees." It does not seem expedient, however, 
to abandon the widely-recognized measure of capacity of stationary or 
land boilers expressed in terms of "boiler horse-power." 

The unit of commercial boiler horse-power adopted by the Committee 
of 1885 was the same as that used in the reports of the boiler tests made at 
the Centennial Exhibition in 1876. The Committee of 1885 reported in 
favor of this standard in language of which the following is an extract : 

"The Committee, after due consideration, has determined to accept the 
Centennial standard, and to recommend that in all standard trials the 
commercial horse-power be taken as an evaporation of 30 pounds of water 
per hour from a feed-water temperature of 100° F. into steam at 70 pounds 
gauge pressure, which shall be considered to be equal to 34% units of 
evaporation,— that is, to 34% pounds of water evaporated from a feed- 
water temperature of 212° FT into steam at the same temperature. This 
standard is equal to 33,305 thermal units per hour." 

The present Committee accepts the same standard, but reverses the 
order of two clauses in the statement, and slightly modifies them to read 
as follows : 

"The unit of commercial horse-power developed by a boiler shall be 
taken as 34% units of evaporation per hour, — that is, 34% pounds of water 
evaporated per hour from a feed-water temperature of 212° F. into dry 
steam of the same temperature. This standard is equivalent to 33,317 
British thermal units per hour. It is also practically equivalent to an 
evaporation of 30 pounds of water from a feed-water temperature of 100° F. 
into steam at 70 pounds gauge pressure. 

38 



594 



Evaporation Factors. 






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Boiler Trials. 595 



Rules for Conducting Boiler Trials. 

Code of 1899. 

9 I. Determine at the outset the specific object of the proposed trial, 
whether it be to ascertain the capacity of the boiler, its efficiency as a 
steam generator, its efficiency and its defects under usual working con- 
ditions, the economy of some particular kind of fuel, or the effect of 
changes of design, proportion, or operation ; and prepare for the trial 
accordingly. 

II. Examine the boiler, both outside and inside ; ascertain the dimen- 
sions of grates, heating surfaces, and all important parts ; and make a 
full record, describing the same, and illustrating special features by 
sketches. The area of heating surface is to be computed from the surfaces 
of shells, tubes, furnaces, and fire-boxes in contact with the fire or hot 
gases. The outside diameter of water-tubes and the inside diameter of 
fire-tubes are to be used in the computation. All surfaces below the mean 
water-level which have water on one side and products of combustion on 
the other are to be considered as water-heating surface, and all surfaces 
above the mean water-level which have steam on one side and products of 
combustion on the other are to be considered as superheating surface. 

III. Notice the general condition of the boiler and its equipment, 
and record such facts in relation thereto as bear upon the objects in view. 

If the object of the trial is to ascertain the maximum economy or 
capacity of the boiler as a steam generator, the boiler and all its appur- 
tenances should be put in first-class condition. Clean the heating surface 
inside and outside, remove clinkers from the grates and from the sides of 
the furnace. Remove all dust, soot, and ashes from the chambers, smoke 
connections, and flues. Close air-leaks in the masonry and poorly-fitted 
cleaning doors. See that the damper will open wide and close tight. Test 
for air-leaks by firing a few shovels of smoky fuel and immediately closing 
the damper, observing the escape of smoke through the crevices, or by 
passing the flame of a candle over cracks in the brickwork. 

IV. Determine the character of the coal to be used. For tests of 
the efficiency or capacity of the boiler for comparison with other boilers 
the coal should, if possible, be of some kind which is commercially re- 
garded as a standard. For New England and that portion of the country 
east of the Allegheny Mountains, good anthracite egg coal, containing not 
over 10 per cent, of ash, and semi-bituminous Clearfield (Pennsylvania), 
Cumberland (Maryland), and Pocahontas (Virginia) coals are thus re- 
garded. West of the Allegheny Mountains, Pocahontas (Virginia) and 
New River (West Virginia) semi-bituminous and Youghiogheny or Pitts- 
burg bituminous coals are recognized as standards.* There is no special 
grade of coal mined in the Western States which is widely recognized as 
of superior quality or considered as a standard coal for boiler testing. Big 
Muddy lump, an Illinois coal mined in Jackson County, Illinois, is sug- 
gested as being of sufficiently high grade to answer these requirements in 
districts where it is more conveniently obtainable than the other coals 
mentioned above. 

For tests made to determine the performance of a boiler with a particu- 
lar kind of coal, such as may be specified in a contract for the sale of a 
boiler, the coal used should not be higher in ash and in moisture than that 
specified, since increase in ash and moisture above a stated amount is apt 
to cause a falling off of both capacity and economy in greater proportion 
than the proportion of such increase. 

V. Establish the correctness of all apparatus used in the test for 
weighing and measuring. These are : 

1. Scales for weighing coal, ashes, and water. 

2. Tanks or water-meters for measuring water. Water-meters, as a rule, 



* These coals are selected because they are about the only coals which possess 
the essentials of excellence of quality, adaptability to various kinds of furnaces, 
grates, boilers, and methods of firing, and wide distribution and general accessi- 
bility in the markets. 



596 Boiler Trials. 



should only be used as a check on other measurements. For accurate 
work, the water should be weighed or measured in a tank. 

3. Thermometers and pyrometers for taking temperatures of air, steam, 
feed water, waste gases, etc. 

4. Pressure gauges, draught gauges, etc. «< 
The kind and location of the various pieces of testing apparatus must 

be left to the judgment of the person conducting the test, always keeping 
in mind the main object, — i.e., to obtain authentic data. 

VI. See that the boiler is thoroughly heated before the trial to its 
usual working temperature. If the boiler is new and of a form provided 
with a brick setting, it should be in regular use at least a week before the 
trial, so as to dry and heat the walls. If it has been laid off and become 
cold, it should be worked before the trial until the walls are well heated. 

VII. The boiler and connections should be proved to be free from 
leaks before beginning a test, and all water connections, including blow 
and extra feed pipes, should be disconnected, stopped with blank flanges, 
or bled through special openings beyond the valves, except the particular 
pipe through which water is to be fed to the boiler during the trial. 
During the test the t»low-off and feed pipes should remain exposed to 
view. 

If an injector is used, it should receive steam directly through a felted 
pipe from the boiler being tested.* 

If the water is metered after it passes the injector, its temperature should 
be taken at the point where it leaves the injector. If the quantity is 
determined before it goes to the injector, the temperature should be deter- 
mined on the suction side of the injector, and if no change of temperature 
occurs other than that due to the injector, the temperature thus deter- 
mined is properly that of the feed water. When the temperature changes 
between the injector and the boiler, as by the use of a heater or by radi- 
ation, the temperature at which the water enters and leaves the injector 
and that at which it enters the boiler should all be taken. In that case 
the weight to be used is that of the water leaving the injector, computed 
from the heat units if not directly measured, and the temperature, that of 
the water entering the boiler. 
Let 

w = weight of water entering the injector ; 

x = weight of steam entering the injector; 

hi = heat units per pound of water entering injector : 

h 2 = heat unitsper pound of steam entering injector ; 

h 3 = heat units per pound of water leaving injector. 

Then w + x = weight of water leaving injector, 

h 3 — hi 



ho — h s ' 

See that the steam main is so arranged that water of condensation can- 
not run back into the boiler. 

VIII. Duration of the Test.— For tests made to ascertain either the 
maximum economy or the maximum capacity of a boiler, irrespective of 
the particular class of service for which it is regularly used, the duration 
should be at least 10 hours of continuous running. If the rate of com- 
bustion exceeds 25 pounds of coal per square foot of grate surface per 
hour, it may be stopped when a total of 250 pounds of coal has been 
burned per square foot of grate. 



* Id feeding a boiler undergoing test with an injector taking steam from 
another 1 >< »i lor, or from the main steam pipe from several boilers, the evaporative 
results may be modified by a difference in the quality of the steam from such 
source compared with that supplied by the boiler being tested, and in some cases 

the connection tO the injector may act as a drip i'nv the main steam pipe. If it is 
known that the -team from the main pipe is of the same pressure and quality as 
that furnished by the boiler undergoing the test, the steam may be taken from 
such main pipe. 



Boiler Trials. 597 



In cases where the service requires continuous running for the whole 
24 hours of the day, with shifts of fireman a number of times during that 
period, it is well to continue the test for at least 24 hours. 

When it is desired to ascertain the performance under the working 
conditions of practical running, whether the boiler be regularly in use 24 
hours a day or only a certain number of hours out of each 24, the fires 
being banked the balance of the time, the duration should not be less 
than 24 hours. 

IX. Starting and Stopping a Test.— The conditions of the boiler and 
furnace in all respects should be, as nearly as possible, the same at the end 
as at the beginning of the test. The steam pressure should be the same, 
the water-level the same, the fire upon the grates should be the same in 
quantity and condition, and the walls, flues, etc., should be of the same 
temperature. Two methods of obtaining the desired equality of condi- 
tions of the fire may be used,— viz., those which were called in the Code 
of 1885 "the standard method" and "the alternate method," the latter 
being employed where it is inconvenient to make use of the standard 
method.* 

X. Standard Method of Starting and Stopping a Test.— Steam 
being raised to the working pressure, remove rapidly all the fire from the 
grate, close the damper, clean the ash-pit, and as quickly as possible start 
a new fire with weighed wood and coal, noting the time and the water- 
level,! while the water is in a quiescent state, just before lighting the fire. 

At the end of the test remove the whole fire, which has been burned 
low, clean the grates and ash-pit, and note the water-level when the water 
is in a quiescent state, and record the time of hauling the fire. The water- 
level should be as nearly as possible the same as at the beginning of the 
test. If it is not the same, a correction should be made by computation, 
and not by operating the pump after the test is completed. 

XI. Alternate Method of Starting and Stopping a Test.— The boiler 
being thoroughly heated by a preliminary run, the fires are to be burned 
low and well cleaned. Note the amount of coal left on the grate as nearly 
as it can be estimated ; note the pressure of steam and the water-level ; 
note the time, and record it as the starting time. Fresh coal, which has 
been weighed, should now be fired. The ash-pits should be thoroughly 
cleaned at once after starting. Before the end of the test the fires should 
be burned low, just as before the start, and the fires cleaned in such a 
manner as to leave a bed of coal on the grates of the same depth, and in 
the same condition, as at the start. When this stage is reached, note the 
time, and record it as the stopping time. The water-level and steam press- 
ures should previously be brought as nearly as possible to the same point 
as at the start. If the water-level is not the same as at the start, a cor- 
rection should be made by computation, and not by operating the pump 
after the test is completed. 

XII. Uniformity of Conditions.— In all trials made to ascertain maxi- 
mum economy or capacity the conditions should be maintained uniformly 
constant. Arrangements should be made to dispose of the steam so that 
the rate of evaporation may be kept the same from beginning to end. 
This may be accomplished in a single boiler by carrying the steam through 
a waste steam pipe, the discharge from which can be regulated as desired. 
In a battery of boilers, in which only one is tested, the draught may be 
regulated on the remaining boilers, leaving the test boiler to work under 
a constant rate of production. 

Uniformity of conditions should prevail as to the pressure of steam, 



* The Committee concludes that it is best to retain the designations " stand- 

'" ard" and "alternate," since they have become widely known and established in 

! the minds of engineers and in the reprints of the Code of 1885. Many engineers 

prefer the "alternate" to the "standard" method, on account of its being less 

liable to error due to cooling of the boiler at the beginning and end of a test. 

f The gauge glass should not be blown out within an hour before the water- 
level is taken at the beginning and end of a test, otherwise an error in the read- 
ing of the water-level may be caused by a change in the temperature and density 
of the water in the pipe leading from the bottom of the glass into the boiler. 



598 Boiler Trials. 



the height of water, the rate of evaporation, the thickness of fire, the 
times of firing and quantity of coal fired at one time, and as to the inter- 
vals between the times of cleaning the fires. 

The method of firing to be carried on in such tests should be dictated i 
by the expert or person in responsible charge of the test, and the method**' 
adopted should be adhered to by the fireman throughout the test. 

XIII. Keeping the Records. — Take note of every event connected 
with the progress of the trial, however unimportant it may appear. 
Record the time of every occurrence and the time of taking every weight 
and every observation. 

The coal should be weighed and delivered to the fireman in equal pro- 
portions, each sufficient for not more than one hour's run, and a fresh 
portion should not be delivered until the previous one has all been fired. 
The time required to consume each portion should be noted, the time 
being recorded at the instant of firing the last of each portion. It is desir- 
able that at the same time the amount of water fed into the boiler should 
be accurately noted and recorded, including the height of the water in the 
boiler and the average pressure of steam and temperature of feed during 
the time. By thus recording the amount of water evaporated by successive 
portions of coal, the test may be divided into several periods, if desired, 
and the degree of uniformity of combustion, evaporation, and economy 
analyzed for each period. In addition to these records of the coal and the 
feed water, half-hourly observations should be made of the temperature 
of the feed water, of the flue gases, of the external air in the boiler-room, 
of the temperature of the furnace when a furnace pyrometer is used; also, 
of the pressure of steam and of the readings of the Instruments for deter- 
mining the moisture in the steam. A log should be kept on properly- 
prepared blanks containing columns for record of the various observations. 

When the "standard method" of starting and stopping the test is used 
the hourly rate of combustion and of evaporation and the horse-power 
should be computed from the records taken during the time when the fires 
are in active condition. This time is somewhat less than the actual time 
which elapses between the beginning and end of the run. The loss of 
time due to kindling the fire at the beginning and burning it out at the 
end makes this course necessary. 

XIV. Quality of Steam.— The percentage of moisture in the steam 
should be determined by the use of either a throttling or a separating 
steam calorimeter The sampling nozzle should be placed in the vertical 
steam pipe rising from the boiler. It should be made of %-inch pipe, and 
should extend across the diameter of the steam pipe to within half an 
inch of the opposite side, being closed at the end and perforated with not 
less than twenty %-inch holes equally distributed along and around its 
cylindrical surface, but none of these 'holes should be nearer than % inch 
to the inner side of the steam pipe. The calorimeter and the pipe leading 
to it should be well covered with felting. Whenever the indications of 
the throttling or separating calorimeter show that the percentage of moist- 
ure is irregular, or occasionally in excess of 3 per cent., the results should 
be checked by a steam separator placed in the steam pipe as close to the 
boiler as convenient, with a calorimeter in the steam pipe just beyond 
the outlet from the separator. The drip from the separator should be 
caught and weighed, and the percentage of moisture computed therefrom 
added to that shown by the calorimeter. 

Superheating should be determined by means of a thermometer placed 
in a mercury-well inserted in the steam pipe. The degree of superheating 
should be taken as the difference between the reading of the thermometer 
for superheated steam and the readings of the same thermometer for satu- 
rated steam at the same pressure, as determined by a special experiment, 
and not by reference to steam tables. 

XV. Sampling the Coal and Determining its Moisture.— As each 
barrow-load or fresh portion of coal is taken from the coal-pile a repre- a 
sentative shovelful is selected from it and placed in a barrel or box In a m 
cool place and kept until the end of the trial. The samples are then mixed ™ 
and broken into piec< a not exceeding 1 inch in diameter, and reduced by 
the process of repeated quartering and crushing until a final sample 
weighing about 5 pounds is obtained and the size of the larger pieces is 
such that they will pass through a sieve with %-inch meshes. From this 



Boiler Trials. 590 



sample two 1-quart, air-tight glass preserving jars, or other air-tight vessels 
which will prevent the escape of moisture from the sample, are to be 
promptly filled, and these samples are to be kept for subsequent determi- 
nations of moisture and of heating value and for chemical analyses. 
^During the process of quartering, when the sample has been reduced to 
about 100 pounds, a quarter to a half of it may be taken for an approxi- 
mate determination of moisture. This may be made by placing it in a 
shallow iron pan not over 3 inches deep, carefully weighing it, and set- 
ting the pan in the hottest place that can be found on the brickwork of 
the boiler setting or flues, keeping it there for at least 12 hours, and then 
weighing it. The determination of moisture thus made is believed to be 
approximately accurate for anthracite and semi-bituminous coals, and 
also for Pittsburg or Youghiogheny coal ; but it cannot be relied upon for 
coals mined west of Pittsburg, or for other coals containing inherent 
moisture. For these latter coals it is important that a more accurate 
method be adopted. The method recommended by the Committee for all 
accurate tests, whatever the character of the coal, is described as follows : 
Take one of the samples contained in the glass jars and subject it to a 
thorough air-drying by spreading it in a thin layer and exposing it for 
several hours to the atmosphere of a warm room, weighing it before and 
after, thereby determining the quantity of surface moisture it contains. 
Then crush the whole of it by running it through an ordinary coffee-mill, 
adjusted so as to produce somewhat coarse grains (less than ^inch), 
thoroughly mix the crushed sample, select from it a portion of from 10 to 
50 grams, weigh it in a balance which will easily show a variation as small 
as 1 part in 1000, and dry it in an air- or sand-bath at a temperature 
between 240° and 280° F. for one hour. Weigh it and record the loss, then 
heat and weigh it again repeatedly, at intervals of an hour or less, until 
the minimum weight has been reached and the weight begins to increase 
by oxidation of a portion of the coal. The difference between the original 
and the minimum weight is taken as the moisture in the air-dried coal. 
This moisture test should preferably be made on duplicate samples, and 
the results should agree within 0.3 to 0.4 of one per cent., the mean of the 
two determinations being taken as the correct result. The sum of the 
percentage of moisture thus found and the percentage of surface moisture 
previously determined is the total moisture. 

XVI. Treatment of Ashes and Refuse.— The ashes and refuse are to 
be weighed in a dry state. If it is found desirable to show the principal 
characteristics of the ash, a sample should be subjected to a proximate 
analysis and the actual amount of incombustible material determined. 
For elaborate trials a complete analysis of the ash and refuse should be 
made. 

XVII. Calorific Tests and Analysis of Coal.— The quality of the 
fuel should be determined either by heat test or by analysis, or by both. 

The rational method of determining the total heat of combustion is to 
burn the sample of coal in an atmosphere of oxygen gas, the coal to be 
sampled as directed in Article XV. of this code. 

The chemical analysis of the coal should be made only by an expert 
chemist. The total heat of combustion computed from the results of the 
ultimate analysis may be obtained by the use of Dulong's formula (with 

constants modified by recent determinations) ,— viz. , 14600 C-f 62000 ( H-— \ 

+ 4000S, in which C, H, 0, and S refer to the proportions of carbon, hydro- 
gen, oxygen, and sulphur, respectively, as determined by the ultimate 
analysis.* 

It is desirable that a proximate analysis should be made, thereby deter- 
mining the relative proportions of volatile matter and fixed carbon. 
These proportions furnish an indication of the leading characteristics of 
the fuel, and serve to fix the class to which it belongs. As an additional 
indication of the characteristics of the fuel the specific gravity should be 
determined. 



* Favre and Silberman give 14,544 B. T. U. per pound carbon ; Berthelot, 14,647 
B. T. U. Favre and Silberman give 62,032 B. T. U. per pound bydrogen ; Thomsen, 
61,816 B. T. U. 



600 Boiler Trials. 



XVIII. Analysis of Flue Gases.— The analysis of the flue gases is an 
especially valuable method of determining the relative value of different 
methods of firing or of different kinds of furnaces. In making these 
analyses great care should be taken to procure average samples, since the 
composition is apt to vary at different points of the flue. The composition^ 
is also apt to vary from minute to minute, and for this reason the drawings 
of gas should last a considerable period of time. Where complete deter- 
minations are desired, the analyses should be intrusted to an expert 
chemist. For approximate determinations the Orsat* or the Hempelf 
apparatus may be used by the engineer. 

For the continuous indication of the amount of carbonic acid present 
in the flue gases an instrument may be employed which shows the weight 
of the sample of gas passing through it. 

XIX. Smoke Observations.— It is desirable to have a uniform system 
of determining and recording the quantity of smoke produced where bitu- 
minous coal is used. The system commonly employed is to express the 
degree of smokiness by means of percentages dependent upon the judg- 
ment of the observer. The Committee does not place much value upon a 
percentage method, because it depends so largely upon the personal ele- 
ment, but if this method is used it is desirable that, so far as possible, a 
definition lie given in explicit terms as to the basis and method employed 
in arriving at the percentage. The actual measurement of a sample of 
soot and smoke by some form of meter is to be preferred. 

XX. Miscellaneous.— In tests for purposes of scientific research, in 
which the determination of all the variables entering into the test is de- 
sired, certain observations should be made which are in general unneces- 
sary for ordinary tests. These are the measurement of the air-supply, the 
determination of its contained moisture, the determination of the amount 
of heat lost by radiation, of the amount of infiltration of air through the 
setting, and (by condensation of all the steam made by the boiler) of the 
total heat imparted to the water. 

As these determinations are rarely undertaken, it is not deemed ad- 
visable to give directions for making them. 

XXI. Calculations of Efficiency.— Two methods of defining and cal- 
culating the efficiency of a boiler are recommended. They are, — 

„ ■«,«. • ^ •! Heat absorbed per pound of combustible 

1. Efficiency of the boiler — 



2. Efficiency of the boiler and grate 



Calorific value of 1 pound of combustible ' 
Heat absorbed per pound of coal 



Calorific value of 1 pound of coal * 



The first of these is sometimes called the efficiency based on combusti- 
ble, and the second the efficiency based on coal. The first is recommended 
as a standard of comparison for all tests, and this is the one which is 
understood to be referred to when the word "efficiency" alone is used 
without qualification. The second, however, should be included in a 
report of a test, together with the first, whenever the object of the test is 
to determine the efficiency of the boiler and furnace together with the 
grate (or mechanical stoker), or to compare different furnaces, grates, 
fuels, or methods of firing. 

The heat absorbed per pound of combustible (or per pound of coal) is 
to be calculated by multiplying the equivalent evaporation from and at 
212° per pound of combustible (or of coal) by 965.7. 

XXII. The Heat Balance.— An approximate " heat balance," or state- 
ment of the distribution of the heating value of the coal among the 
several items of heat utilized and heat lost, may be included in the report 
of a test when analyses of the fuel and of the chimney gases have been 
made. It should bereported in the following form : 

*See R. S. Hales paper od "Flue Gas Analysis," "Transactions of the Ameri- 
can Society of Mechanical Engineers," Vol. XVIII., p. 109. 

f See Henipel's "Methods of Gas Analysis" (Macmillan & Co.). 



BOILEK TKIALS. 



601 



Heat Balance, or Distribution of the Heating Value of the 
Combustible. 

m9 Total heat value of 1 pound of combustible B. T. U. 



B.T.TJ. 



Per cent. 



2. 



3. 



Heat absorbed by the boiler = evaporation from 

and at 212° per pound of combustible X 965.7. 
Loss due to moisture in coal = per cent, of moisture 
referred to combustible -4- 100 X [(212 — t) + 966 + 
0.48 ( T— 212) ] (t= temperature of air in the boiler- 
room, T = that of the flue gases). 
Loss due to moisture formed by the burning of hy- 
drogen =3 per cent, of hydrogen to combustible -j- 
100 X 9X [(212 — +966 +0.48 (T— 212)]. 
4.* Loss due to heat carried away in the dry chimney 
gases = weight of gas per pound of combustible X 
0.24 X (T—t). 
5.f Loss due to incomplete combustion of carbon = 
CO per cent. C in combustible 

co 2 + co x loo x 10150 ' 

6. Loss due to unconsumed hydrogen and hydro- 
carbons, to heating the moisture in the air, to 
radiation, and unaccounted for. (Some of these 
losses may be separately itemized if data are ob- 
tained from which they may be calculated.) 

Totals 



100 



XXIII. Report of the Trial.— The data and results should be reported 
in the manner given in either one of the two following tables, omitting 
lines where the tests have not been made as elaborately as provided for in 
such tables. Additional lines may be added for data relating to the 
specific object of the test. The extra lines should be classified under the 
headings provided in the tables, and numbered as per preceding line, with 
sub-letters a, b, etc. The short form of report, Table No. 2, is recom- 
mended for commercial tests and as a convenient form of abridging the 
longer form for publication when saving of space is desirable. For elabo- 
rate trials it is recommended that the full log of the trial be shown 
graphically, by means of a chart. 



* The weight of gas per pound of carbon burned may be calculated from the 
gas analyses, as follows : 

11C0 2 + 80 + 7(CO + N) . _. _ v _,_ _ 

Dry gas per pound carbon = Q /^r> nTvi » m wnicn C0 2 , CO, 0, 

0(002 "i" CO) 
and N are the percentages by volume of the several gases. As the sampling and 
analyses of the gases in the present state of the art are liable to considerable 
errors, the result of this calculation is usually only an approximate one. The 
heat balance itself is also only approximate for this reason, as well as for the fact 
that it is not possible to determine accurately the percentage of unburned hydro- 
gen or hydrocarbons in the flue gases. 

The weight of dry gas per pound of combustible is found by multiplying the 
dry gas per pound of carbon by the percentage of carbon in the combustible, and 
dividing by 100. 

f COo and CO are respectively the percentage by volume of carbonic acid and 
carbonic oxide in the flue gases. The quantity 10150 = number of heat units 
generated by burning to carbonic acid 1 pound of carbon contained in carbonic 
oxide. 



602 Boiler Trials. 



Table No. 2. 
Data and Results of Evaporative Test. 

Arranged in accordance with the Short Form advised by the Boiler Test " 
Committee of the American Society of Mechanical Engineers. 
Code of 1899. 

Made by on boiler, at to 

determine 

Kind of f nel 

Kind of furnace. . . „ 

Method of starting and stopping the test (" standard" or 
11 alternate," Art. X. and XL, Code) 

Grate surface sq. ft. 

Water-heating surface sq. ft. 

Superheating surface sq. ft. 

Total Quantities. 

1. Date of trial 

2. Duration of trial hours. 

3. Weight of coal as fired lbs. 

4. Percentage of moisture in coal per cent. 

5. Total weight of dry coal consumed lbs. 

6. Total ash and refuse lbs. 

7. Percentage of ash and refuse in dry coal per cent. 

8. Total weight of water fed to the boiler lbs. 

9. Water actually evaporated, corrected for moisture or 

superheat in steam lbs. 

10. Equivalent water evaporated into dry steam from and 

at 212° lbs. 

Hourly Quantities. 

11. Dry coal consumed per hour lbs. 

12. Dry coal per square foot of grate surface per hour lbs. 

13. Water evaporated per hour, corrected for quality of 

steam lbs. 

14. Equivalent evaporation per hour from and at 212° lbs. 

15. Equivalent evaporation per hour from and at 212° per 

square foot of water-heating surface lbs. 

Average Pressures, Temperatures, etc. 

16. Steam pressure by gauge lbs. per sq. in. 

17. Temperature of feed water entering boiler deg. 

18. Temperature of escaping gases from boiler deg. 

19. Force of draft between damper and boiler ins. of water. 

20. Percentage of moisture in steam, or number of degrees 

of superheating per cent, or deg. 

Horse=power. 

21. Horse-power developed (Item 14 -=- 34%) H. P. 

22. Builders' rated horse-power H. P. 

23. Percentage of builders' rated horse-power developed . per cent. 

Economic Results. 

24. Water apparently evaporated under actual conditions 

per pound of coal as fired (Item 8 -*■ Item 3) lbs. 

25. Equivalent evaporation from and at 212° per pound of 

coal as fired (Item 10 ■— Item 3) lbs. 

26. Equivalent evaporation from and at 212° per pound of 

dry coal (Item 10 -^ Item 5) lbs. 

27. Equivalent evaporation from and at 212° per pound of 

combustible [Item 10 -f- (Item 5 — Item 6)] lbs. 

(If Items 25, 26, and 27 are not corrected for quality of 
steam, the fact should be stated. ) 



Boiler Trials. 603 



Efficiency. 

28. Calorific value of the dry coal per pound B. T. U. 

29. Calorific value of the combustible per pound B. T. U. 

30. Efficiency of boiler (based on combustible) per cent. 

* 31. Efficiency of boiler, including grate (based on dry coal) per cent. 

Cost of Evaporation. 

32. Cost of coal per ton of pounds, delivered in boiler- 

room $ 

33. Cost of coal required for evaporating 1000 pounds of 

water from and at 212° $ 

Although the capacity of a boiler should be specified by the quantity 
of water it is capable of evaporating per hour, it is often necessary to 
state the dimensions and proportions to be furnished. Certain general 
relations of heating and grate surface have come to be recognized as 
representing evaporative capacity, and although this practice is to be 
discouraged, it cannot be ignored. 

The area of heating surface allowed for 1 horse-power, or the evapora- 
tion of 30 pounds of water from 100° F. per hour, is usually about 12 square 
feet for return tubular or water-tube boilers, with about % of a square foot 
of grate surface per horse-power. The proportion of heating surface varies, 
however, for various kinds of boilers, and the following formula may be 
used for the determination of the heating surface in designing boilers : 
Let 

S = heating surface, in square feet ; 
Q — quantity of water evaporated per hour ; 
t = total heat of steam at the working pressure in the boiler ; 
C = constant, as per table below. 



Then 



Values of constant C: 



s =4 



Locomotive boilers C = 90 

Marine Scotch boilers G= 180 

Cornish boilers C = 220 

Plain cylinder boilers , C = 280 

Return tubular boilers C = 400 

Water- tube boilers C = 400 

Thus, for a return tubular boiler to evaporate 5000 pounds of water per 
hour into steam at 160 pounds pressure, we have Q = 5000 ; t, by steam 
table, = 1195.4 ; C = 400 ; hence, 

5 = 400-^L = 1677 square feet. 
119o.4 

Since 5000 pounds of water, at 30 pounds to the horse-power, is 166 horse- 
power, this corresponds to about 10 square feet per horse-power. 

In estimating the heating surface, all parts of a boiler are not equally 
efficient. Rankin e says that, on an average, from % to § of the total 
heating surface may be taken as effective heating surface. In computing 
the heating surface of tubes the side next to the heated gases should be 
taken. 

The relative value of different forms of heating surface, compared with 
flat horizontal surface above the fire, is as follows : 

1 square foot of flat horizontal surface above the fire, such 
as the crown-plate of the fire-box of the boiler of a loco- 
motive engine 1.00 

1 square foot of circular surface above and concave to the 
fire, such as the crown-plates of the circular furnace of 
an internally-fired boiler 95 



604 



Boiler Trials. 



1 square foot of circular surface above and convex to the 
fire, such as the surface plates of an externally-fired 
plain cylindrical boiler 90 

1 square foot of fiat surface at right angles to the current 
of gases, exposed to direct impingement of flame, such * 

as the fire-box tube- plate of a locomotive boiler 80 

1 square foot of water-tube surface at right angles to the 
current of hot gases 70 

1 square foot of sloping surface at the side of and inclined 
towards the fire, such as the sides of a fire-box when in- 
clined sufficiently to facilitate evaporation 65 

1 square foot of vertical surface at the side of the fire, such 
as the sides of a fire-box when vertical 50 

1 square foot of the surface of the tubes of a locomotive 
boiler, contained in a length not exceeding 3 feet from 
the fire-box tube- plate 30 

Horizontal surfaces below the fire and the under portions of internally- 
heated tubes have practically no evaporative value, and cannot be con- 
sidered as effective heating surface, therefore the lower half of a furnace 
tube below the grate bars should not be included in calculating the heating 
surface of a steam boiler. 

The draught area through the tubes of a boiler should be proportional to 
the area of the grate, and also depends upon the intensity of the draught. 
For natural chimney draught the area is usually made about 0.2 of the 
grate area ; it may reach 0.25, or fall as low as 0.125, but these are extremes. 

The ratio of grate surface to heating surface varies according to the 
type of boilers. Accepted proportions are as follows : 



Ratio of Grate to Heating Surface. 



Type of boiler. 



Ratio. 



Scotch marine boiler 

Lancashire 

Cornish 

Horizontal return tubular. 

Water tube 

Locomotive 

Plain cylinder 



25 to 38 

26 to 33 
25 to 40 
30 to 50 
35 to 65 
60 to 90 
10 to 15 



The quantity of water evaporated for a given combustion of fuel, when 
the proportions of heating and grate surface are given, may be determined 
by the formulas of D. K. Clark. 

Let 

w = weight of water, in pounds, per square foot of grate per hour ; 
c = pounds of fuel per square foot of grate per hour ; 
r = ratio of heating to grate surface. 



Then we have for 

Stationary boilers w = 0.0222r 2 + 9.56c 

Marine boilers w = 0.016r 2 + io.25c 

Portable engine boilers w = 0.008r 2 + 8.6c 

Locomotive boilers w = 0.009r 2 + 9.7c 



Chimneys. 



605 



CHIMNEYS. 



The proportions of chimneys to furnish proper draught for steam boilers 
^depend upon so many variables that it is impracticable to give absolute 
^rational formulas, and hence ernpirical rules are used. 

It is generally assumed that the area should bear a direct proportion to 
the quantity of fuel burned, and an inverse proportion to the square root 
of the height. The force of draught, however, has not only to draw the air 
in to maintain combustion, but must enable it to overcome the resistance 
of the fuel bed upon the grate, and this is always an indeterminate resist- 
ance. Moreover, the force of the draught depends upon the temperature of 
the discharge gases ; but this latter should not be too high, or heat will 
be lost which should have been absorbed by the boiler. The entire sub- 
ject will be found very fully discussed in the " Transactions of the Ameri- 
can Society of Mechanical Engineers," Vol. XL, pp. 451, 974, and' 984. 

A common, ready rule for chimney area is to make it equal to one-tenth 
of the area of the grate. Mr. A. F. Nagle gives the rule to allow 2 square 
inches of chimney area for every pound of coal burned per hour. 

The following formulas are given by their respective authors, as based 
upon the results of experience, taking into account the investigations of 
Peclet, Rankine, and others. 

Let 

A = area of chimney, in square feet ; 

h = height, in feet ; 

F = total number of pounds of coal burned per hour ; 

t •= temperature of discharge gases ; 
G = grate area, in square feet. 



Then 



0.0825J 7 

A = 7=" 



V h 



/0.0825i^\2 

h r\—r-) 



Smith ; 



A = 



0.06P 



V h 



/Q.j 



06.F\2 



Kent ; 



A == 0.07F? 



»-?(^ 



) 



Gale. 



The last formulas, it will be observed, do not make the height and area 
interchangeable, and for that reason they are to be preferred. Colonel 
E. D. Meier suggests the use of Gale's formula for heights, so modified as 
to read : 



t V QJ ' 



after which any other formula, such as Kent's, may be used to find the 
area. The following table has been computed by Colonel Meier for heights 
and areas of boiler chimneys, based on an assumed evaporation of 7 
pounds of waterier pound of coal, which is equivalent to the combustion 
of 5 pounds of coal per horse-power per hour. If the coal burned per hour 
is given, divide by 5, and take the chimney dimensions for the corre- 
sponding horse-power. 



606 



Chimneys. 



Table of Chimney Dimensions. 



(-1 

g. 

GO 

.2 


a 
u 


Heights, in feet. , 


75 


80 


85 


90 


95 


100 110 


120 


130 


140 


150 


175 


200 


< 


Commercial horse-power. 


3.14 
3.69 
4.28 
4.91 
5.59 


24 
26 
28 
30 
32 
34 
36 
40 
44 
48 
54 
60 
66 
72 
84 
96 
108 
120 


75 
90 


78 

92 

106 

122 


81 
95 
110 
127 
144 
162 


98 
114 
130 
149 

168 
188 


117 
133 
152 
171 
192 
237 
287 


120 

137 
156 
176 
198 
244 
296 
352 
445 


164 
185 
208 
257 
310 
370 
468 
577 
697 


215 

267 
322 
384 
484 
600 
725 
862 
1173 


279 

337 

400 

507 

627 

758 

902 

1229 

1584 

2058 


413 

526 

650 

784 

932 

1270 

1660 

2102 

2596 


672 

815 
969 
1319 
1725 
2181 
2693 


1044 
1422 
1859 
2352 
2904 






6.31 








7.07 








8.73 










10.56 












12.57 












15.90 














19.63 














23.76 
















28.27 
















38.48 


















50.27 
















1983 


63.62 


















2511 


78.54 


















31C0 



























The following formulas for chimney dimensions, for use in the metric 
system, are given in the " Ingenieurs Taschenbuch :" 
Let 

d = internal diameter, in metres ; 
h = height, in metres ; 
R = grate area, in square metres ; 
B a coal burned per hour, in kilogrammes. 
Then 

d = 0.1# - 4 metres, 



h = 0.00277 
For use in British measures we have 



(4)+' 



d = internal diameter, in feet ; 

h = height, in feet ; 

R = grate area, in square feet ; 

B = coal burned per hour, in pounds. 



Then 



d = 0.2422?o-4 feet, 



h = 0.216 



(!-)" 



+ 6d. 



These appear to be the most satisfactory formulas of all. The diameter, 
and hence the area, is dependent solely upon the quantity of coal burned 
per hour, and the height is determined mainly by the rate of combustion 



Chimneys. 



607 



per square foot of grate, plus 6 diameters ; the latter member providing for 
the relation of height to diameter. With these formulas no absurd rela- 
tions of height to diameter are possible, and the range of heights for 
various rates of combustion accord well with practice. 
» When the rate of combustion is not known it may be taken according 
to the character of the boiler and furnace. Taking the grate surface at 
0.01 square foot per pound of water evaporated per hour, or about % square 
foot per horse-power, and the quantity of water evaporated per pound of 
coal from 5 to 10 pounds,— that is, 0.20 to 0.1 pound of coal per pound of 
water,— we have corresponding ratios of 10 to 20 pounds of coal per square 
foot of grate. It is advisable to make the chimney capable of maintaining 

a rate of 20 pounds per square foot of grate, so that ( — j = 20 2 = 400, and 

this gives a minimum height of chimney for that rate as 

h = 0.216 X 400 == 86.4 feet, 

plus 6 diameters. The diameter is then determined by the total quantity 
of coal burned per hour ; and taking this at 0.2 pound of coal per pound 
of water, or 6 pounds per horse-power, we have all the data necessary to 
determine the size of a chimney for any given evaporation of water. 
Thus, for 3000 pounds of water per hour, or 100 horse-power, we have 

B = 3000 X 0.2 = 600, 

and d = 0.242 X 600 - 4 = 3.13, 

or, say, 3 feet diameter, and the height will be 

86.4 + 18 = 104.4 feet. 

The areas given by these formulas are somewhat larger than many 
rules, and may serve for boilers of at least 25 per cent, greater capacity 
than close computation will indicate. 

Theoretically, about 12 pounds of air are required for the combustion 
of 1 pound of coal ; but, in practice, from 18 to 24 pounds actually pass 
through the furnace. This excess is found necessary to insure combustion, 
owing to the imperfect mixture of the air and the gases. 



Draught Pressure Required for Combustion of 
Different Fuels. 



Kind of fuel. 


Total 
draught, 
in inches, 
of water. 


Kind of fuel. 


Total 
draught, 
in inches, 
of water. 


Straw 


.20 

.30 

.35 

.4 

.5 

.6 
.4 to .7 
.6 to .9 


Slack, very small 

Coal-dust 


7 to 1 1 


Wood 


8 to 1 1 


Sawdust 


Semi-anthracite coal 

Mixture of breeze and 
slack 


9 to 1 2 


Peat, light 




Peat, heavy 


1 to 1 3 


Sawdust mixed with 
small coal 


Anthracite, round 

Mixture of breeze and 
coal-dust 


1.2 to 1.4 


Steam coal, round 


1.2 to 1.5 


Slack, ordinary 


Anthracite slack 


1 3 to 1 8 







608 



Chimney Flues. 



Flue Area, in Square Inches, Required for the Passage 
of a Given Volume of Air at a Given Velocity, 

(B. F. Sturtevant.) 



Volume, in 










Velocity, 


in feet, per mi 


lute. 










cubic feet, 
per minute. 


300 


400 


500 


600 


700 


800 


900 


1000 


1100 


1200 


1300 


1400 


1500 


1600 


100 


48 


36 


29 


24 


21 


18 


16 


14 


13 


12 


11 


10 


9.6 


9.0 


125 


60 


45 


36 


30 


26 


23 


20 


18 


16 


15 


14 


13 


12.0 


11.3 


150 


72 


54 


43 


36 


31 


27 


24 


22 


20 


18 


16 


15 


14.4 


13.5 


175 


84 


63 


50 


42 


36 


32 


28 


25 


23 


21 


19 


18 


16.8 


15.8 


200 


96 


72 


58 


48 


41 


36 


32 


29 


26 


24 


22 


21 


19.2 


18.0 


225 


108 


81 


65 


54 


46 


41 


36 


32 


29 


27 


25 


23 


21.6 


20.3 


250 


120 


90 


72 


60 


51 


45 


40 


36 


33 


30 


28 


26 


24.0 


22.5 


275 


132 


99 


79 


66 


57 


50 


44 


40 


36 


33 


30 


28 


26.4 


24.8 


300 


144 


108 


86 


72 


62 


54 


48 


43 


39 


36 


33 


31 


28.8 


27.0 


325 


156 


117 


94 


78 


67 


59 


52 


47 


43 


39 


36 


33 


31.2 


29.3 


350 


168 


126 


101 


84 


72 


63 


56 


50 


46 


42 


39 


36 


33.6 


31.5 


375 


180 


135 


108 


90 


77 


68 


60 


54 


49 


45 


42 


39 


36.0 


33.8 


400 


192 


144 


115 


96 


82 


72 


64 


58 


52 


48 


44 


41 


38.4 


36.0 


425 


204 


153 


122 


102 


87 


77 


68 


61 


56 


51 


47 


44 


40.8 


38.3 


450 


216 


162 


130 


108 


93 


81 


72 


65 


59 


54 


50 


46 


43.2 


40.5 


475 


228 


171 


137 


114 


98 


86 


76 


68 


62 


57 


53 


49 


45.6 


42.8 


500 


240 


180 


144 


120 


103 


90 


80 


72 


65 


60 


55 


51 


48.0 


45.0 


525 


252 


189 


151 


126 


108 


95 


84 


76 


69 


63 


58 


54 


50.4 


47.3 


550 


264 


198 


158 


132 


113 


99 


88 


79 


72 


66 


61 


57 


52.8 


49.5 


575 


276 


207 


166 


138 


118 


104 


92 


83 


75 


69 


64 


59 


55.2 


51.8 


600 


288 


216 


173 


144 


123 


108 


96 


86 


79 


72 


66 


62 


57.6 


54.0 


625 


300 


225 


180 


150 


129 


113 


100 


90 


82 


75 


69 


64 


60.0 


56.3 


650 


312 


234 


187 


156 


134 


117 


104 


94 


85 


78 


72 


67 


62.4 


58.5 


675 


324 


243 


194 


162 


139 


122 


108 


97 


88 


81 


75 


69 


64.8 


60.8 


700 


336 


252 


202 


168 


144 


126 


112 


101 


92 


84 


78 


72 


67.2 


63.0 


725 


348 


261 


209 


174 


149 


131 


116 


104 


95 


87 


80 


75 


69.6 


65.3 


750 


360 


270 


216 


180 


154 


135 


120 


108 


98 


90 


83 


77 


72.0 


67.5 


775 


372 


279 


223 


186 


159 


140 


124 


112 


101 


93 


86 


80 


74.4 


69.8 


800 


384 


288 


230 


192 


165 


144 


128 


115 


105 


96 


89 


82 


76.8 


72.0 


825 


396 


297 


238 


198 


170 


149 


132 


119 


108 


99 


91 


85 


79.2 74.3 


850 


408 


306 


245 


204 


175 


153 


136 


122 


111 


102 


94 


87 


81.6 76.5 


875 


420 


315 


252 


210 


180 


158 


140 


126 


115 


105 


97 


90 


84.0 


78.8 


900 


432 


324 


259 


216 


185 


162 


144 


130 


118 


108 


100 


93 


86.4 


81.0 


925 


444 


333 


266 


222 


190 


1(57 


148 


133 


121 


111 


103 


95 


88.8 


83.3 


950 


456 


342 


274 


228 


195 


171 


152 


137 


124 


114 


105 


98 


91.2 


85.5 


975 


468 


351 


281 


234 


201 


176 


156 


140 


128 


117 


108 


100 


93.6 


87.8 


1000 


480 


360 


288 


240 


206 


180 


160 


144 


131 


120 


111 


103 96.0 


90.0 



*ft. 



Chimney Flues. 



609 



Flue Area, in Square Inches, Required for the Passage 
of a Given Volume of Air at a Given Velocity, 

(B. F. Sturtevant.) 



Velocity, in feet, per minute. 


Volume, in 


1700 


1800 


1900 


2000 


2100 


2200 


2300 


2400 


2600 


2700 


2800 


2900 


3000 


3100 


cubic feet, 
per minute. 


8.5 


8 


7.6 


7.2 


6.9 


6.6 


6.3 


6.0 


5.5 


5.3 


5.1 


5.0 


4.8 


4.6 


100 


10.6 


10 


9.5 


9.0 


8.6 


8.2 


7.8 


7.5 


6.9 


6.7 


6.4 


6.2 


6.0 


5.8 


125 


12.7 


12 


11.4 


10.8 


10.3 


9.8 


9.4 


9.0 


8.0 


8.0 


7.7 


7.5 


7.2 


7.0 


150 


14.8 


14 


13.3 


12.6 


12.0 


11.5 


11.0 


10.5 


9.7 


9.3 


9.0 


8.7 


8.4 


8.1 


175 


16.9 


16 


15.2 


14.4 


13.7 


13.1 


12.5 


12.0 


11.1 


10.7 


10.3 


9.9 


9.6 


9.3 


200 


19.1 


18 


17.1 


16.2 


15.6 


14.7 


14.1 


13.5 


12.5 


12.0 


11.6 


11.2 


10.8 


10.4 


225 


21.2 


20 


19.0 


18.0 


17.1 


16.4 


15.7 


15.0 


13.9 


13.3 


12.9 


12.4 


12.0 


11.6 


250 


23.3 


22 


21.8 


19.8 


18.9 


18.0 


17.2 


16.5 


15.2 


14.7 


14.1 


13.7 


13.2 


12.8 


275 


25.4 


24 


22.7 


21.6 


20.6 


19.6 


18.8 


18.0 


16.6 


16.0 


15.4 


14.9 


14.4 


13.9 


300 


27.5 


26 


24.6 


23.4 


22.3 


21.3 


20.6 


19.5 


18.0 


17.3 


16.7 


16.1 


15.6 


15.1 


325 


29.6 


28 


26.5 


25.2 


24.0 


22.9 


21.9 


21.0 


19.4 


18.7 


18.0 


17.416.8 


16.3 


350 


31.8 


30 


28.4 


27.0 


25.7 


24.5 


23.5 


22.5 


20.8 


20.0 


19.3 


18.6 18.0 


17.4 


375 


33.9 


32 


30.3 


28.8 


27.4 


26.2 


25.0 


24.0 


22.2 


21.3 


20.6 


19.8 


19.2 


18.6 


400 


36.0 


34 


32.2 


30.6 


29.1 


27.8 


26.6 


25.5 


23.5 


22.7 


21.9 


21.1 


20.4 


19.7 


425 


38.1 


36 


34.1 


32.4 


30.9 


29.5 


28.2 


27.0 


24.9 


24.0 


23.1 


22.3 


21.6 


20.9 


450 


40.2 


38 


36.0 


34.2 


32.6 


31.1 


29.7 


28.5 


26.3 


25.3 


24.4 


23.6 


22.8 


22.1 


475 


42.4 


40 


37.9 


36.0 


34.3 


32.7 


31.3 


30.0 


27.7 


26.7 


25.7 


24.8 24.0 


23.2 


500 


44.5 


42 


39.8 


37.8 


36.0 


34.4 


32.9 


31.5 


29.1 


28.0 


26.9 


25.025.2 


24.4 


525 


46.6 


44 


41.7 


38.6 


37.7 


36.0 


34.4 


33.0 


30.5 


29.3 


28.3 


27.3,26.4 


25.5 


550 


48.7 


46 


43.6 


41.4 


39.4 


37.6 


36.0 


34.5 


31.9 


30.7 


29.6 


28.527.6 


26.7 


575 


50.8 


48 


45.5 


43.2 


41.1 


39.3 


37.6 


36.0 


33.2 


32.0 


30.8 


29.8 28.8 


27.8 


600 


52.9 


50 


47.4 


45.0 


42.9 


40.9 


39.1 


37.5 


34.6 


33.3 


32.1 


31.0,30.0 


29.0 


625 


55.1 


52 


49.3 


46.8 


44.6 


42.5 


40.7 


39.0 


36.0 


34.7 


33.4 


32.2 31.2 


30.2 


650 


57.2 


54 


51.2 


48.6 


46.3 


44.1 


42.3 


40.5 


37.5 


36.0 


34.7 


33.5 32.4 


31.3 


675 


59.3 


56 


53.1 


50.4 


48.0 


45.8 


43.8 


42.0 


38.8 


37.3 


36.0 


34.7 


33.6 


32.5 


700 


61.4 


58 


55.0 


52.2 


49.7 


47.4 


45.4 


43.5 


40.2 


38.7 


37.3 


36.0 


34.8 


33.6 


725 


63.5 


60 


56.9 


54.0 


51.4 


49.1 


47.0 


45.0 


41.5 


40.0 


38.6 


37.2 


36.0 


34.8 


750 


65.6 


62 


58.8 


56.3 


53.1 


50.7 


48.5 


46.5 


42.9 


41.3 


39.9 


38.5 


37.2 


36.0 


775 


67.8 


64 


60.6 


57.6 


54.9 


52.4 


50.1 


48.0 


44.3 


42.7 


41.2 


39.7 


38.4 


37.1 


800 


69.9 


66 


62.5 


59.4 


56.6 


54.0 


51.7 


49.5 


45.7 


44.0 


42.4 


40.9 


39.6 


38.3 


825 


72.0 


68 


64.4 


61.2 


58.4 


55.6 


53.2 


51.0 


47.1 


45.3 


43.7 


42.2 


40.8 


39.4 


850 


74.0 


70 


67.3 


63.0 


60.0 


57.3 


54.8 


52.5 


48.5 


46.7 


45.0 


43.4 


42.0 


40.6 


875 


76.2 


72 


68.2 


64.8 


61.7 


58.9 


56.3 


54.0 


49.9 


48.0 


46.3 


44.6 


43.2 


41.8 


900 


78.4 


74 


70.1 


66.6 


63.4 


60.5 


57.9 


55.5 


51.3 


49.3 


47.6 


46.0 


44.4 


42.9 


925 


SO. 5 


76 


72.0 


68.4 


65.1 


62.2 


59.5 


57.0 


52.6 


50.7 


48.8 


47.1 


45.6 


44.1 


950 


(52.6 


78 


73.9 


70.2 


66.8 


63.8 


61.0 


58.5 


54.0 


52.0 


50.2 


48.4 


46.8 


45.3 


975 


$4.7 


80 


75.8 


72.0 


68.7 


66.0 


62.6 


60.0 


55.4 


53.3 


51.4 


49.6 


48.0 


46.4 


1000 



610 



Air Pressure. 



Pressure, in Ounces, per Square Inch. 

Corresponding to Various Heads of Water, in Inches. 



Head, in 
inches. 








Decimal parts of an 


inch. 






























.0 


- 1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 







.06 


.12 


.17 


.23 


.29 


.35 


.40 


.46 


.52 


1 


.58 


.63 


.69 


.75 


.81 


.87 


.93 


.98 


1.04 


1.09 


2 


1.16 


1.21 


1.27 


1.33 


1.39 


1.44 


1.50 


1.56 


1.62 


1.67 


3 


1.73 


1.79 


1.85 


1.91 


1.96 


2.02 


2.08 


2.14 


2.19 


2.25 


4 


2.31 


2.37 


2.42 


2.48 


2.54 


2.60 


2.66 


2.72 


2.77 


2.83 


5 


2.89 


2.94 


3.00 


3.06 


3.12 


3.18 


3.24 


3.29 


3.35 


3.41 


6 


3.47 


3.52 


3.58 


3.64 


3.70 


3.75 


3.81 


3.87 


3.92 


3.98 


7 


4.04 


4.10 


4.16 


4.22 


4.28 


4.33 


4.39 


4.45 


4.50 


4.56 


8 


4.62 


4.67 


4.73 


4.79 


4.85 


4.91 


4.97 


5.03 


5.08 


5.14 


9 


5.20 


5.26 


5.31 


5.37 


5.42 


5.48 


5.54 


5.60 


5.66 


5.72 



Height of Water Column, in Inches. 

Corresponding to Pressures, in Ounces, per Square Inch. 



Pressure, 








Decimal part 


3 of an ounce. 








in ounces, 






















inch. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 







.17 


.35 


.52 


.69 


.87 


1.04 


1.21 


1.38 


1.56 


1 


1.73 


1.90 


2.08 


2.25 


2.42 


2.60 


2.77 


2.94 


3.11 


3.29 


2 


3.46 


3.63 


3.81 


3.98 


4.15 


4.33 


4.50 


4.67 


4.84 


5.01 


3 


5.19 


5.36 


5.54 


5.71 


5.88 


6.06 


6.23 


6.40 


6.57 


6.75 


4 


6.92 


7.09 


7.27 


7.44 


7.61 


7.79 


7.96 


8.13 


8.30 


8.48 


5 


8.65 


8.82 


9.00 


9.17 


9.34 


9.52 


9.69 


9.86 


10.03 


10.21 


6 


10.38 


10.55 


10.73 


10.90 


11.07 


11.26 


11.43 


11.60 


11.77 


11.95 


7 


12.11 


12.28 


12.46 


12.63 


12.80 


12.97 


13.15 


13.32 


13.49 


13.67 


8 


13.84 


14.01 


14.19 


14.36 


14.53 


14.71 


14.88 


15.05 


15.22 


15.40 


9 


15.57 


15.74 


15.92 


16.09 


16.26 


16.45 


16.62 


16.76 


16.96 


17.14^|| 



Steam Boileks. 



611 



STEAM=BOILER DETAILS. 
Material for Riveting. 

Board of Trade.— Tensile strength of rivet bars between 26 and 30 tons ; 
Elongation in 10 inches not less than 25 per cent., and contraction of area 
not less than 50 per cent, 

Lloyd's.— Tensile strength, 26 to 30 tons; elongation not less than 20 
per cent, in 8 inches. The material must stand bending to a curve, the 
inner radius of which is not greater than 1% times the thickness of the 
plate, after having been uniformly heated to a low cherry-red, and 
quenched in water at 82° F. 

United States Statutes.— No special provision. 

Bureau Veritas.— Tensile strength, 53,000 pounds. 

German Lloyd's.— Tensile strength, 45,000 to 51,000 pounds; elonga- 
tion, 23.5 per cent, to 26 per cent., depending on thickness of plate. 

Rules Connected with Riveting. 

Board of Trade.— The shearing resistance of the rivet steel to be taken 
at 23 tons per square inch, 5 to be used for the factor of safety indepen- 
dently of any addition to this factor for the plating. Rivets in double 
shear to have only 1.75 times the single section taken in the calculation, 
instead of 2. The diameter must not be less than the thickness of the 
plate, and the pitch never greater than 8% inches. The thickness of 
double butt straps (each) not to be less than % the thickness of the plate ; 
single butt straps not less than f . 

Distance from centre of rivet to edge of hole = diameter of rivet X VA- 

Distance between rows of rivets 
= 2 X diameter of rivet or = [(diameter X 4) + 1] h- 2, if chain, and 



y [(pitch X 11) + (dia meter X 4)] X (pitch -=- diameter X 4) 



if zigzag. 

Diagonal pitch = (pitch X 6 + diameter X 4) -=- 10. 

Lloyd's.— Rivets in double shear to have only 1.75 times the single 
section taken in the calculation, instead of 2. The shearing strength of 
rivet steel to be taken at 85 per cent, of the tensile strength of the material 
of shell plates. In any case where the strength of the longitudinal joint 
is satisfactorily shown by experiment to be greater than given by the 
formula, the actual strength may be taken in the calculation. 

United States Statutes.— No rules. 

Bureau Veritas.— Shearing strength assumed = 0.8 tensile strength; 

at working pressure shearing strength to be — r part of full shearing 

strength. Double shear twice single section. Circular seams to be double- 
riveted if plates exceed % inch. 

German Lloyd's.— Shearing assumed = 0.8 tensile strength of plates,— 
factor of safety = 5 for lap joints and 1.15 X 5 for double butt joints,— total 
rivet area to be taken. Butt straps at least 0.75 of plate diameter of rivets 
not over twice, or less than thickness of plate for thin and thick plates, 
respectively. Pitch of rivets not over 8 times thickness of plate strap. 

Proportions of Rivets. 

(Thurston.) 



Thickness of plate 

Diameter of rivet 

Diameter of rivet-hole 

JaPitch— single riveting 

j Pitch— double riveting 

j Strength of single-riveted joint. . 

Strength of double-riveted joint. 



w 


16 


%" 


tV 


Vs" 


\\" 


Y4> 


M" 


w 


%" 


13// 
16 


Vs" 


2" 


2i¥' 


y/s" 


2tV 


3" 


3%" 


3K" 


S%" 


.66$ 


.64$ 


.62$ 


.60$ 


.77$ 


.76$ 


.75$ 


.74$ 



Vs" 

w 

2M" 

.58^ 
.73$ 



612 



Steam Boilers. 



Lloyd's Proportions for Riveted Joints. 

Single-riveted Joints. 











r 
Percentage. 


Thickness of 


Diameter of 
iron rivet. 


Pitch. 


Lap. 




iron plate. 














Plate. 


Rivet. 


Inch. 


Inch. 


Inch. 


Inch. 






A 


11 

T6 


m 


2ft 


67.9 


60.7 


% 


H 


2ft 


2ft 


67.0 


60.0 


ft 


15 

16 


2}£ and & 


2it 


67.4 


60.1 


% 


1 


2ft 


3 


64.9 


58.9 


ft 


i» 


2% and & 


3% 


65.1 


58.6 


% 


1t 3 b 


2% and & 


3ft 


63.7 


57.3 



Double-riveted Joints. 











Percentage. 


Thickness of 


Diameter of 
iron rivet. 


Pitch. 


Lap. 




iron plate. 














Rivet. 


Plate. 


Inch. 


Inch. 


Inch. 


Inch. 






% 


ii 

T5 


2% and & 


q 7 


80.2 


72.1 


ft 


% 


2% and & 


33^ 


77.7 


71.1 


% 


ii 


2% and ^ 


4ft 


77.2 


69.4 


ft 


H 


3>^ 


*H 


78.5 


70.0 


% 


l 


3M 


5 


77.3 


69.2 


ii 
is 


It 1 . 


3% 


°ts 


76.4 


68.5 


% 


1% 


3% and & 


5% 


75.0 


68.1 



Treble-riveted Joints. 











Percentage. 


Thickness of 


Diameter of 
iron rivet. 


Pitch. 


Lap. 




iron plate. 














Rivet. 


Plate. 


Inch. 


Inch. 


Inch. 


Inch. 






y* 


% 


3% and 5 V 


4^ 


83.9 


76.2 


ft 


tt 


314 and & 


4% 


84.3 


75.2 


% 


Vs 


3ft 


5^ 


83.9 


75.4 


H 


1 


4ft 


6 


84.4 


75.4 


% 


1ft 


VA 


6% 


83.4 


75.0 


B 


iy* 


4% and & 


6% 


83.3 


74.5 


% 


1ft 


4% and ^ 


w* 


82.5 


74.1 


H 


1M 


4% and & 


1% 


82.1 


73.8 


1 


1ft 


411 


Wb 


82.2 


73.4 



Steam Boilers. 613 



j. 



Materials for Boiler Shells. 



Board of Trade.— Tensile strength between 27 and 32 tons. In the 
. Lorroal condition, elongation not less than 18 per cent, in 10 inches, but 
should be about 25 per cent. ; if annealed, not less than 20 per cent. Strips 
2 inches wide should stand bending until the sides are parallel at a dis- 
tance from each other of not more than 3 times the plate's thickness. 

Lloyd's. — Tensile strength between the limits of 26 and 30 tons per 
square inch. Elongation not less than 20 per cent, in 8 inches. Test strips 
heated to a low cherry-red and plunged into water at 82° F. must stand 
bending to a curve, the inner radius of which is not greater than V/% times 
the plate's thickness. 

United States Statutes.— Plates of % inch thickness and under shall 
show a contraction of not less than 50 per cent. ; when over % inch, and 
up to % inch, not less than 45 per cent. ; when over % inch, not less than 
40 per cent. 

Bureau Veritas.— Tensile strength not over 61,000 pounds. Elongation, 
20 to 31 per cent, for various tensile strengths. Quench strips must bend 
180° around diameter = 3t. 

German Lloyd's.— Tensile strength not over 61,000 pounds. Elonga- 
tion, 20 to 26 per cent, for various tensile strengths. Quench strips must 
bend 180° around diameter = U. 

Proportions of Boiler Shells. 

TX BXtX2 



Board of Trade.— P = 



DXF 



J) = diameter of boiler, in inches ; 
P = working pressure, in pounds, per square inch ; 
t = thickness, in inches ; 

B = percentage of strength of joint compared to solid plate ; 
T= tensile strength allowed for the material, in pounds, per square 

inch; 
F= a factor of safety, being 4.5, with certain additions depending 

on method of construction. 

Lio yt rs.-p= Cx(< - 2 > xi; . 

t = thickness of plate, in sixteenths of an inch ; 
B and D as before ; 
C= a constant depending on the kind of joint. 

When longitudinal seams have double butt straps, C = 20. When longi- 
tudinal seams have double butt straps of unequal width, only covering on 
one side, the reduced section of plate at the outer line of rivets, C = 19.5. 

When the longitudinal seams are lap-jointed, 6— 18.5. 

United States Statutes.— Using same notation as for Board of Trade. 

t X 2 X T 

P = — -— for single riveting ; add 20 per cent, for double riveting 

D x 6 
where T is the lowest tensile strength stamped on any plate. 

r v ,* rx BX (* — 0.042)2 

Bureau Ver,tas.-P = Dx ^ xm • 

B = per cent, of plate section at joint. 
P also depends on rivet section. 

German Lloyd's.-P ^ X2X5Xr 



D X F X 100 * 
F varies from 4.65 to 5, depending on thickness of plate. 



614 Steam Boileks. 



Board of Trade.— P 



Proportions for Flat Plates. 

(7(< + l)« 



8- 



p = working pressure, in pounds, per square inch ; 
S = surface supported, in square inches ; 
t = thickness, in sixteenths of an inch ; 
C = a constant, as per following table. 

C = 125 for plates not exposed to heat or flame, the stays fitted with 
nuts and washers, the latter at least 3 times the diameter of 
the stay and % the thickness of the plate. 

(7=187.5 for the same condition, but the washers % the pitch of 
stays in diameter, and thickness not less than plate. 

C=200 for the same condition, but doubling plates in place of 
washers, the width of which is % the pitch and thickness 
the same as the plate. 

C= 112.5 for the same condition, but stays fitted with nuts only. 

0= 75 when exposed to impact of heat or flame and steam in con- 
tact with the plates, and the stays fitted with nuts and 
washers 3 times the diameter of the stay and % the plate's 
thickness. 

C= 67.5 for the same condition, but stays fitted with nuts only. 

(7=100 when exposed to heat or flame and water in contact with 
the plates, and stays screwed into the plates and fitted with 
nuts. 

C = 66 for the same condition, but stays with riveted heads. 

United States Statutes.— Using same notations as for Board of Trade. 

C X t 
P= — , where p = greatest pitch, in inches, Pand t as above. 

(7= 112 for plates T 7 g of an inch thick and under, fitted with screw 
stay-bolts and nuts, or plain bolt fitted with single nut and 
socket or riveted head and socket. 

(7= 120 for plates above T 7 n of an inch, under the same conditions. 

C— 140 for flat surfaces where the stays are fitted with nuts inside 
and outside. 

(7= 200 for flat surfaces under the same condition, but with the addi- 
tion of a washer riveted to the plate at least half the plate's 
thickness and of a diameter equal to § pitch. 

N.B.— Plates fitted with double angle-irons and riveted to plate, with 
leaf at least % the thickness of plate and depth at least % of pitch, would 
be allowed the same pressure as determined by formula for plate with 
washer riveted on. 

N.B.— No brace or stay-bolt used in marine boilers to have a greater 
pitch than 10^ inches on fire-boxes and back connections. 

Certain experiments were carried out by the Board of Trade which 
showed that the resistance to bulging does not vary as the square of the 
plate's thickness. There seems, also, good reason to believe that it is not 
inversely as the square of the greatest pitch. 

(I 1\2 t 

Bureau Veritas.— P = i-= ~ X -77. 

a 2 -p o 2 6 

T = tensile strength, in tons, per square inch ; 
a = pitch in one row, in inches ; 
b = distance between rows ; 

C = factor depending on method of supporting, and varies from 0.055 
to 0.084. 

t 2 
German Lloyd's.— P= ——— — -. 

C 2 X v- 

C varies from 0.00425 to 0.00639, depending on exposure and method of 
supporting. 



- 



Steam Boilers. 615 






Plates for Flanging. 

The Board of Trade gives the following rule for the strength of fur- 
naces Stiff ened with flanged seams, provided the pitch of the flanges does 
fhot exceed 120 T— 12, and the flanging is of suitable design and effected at 
one heat : 



P = 



9900 X T / L j- 12 \ 

3Xi> V 60 X T)' 



p = working pressure per square inch ; 

T = thickness of plate, in inches ; 

L = pitch of flanges, in inches ; 

D — outside diameter of tubes, in inches. 

Bureau Veritas.— Tensile strength not over 61,000 pounds. Elongation, 
20 to 31 per cent, for various tensile strengths. Quench strips must bend 
180° around diameter = St. 

German Lloyd's. — Tensile strength not over 53,000 pounds. Elonga- 
tion not under 22% per cent. Quench strips must bend 180° around diam- 
eter = U. 

Furnace Flues. 

ex t 2 

Board of Trade. Long Furnaces.— P = — , but not where 

\Li -f- L) X JJ 

L is shorter than (11.5Z — 1), at which length the rule for short furnaces 
comes into use. 

p = working pressure, in pounds, per square inch ; 

t = thickness, in inches ; 
D = outside diameter, in feet ; 
L = length of furnaces, in feet, up to 10 feet ; 

C= a constant, as below, for drilled holes. 

C= 99,000 for welded or butt-jointed, with single straps, double- 
riveted. 
C— 88,000 for butts with single straps, single-riveted. 
C = 99,000 for butts with double straps, single-riveted. 

Provided, always, that the pressure so found does not exceed that given 
by the following formulas, which apply also to short furnaces : 

CX t 
P = — =-— for all the patent furnaces named. 

C= 8800 for plain furnaces. 

C= 14,000 for Fox. Minimum thickness, T % inch ; greatest, % inch ; 

plain part not to exceed 6 inches in length. 
C = 13,500 for Morison. Minimum thickness, f% inch ; greatest, % 

inch ; plain part not to exceed 6 inches in length. 
C= 14,000 for Purves-Brown. Limits of thickness, T 7 s and % inch; 

plain part 9 inches in length. 

United States Statutes. Long Furnaces.— Same notation. 

89,600 X P , . _ , _ . , 

P = — j^ — , but L not to exceed 8 feet. 

Li X -D 

Short Furnaces, Plain and Patent.— P as before, when not 8 feet 

- 89,600 X & 

long -- 



LXD 



P = — =r — , when 



C= 14,000 for Fox corrugations, where D = mean diameter. 
C= 14,000 for Purves-Brown, where D = diameter of flue. 
(7=5677 for plain flues over 16 inches diameter and less than 40 
inches, when not over 3-foot lengths. 



616 Steam Boilers. 



Lloyd's and Bureau Veritas for Morison Suspension Furnaces. — 

1259(T-2) 

WJT D 

T = thickness, in sixteenths of an inch ; 
D = greatest diameter, in inches ; • 

WP = working pressure. 



Stays. 

MATERIAL. 

Board of Trade.— The tensile strength to lie between the limits of 27 
and 32 tons per square inch, and to have an elongation of not less than 20 
per cent, in 10 inches. Steel stays which have been welded or worked in 
the fire should not be used. 

Lloyd's.— 26 to 30 ton steel, with elongation not less than 20 per cent, 
in 8 inches. 

United States Statutes.— Reduction of area must not be less than 40 
per cent, if the test bar is more than % of an inch in diameter. 

Bureau Veritas.— Same as for shell plates. 

German Lloyd's.— Large stays, tensile strength 45,800 to 61,200 pounds. 
Elongation same as shell plates. Screwed stays, tensile strength 44,600 to 
53,400 pounds, and corresponding elongation. 



Loads on Stays. 

Board of Trade.— 9000 pounds per square inch is allowed on the net 
section, provided the tensile strength ranges from 27 to 32 tons. Steel stays 
are not to be welded or worked in the fire. 

Lloyd's.— For screwed and other stays not exceeding 1% inches in 
diameter effective, 8000 pounds per square inch is allowed ; for stays above 
1% inches, 9000 pounds. No stays are to be welded. 

United States Statutes.— Braces and stays shall not be subjected to a 
greater stress than 6000 pounds per square inch. 

Bureau Veritas. — r^p- of lower test limit on net section. Then add 

y% inch to diameter of stay. 

German Lloyd's.— Not to exceed \ of tensile strength, or about 8500 
pounds per square inch. 



Board of Trade.— P - 



Stay Girders. 

(W-p)DX L' 



p = working pressure, in pounds, per square inch ; 

W = width of name-box, in inches ; 

L = length of girder, in inches ; 

p — pitch of bolts, in inches ; 

D = distance between girders from centre to centre, in inches ; 

d = depth of girder, in inches ; 

t = thickness of sum of same, in inches ; 

C= a constant = 6600 for 1 bolt, 9900 for 2 or 3 bolts, and 11,220 for 
4 bolts. 

Lloyd's.— The same formula and constants, except that C= 11,000 for 
4 or 5 bolts, 11,550 for 6 or 7 bolts, and 11,880 for 8 or more. 



Steam Boilers. 



617 



Board of Trade.— P = 



Tube Plates, 

t(D -d)X 20000 
WXB 



D = least horizontal distance between centres of tubes, in inches ; 

d = inside diameter of ordinary tubes ; 
t = thickness of tube plate, in inches ; 

W= extreme width of combustion-box, in inches, from front of tube 
plate to back of fire-box, or distance between combustion-box 
tube plates, when the boiler is double-ended, and the box 
common to both ends. 

The crushing stress on tube plates caused by the pressure on the flame- 
box top is to be limited to 10,000 pounds per square inch. 



Fox and Purves Furnace Tubes. 

Working Pressures allowed by Board of Trade and Lloyd's. 



° o 








\\ 


'orking pressure, 


in pounds, per square inch. 










%inch 
thick. 


J! inch 
thick. 


T ynch 
thick. 


§f inch 
thick. 


% inch 
thick. 


M incn 

thick. 


T 9 ginch 
thick. 


if inch 
thick. 


%inch 
thick. 


i'l 
5 


H 
o 


00 

o 

3 


<M 

o 


no 

>> 

O 

3 


O 


no 

m: 


H 

«M 

o 


to 

o 

3 


H 

O 


00 

>» 
o 

3 


o 


JO 

3 


O 

PQ 


O 

3 


O 


C 

3 


o 


T3 
■>> 

o 

3 


Ft. In. 

2 6 
2 7 
2 8 
2 9 
2 10 

2 11 

3 
3 1 
3 .2 
3 3 

3 4 
3 5 
3 6 

3 7 
3 8 

3 9 
3 10 

3 11 

4 
4 1 

4 2 
4 3 

I 4 4 

i 4 5 

4 6 


164 
159 
154 
150 
145 

141 
138 
134 
131 

128 

125 
122 
119 
116 
114 

111 

109 
107 
105 
102 

100 
99 
97 
95 
93 


145 
141 
137 
133 
129 

126 
123 
120 
117 
114 

112 

109 
107 
105 
102 

100 
98 
96 
94 
93 

91 

89 
88 
86 
85 


177 

172 
167 
162 
157 

153 
149 
145 
142 

138 

135 
132 
129 
126 
123 

121 
118 
116 
113 
111 

109 
107 
105 
103 
101 


163 
158 
154 
150 
146 

142 

138 
135 
132 
129 

126 
123 
120 
118 
115 

113 
111 
108 
106 
104 

102 

100 

99 

97 

95 


191 
185 
180 
175 
170 

165 
161 
157 
153 
149 

145 

142 
139 
136 
133 

130 
127 
125 

122 
120 

117 
115 
113 
111 
109 


181 
176 
171 
166 
162 

158 
154 
150 
146 
143 

140 
137 
134 
131 
128 

125 
123 
120 
118 
116 

114 
112 
110 
108 
106 


205 
198 
193 
187 
182 

177 
172 

168 
164 
160 

156 
152 
149 
145 
142 

139 
136 
133 
131 
128 

126 
123 
121 
119 
117 


199 
193 

188 
183 
178 

174 
169 
165 
161 
157 

154 
150 
147 
144 
141 

138 
135 
133 
130 

128 

125 
123 
121 
119 
117 


218 
212 
205 
200 
194 

189 
184 
179 
175 
170 

166 
162 
159 
155 
152 

148 
145 
142 
140 
137 

134 
132 
129 
127 
125 


217 
211 
205 
200 
194 

189 
185 
180 
176 
172 

168 
164 
160 
157 
154 

151 

148 
145 
142 
139 

137 
134 
132 
129 
127 


232 
225 
218 
212 
206 

201 
195 
190 
185 
181 

177 
172 
169 
165 
161 

158 
154 
151 
148 
145 

143 
140 
137 
135 
132 


235 
229 
222 
216 
211 

205 
200 
195 
190 
186 

182 
178 
174 
170 
167 

163 
160 
157 
154 
151 

148 
145 
143 
140 
138 


246 
238 
231 
225 

218 

212 
207 
201 
196 
192 

187 
183 
178 
175 
171 

167 
164 
160 
157 
154 

151 
148 
145 
143 
140 


254 
246 
239 
233 
227 

221 
215 
210 
205 
200 

196 
191 

187 
183 
179 

176 
172 
169 
166 
162 

159 
157 
154 
151 
148 


259 
251 
244 
237 
230 

224 
218 
213 
207 
202 

197 
193 
188 
184 
180 

176 
173 
169 
166 
162 

159 
156 
153 
151 

148 


272 
264 
257 
250 
243 

237 
231 
225 
220 
215 

210 
205 
201 
196 
192 

188 
185 
181 
177 
174 

171 
168 
165 
162 
159 


273 
265 
257 
250 
243 

236 
230 
224 
218 
213 

208 
203 
198 
194 
190 

186 
182 
178 
175 
171 

168 
165 
162 
159 
156 


290 
282 
274 
266 
259 

253 
246 
240 
235 
229 

224 
219 
214 
210 
205 

201 
197 
193 
189 
186 

182 
179 
176 
173 
170 



Internal flues should be so constructed as to allow for expansion. 



618 



Boiler Furnaces. 



Table Showing Working Pressure and Thickness of 
Morison Suspension Furnaces. 



© 






Working pressure, in pounds, 


per square 


$ inch. 






a & 
P © 












Thickness of furnace. 


© 3 


"o 


d 

© 


d 
© 


d 
© 


d 

© 

.5 


d 
o 

p 


d 
© 
p 


d 

© 
p 


d 

© 

a 


d 

© 


rP 
© 
P 


d 

p 


d 

© 

# p 


— 
© 
^p 


© 


p o 


H 


$ 


S00 


«|N 


5 


H)M 


5 


»im 


* 


0*1 

h|« 


\C0 




HfH 


«le» 




Ft. In. 
































2 4 


146 


160 


175 


189 


204 


219 


233 


247 


262 


276 


290 


304 


318 


332 


347 


2 5 


142 


156 


170 


183 


197 


212 


225 


239 


253 


267 


281 


294 


308 


322 


336 


2 6 


137 


151 


164 


178 


191 


205 


218 


232 


245 


259 


272 


285 


299 


312 


325 


2 7 


133 


146 


159 


172 


186 


199 


212 


225 


238 


251 


264 


277 


290 


302 


315 


2 8 


129 


142 


154 


166 


180 


193 


205 


218 


231 


243 


256 


268 


281 


294 


306 


2 9 


125 


138 


150 


162 


175 


187 


200 


212 


224 


236 


249 


261 


273 


285 


297 


2 10 


122 


134 


146 


158 


170 


182 


194 


206 


218 


230 


242 


254 


265 


277 


289 


2 11 


119 


130 


142 


154 


165 


177 


189 


200 


212 


224 


235 


247 


258 


270 


281 


3 


115 


127 


138 


149 


161 


172 


184 


195 


207 


218 


229 


240 


252 


263 


274 


3 1 


112 


123 


135 


146 


157 


168 


179 


190 


201 


212 


223 


234 


245 


256 


267 


3 2 


110 


120 


131 


142 


153 


164 


175 


185 


196 


207 


218 


228 


239 


250 


260 


3 3 


107 


117 


128 


138 


149 


160 


170 


181 


191 


202 


212 


223 


233 


244 


254 


3 4 


104 


115 


125 


135 


146 


156 


166 


176 


187 


197 


207 


217 


228 


238 


248 


3 5 


102 


112 


122 


132 


142 


152 


162 


172 


183 


192 


202 


212 


222 


232 


242 


3 6 


100 


109 


119 


129 


139 


149 


159 


168 


178 


188 


198 


207 


217 


227 


237 


3 7 


97 


107 


116 


126 


136 


146 


155 


165 


174 


184 


193 


203 


213 


222 


232 


3 8 


95 


105 


114 


123 


133 


142 


152 


161 


171 


180 


189 


199 


208 


217 


227 


3 9 


93 


102 


112 


121 


130 


139 


148 


158 


167 


176 


185 


194 


203 


213 


222 


3 10 


91 


100 


109 


118 


127 


137 


145 


154 


163 


172 


181 


190 


199 


209 


217 


3 11 


89 


98 


107 


116 


125 


134 


142 


151 


160 


169 


178 


186 


195 


204 


213 


4 


87 


96 


105 


113 


122 


131 


140 


148 


157 


166 


174 


183 


191 


200 


208 


4 1 


86 


94 


103 


111 


120 


128 


137 


145 


154 


162 


170 


179 


188 


196 


204 


4 2 


84 


92 


101 


109 


118 


126 


134 


142 


151 


159 


167 


176 


184 


192 


200 


4 3 


82 


91 


99 


107 


115 


123 


132 


140 


148 


156 


164 


172 


180 


189 


197 


4 4 


81 


88 


97 


105 


113 


121 


129 


137 


145 


153 


161 


169 


177 


185 


193 


4 5 


79 


87 


95 


103 


111 


119 


127 


135 


143 


150 


158 


166 


174 


182 


190 


4 6 


78 


86 


93 


101 


109 


117 


125 


132 


140 


148 


155 


163 


171 


178 


186 


4 7 


77 


84 


92 


99 


107 


115 


122 


130 


138 


145 


153 


160 


168 


175 


183 


4 8 


75 


83 


90 


98 


105 


113 


120 


128 


135 


143 


150 


157 


165 


172 


180 


4 9 


74 


81 


89 


96 


103 


111 


118 


125 


133 


140 


147 


155 


162 


169 


177 


4 10 


73 


80 


87 


94 


102 


109 


116 


123 


131 


138 


145 


152 


159 


167 


174 


4 11 


71 


78 


86 


93 


100 


107 


114 


121 


129 


136 


143 


150 


157 


164 


171 



Boiler Tubes. 



619 



Dimensions of Standard Boiler Tubes, Lap=welded, 
Wrought=iron. 



Outside. 


Thick- 
ness, in 


Weight 
per foot, 


Heating surface 
1 foot in length. 


Area of 


opening. 














Diameter, 
in inches. 


Circum- 
ference, 
in inches. 


inches. 


pounds. 


Outside, 

square 

feet. 


Inside, 
square 
feet. 


Square 
feet. 


Square 
inches. 


m 


4.71 


.08 


1.25 


.393 


.349 


.0097 


1.40 


1% 


5.50 


.10 


1.67 


.458 


.408 


.0133 


1.91 


2 


6.28 


.10 


1.98 


.524 


.472 


.0177 


2.56 


2M 


7.07 


.10 


2.34 


.589 


.540 


.0230 


3.31 


2^ 


7.85 


.11 


2.76 


.655 


.598 


.0284 


4.09 


2% 


8.64 


.11 


3.05 


.720 


.663 


.0350 


5.04 


3 


9.43 


.11 


3.33 


.785 


.729 


.0422 


6.08 


3M 


10.21 


.12 


3.96 


.851 


.789 


.0495 


7.12 


&A 


11.00 


.12 


4.27 


.916 


.854 


.0580 


8.36 


3% 


11.78 


.12 


4.59 


.982 


.919 


.0673 


9.69 


4 


12.57 


.13 


5.32 


1.047 


.979 


.0763 


10.99 


4K 


14.14 


.13 


6.01 


1.178 


1.110 


.0981 


14,13 


5 


15.71 


.14 


7.23 


1.309 


1.234 


.1215 


17.50 


6 


18.85 


.15 


9.35 


1.571 


1.492 


.1771 


25.51 


7 


21.99 


.17 


12.44 


1.833 


1.743 


.2417 


34.81 


8 


25.13 


.18 


15.11 


2.094 


1.998 


.3180 


45.80 


9 


28.27 


.19 


18.00 


2.356 


2.254 


.4048 


58.29 


10 


31.42 


.21 


22.19 


2.618 


2.506 


.4998 


71.98 


11 


34.56 


.22 


25.49 


2.880 


2.764 


.6075 


87.48 


12 


37.70 


.23 


28.52 


3.142 


3.022 


.7205 


103.75 


13 


40.84 


.24 


32.21 


3.403 


3.279 


.8554 


123.19 


14 


43.98 


.25 


36.27 


3.665 


3.534 


.9943 


143.19 


15 


47.12 


.26 


40.61 


3.927 


3.791 


1.1438 


164.72 


16 


50.27 


.27 


45.20 


4.189 


4.047 


1.3032 


187.67 


17 


53.41 


.28 


49.90 


4.451 


4.305 


1.4738 


212.23 


18 


56.55 


.29 


54.82 


4.712 


4.560 


1.6543 


238.22 


19 


59.69 


.30 


59.48 


4.974 


4.817 


1.8465 


265.90 


20 


62.83 


.32 


66.77 


5.219 


5.068 


2.0443 


294.37 


21 


65.97 


.34 


73.40 


5.498 


5.320 


2.2522 


324.31 



620 



Boiler Stating. 



Proportions for Stay Bolts for Flat Surfaces. 

(Barr.) 



*1 _• 


Centre to centre of stay bolts, in inches. 


3 o 

cB 5 


%-inch plate. 


T 5 5 -inch plate. 
%-inch stay. 


%-inch plate. 


/^-inch plate. 


%-inch plate, 
l^-inch stay. 


u 2 


%-inch stay. 


%-inch stay. 


1-inch stay. 


50 


6 


7 


8 


9 


10 


60 


5% 


6% 


7% 
&A 


8% 


9 


70 
80 


5 

4% 


m 


$ 


| 


90 


4% 


6% 


% 


100 


4% 


5% 
5i| 


6% 


7 


110 


4 


6% 


120 


3% 


4i 


5 


5M 
5% 


°% 


130 
140 


3^ 


4% 
41 




£ 


150 


3% 


4 


4% 


5 


5% 



Working Pressures for Flat Stayed Surfaces. 

Pounds per Square Inch. 



So 






Thickness of 


plates, in 


inches. 








K 


5 


Vs 


t 7 s 


% • 


i% 


% 


H 


% 


In. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


Lb. 


10 


62% 


90 


122 


160 


202 


250 


302 


360 


422 


11 


50 


73 


99 


129 


163 


202 


244 


290 


341 


12 


42 


60 


82 


107 


135 


167 


202 


240 


282 


13 


35 


50 


69 


90 


113 


140 


170 


201 


237 


14 


30 


43 


59 


76 


97 


120 


145 


172 


202 


15 


26 


37 


51 


66 


84 


103 


125 


148 


174 


16 


23 


32 


44 


58 


73 


90 


109 


130 


152 


17 


20 


29 


39 


51 


64 


80 


96 


114 


134 


18 


18 


25 


35 


45 


57 


71 


85 


101 


119 


19 


16 


23 


31 


40 


51 


63 


76 


91 


107 


20 


14 


20 


28 


36 


46 


57 


69 


82 


96 


21 


13 


18 


25 


33 


41 


51 


62 


74 


87 


22 


12 


17 


23 


30 


38 


47 


56 


67 


79 


23 


11 


15 


21 


27 


35 


43 


51 


61 


72 


24 


10 


14 


19 


25 


32 


39 


47 


56 


66 


25 


9 


13 


18 


23 


29 


36 


43 


52 


61 


26 


8 


12 


16% 


21 


27 


33 


40 


48 


56 


27 


7% 


11 


15 


19% 


25 


31 


37 


44 


52 


28 


7 


10 


14 


18 


23 


29 


34% 


41 


48 


29 


6% 


9% 


13 


17 


21% 


27 


32 


38 


45 


30 


6 


9 


12 


16 


20 


25 


30 


35% 


42 


31 


5% 


8% 


11 


15 


19 


23 


28 


33 


39 


32 


8 


10% 


14 


18 


21 


26 


31 


36 


33 


5 


7% 


10 


13 


17 


20 


25 


29% 


34 


34 


4^ 
4>| 


7 


9% 


12 


16 


19 


23 


28 


32 


35 


6% 


9 


11 


15 


18 


22 


26 


30 


36 


6 


8% 


11 


14 


17 


21 


25 


29 


37 


4 


6 


8 


10 


13 


16 


20 


23 


27 


38 




5^ 


7% 


10 


12% 


15 


19 


22 


26 


39 


5 


7 


9% 


12 


14% 


18 


21 


25 


40 


5 


6% 


9 


11 


14 


17 


20 


23% 



Boiler Dimensions. 621 



^ 



The rules of the Boiler Inspection Department of the city of Philadel- 
phia have been extensively used, and are as follows : 

Philadelphia City Rules for Boiler Dimensions. 

In estimating the strength of the longitudinal seams for rating maxi- 
mum working pressure on cylindrical boiler shells, two rules should be 
applied : 

Rule A.— From the pitch of the rivets, in inches, subtract the diameter 
of holes punched to receive the rivets ; divide the remainder by the pitch 
of the rivets. The quotient represents the percentage of strength of the 
solid part of the sheet. 

Rule B.— Multiply the area of the hole filled by the rivet by the number 
of rows of rivets in the seam ; divide the product by the pitch of the rivets 
multiplied by the thickness of the sheet. This product, multiplied by the 
shearing strength of the rivet, divided by the tensile strength of the sheet, 
will give the percentage of the strength of the rivets in the seam as com- 
pared with the strength of the solid part of the sheet. 

The shearing strength of a rivet in a composite joint made of iron rivets 
and steel plates shall not be considered in excess of 40,000 pounds. Take 
the lowest of the percentages as found by Rules A and B and apply that 
percentage as the value of the seam in the following rule (C), which deter- 
mines the strength of the longitudinal seams. 

Rule C— Multiply the thickness of the boiler plate, in parts of an inch, 
by the value of the seam as obtained by Rules A or B and by the ultimate 
tensile strength of the metal used in the plates ; divide this product by the 
internal radius of the boiler, in inches, multiplied by the factor of safety. 
The quotient will be the pressure per square inch at which the safety valve 
may be set. 



622 



Boilek Pressures. 



Working Pressures for Cylindrical Shells of Steam 
Boilers, X,ap Joints, Double=riveted. 

(Ban.) 
Factor of Safety, 5. 



Diameter. 


Thickness. 


Iron shell, 
iron rivets. 


Steel shell, 
iron rivets. 


Steel shell, 
steel rivets. 


Inch. 


Inch. 


Lb. 


Lb. 


Lb. 


36 


I A 


91 


Ill 


Ill 


112 


128 


137 


38 


I A 


86 


105 


105 


106 


121 


129 


40 


1 A 


82 


100 


100 


101 


115 


123 


42 


J 34 

I A 


78 


95 


95 


96 


110 


117 


44 




74 


91 


91 


91 


105 


112 


46 


{** 


71 


87 


87 


87 


100 


107 


48 


{\ 


84 
99 


96 
107 


102 
121 


50 


{\ 


81 
95 


92 
103 


98 
116 


52 


{\ 


77 
92 


89 
99 


95 
112 


54 


{\ 


75 

88 


85 
96 


91 
108 


56 


( A 3 /8 


72 
85 


82 
92 


88 
104 


58 


{\ 


69 

82 


79 
89 


85 
100 


60 


[\ 


67 
79 


77 
85 


82 
97 


62 


{\ 


77 
88 


83 
92 


94 

108 


64 


{\ 


74 
86 


81 
89 


91 
105 


66 


{\ 


72 
83 


78 
87 


88 
102 


68 


l % 7 

I 13 


70 
81 


76 
80 


86 
99 


70 




68 


74 


83 


78 


82 


96 




(% 


66 


72 


81 


72 


1 A 


76 


79 


93 




Ik 


85 


89 


104 



Boiler Pressures. 



Working Pressures for Cylindrical Shells of Steam 
\ Boilers, I^ap Joints, Triple=riveted. 

^ (Barr.) 

Factor of Safety, 5. 



Diameter. 


Thickness. 


Iron shell, 
iron rivets. 


Steel shell, 
iron rivets. 


Steel shell, 
steel rivets. 


Inch. 


Inch. 


Lb. 


Lb. 


Lb. 




36 


{% 


100 
124 


121 
139 


123 
151 




38 


{** 


95 
117 


115 
132 


116 
144 




40 


i\ 


90 
112 


109 
125 


110 
136 




42 


{\ 


86 
106 


104 
119 


105 
130 




44 




83 
101 


99 
114 


100 
124 




46 


{** 


79 

97 


95 
109 


96 
119 




48 


(\ 


93 
110 


104 
118 


114 
135 




50 


i\ 


89 
106 


100 
113 


109 
129 




52 


i\ 


86 
102 


96 
109 


105 
124 




54 


(\ 


83 
98 


93 
105 


101 
120 




56 


i\ 


80 
95 


89 
101 


97 
116 




58 


{% 


77 
91 


86 
98 


94 
112 




60 


{% 


74 
88 


83 
95 


91 
108 




62 


{** 


85 
98 


92 
103 


104 
120 




64 


i\ 


83 
95 


89 
100 


101 
117 




66 


{** 


80 
93 


86 
97 


98 
113 




68 


i\ 


78 
90 


84 
94 


95 
110 




70 




76 

87 


81 
91 


92 

107 




72 




74 
85 
97 


79 

89 
98 


90 
104 
117 





624 



Boiler Pressures. 



Working Pressures for Cylindrical Shells of Steam 
Boilers, Butt Joints, Triple=riveted. 

(Barr.) 
Factor of Safety, 5. 



Diameter. 


Thickness. 


Iron shell, 
iron rivets. 


Steel shell, 
iron rivets. 


Steel shell, 
steel rivets. 


Inch. 


Inch. 


Lb. 


Lb. 


Lb. 


36 




108 
135 
161 


134 
165 
197 


134 
165 
197 


38 




102 
128 
152 


127 
156 

187 


127 
156 
187 


40 


to" 


97 
121 
145 


120 
148 
178 


120 
148 
178 


42 


to" 


93 
116 
138 


115 
141 
169 


115 
141 
169 


44 


to™ 


89 
110 
132 


109 
135 
161 


109 
135 
161 


46 


to" 


85 
106 
126 


105 
129 
154 


105 
129 
154 


48 


Its 


101 
121 
141 


124 
148 
172 


124 
148 
172 


50 




97 
116 
135 


119 
142 
165 


119 
142 
165 


52 


lis 


93 
111 
130 


114 
137 
159 


114 
137 
159 


54 


lie 


90 
107 
125 


110 
132 
153 


110 
132 
153 


56 


ll5 


87 
103 
121 


106 
127 
148 


106 
127 
148 


58 


I IB 


84 
100 
117 


102 
123 
142 


102 
123 
142 


60 


&• 


97 
111 
128 


118 
138 
157 


118 
138 
157 


62 


&- 


93 
109 
124 


115 
133 
152 


115 
133 
152 



Boiler Pressures. 



625 



Working Pressures for Cylindrical Shells of Steam 
Boilers, Butt Joints, Triple=riveted. 

(Barr.) 
Factor of Safety, 5. 



Diameter. 


Thickness. 


Iron shell, 
iron rivets. 


Steel shell, 
iron rivets. 


Steel shell, 
steel rivets. 


Inch. 


Inch. 


Lb. 


Lb. 


•Lb. 






?/* 


90 


Ill 


Ill 




64 


J A 


106 


129 


129 




}% 


120 


147 


147 






( A 


135 


165 


165 






fA 


88 


108 


108 




66 


J A 


102 


125 


125 




}A 


117 


143 


143 






I A 


131 


160 


160 






fA 


85 


105 


105 




68 


J * 


99 


121 


121 




]K 


113 


138 


138 






I A 


127 


155 


155 






(% 


83 


102 


102 




70 


J A 


97 


118 


118 




\A 


110 


134 


134 






I A 


123 


151 


151 






|% 


80 


99 


99 






A 


94 


115 


115 




72 


k 


107 


131 


131 






A 


120 


147 


147 


/ 




k 


134 


163 


163 






fA 


90 


110 


110 




75 


J A 


102 


125 


125 




1a 


115 


141 


141 






I % 


128 


157 


157 






fA 


87 


106 


106 




78 


J « 


99 


121 


121 




|A 


111 


135 


135 






I % 


123 


151 


151 






fA 


83 


102 


102 




81 


J A 


95 


116 


116 




|A 


107 


130 


130 






I % 


119 


145 


145 






[A 


92 


112 


11£ 






A 


103 


126 


126 




84 


% 


115 


140 


140 






tt 


126 


158 


158 






hi 


137 


167 


167 






(A 


89 


108 


108 






A 


99 


121 


121 




87 


k 


111 


135 


135 






H 


121 


148 


148 






1% 


132 


162 


162 






f^ 


86 


105 


105 






A 


96 


117 


117 




90 


k 


107 


131 


131 






ft 


117 


143 


143 






h 


128 


156 


156 





40 



626 



Boiler Pressures. 



Working Pressures for Cylindrical Shells of Steam 
Boilers, Butt Joints, Triple=riveted. 

(Barr.) 
Factor of Safety, 5. 



Diameter. 


Thickness. 


Iron shell, 
iron rivets. 


Steel shell, 
iron rivets. 


Steel shell, 
steel rivets. 


Inch. 


Inch. 


Lb. 


Lb. 


Lb. 


93 


hi 


93 
103 
114 


114 
126 
139 


114 
126 
139 


96 


1% 


100 
110 
120 


123 
134 
146 


123 
134 
146 


99 




97 
107 
116 


119 
130 

142 


119 
130 
142 


102 




94 
104 
113 


115 
127 

138 


115 
127 

138 


105 




92 
101 

110 


112 
123 
134 


112 
123 
134 


1C8 


\% 16 


89 

98 

107 


109 
120 
130 


109 
120 
130 


111 




87 

95 

104 


106 
116 

127 


106 
116 
127 


114 




84 

93 

101 


103 
113 
123 


103 
113 
123 


117 




82 
90 
99 


100 

no 

120 


100 
110 
120 


120 


(Vs 


80 
88 
96 


98 
108 
117 


98 
108 
117 



Boiler Dimensions. 



627 



The formulas for boilers given by Reuleaux in the "Constructor" are as 
follows : 
Let 

D = diameter, in metres; 
a = pressure, in atmospheres : 
8 = thickness of shell, in millimetres ; 
S = fibre stress on material, in kilogrammes, per square millimetre. 

8 = lMaB + 2.6. 

The stress in the longitudinal seams will be 

a D 



200 8 



D being taken in millimetres. 
From these we have 



a = 


4 atmospheres. 


7 atmospheres. 


10 atmospheres. 


13 atmospheres. 


D 
metres. 


S 
mm. 


8 
kg. per 

sq. mm. 


8 
mm. 


S 
kg. per 
sq. mm. 


5 
mm. 


S 
kg. per 
sq. mm. 


5 
mm. 


8 

kg. per 
sq. mm. 


.6 


6.3 


1.90 


9.1 


2.31 


11.8 


2.54 


14.6 


2.67 


.8 


7.5 


2.13 


11.2 


2.50 


14.9 


2.70 


18.6 


2.80 


1.0 


8.8 


2.27 


13.4 


2.61 


18.0 


2.78 


22.6 


2.87 


1.5 


11.8 


2.51 


18.8 


2.79 


25.7 


2.92 


32.6 


2.99 


2.0 


14.9 


2.68 


24.2 


2.89 


33.4 


2.99 


42.6 


3.06 



The stresses in the circumferential seams are one-half those in the 
longitudinal seams ; hence, single riveting may be used when the longi- 
tudinal seams are double-riveted. 

For spherical ends, or boiler heads which are formed in the shape of a 
segment of a sphere, if R is the radius of curvature, we have for the thick- 
ness, 8i. 



200S " 



The above formulas, when adapted for English measures, are as follows: 
Let 

D = diameter, in inches ; 

a = pressure, in atmospheres ; 

p == pressure, in pounds, per square inch ; 

8 = thickness of shell, in inches ; 

S = fibre stress, in pounds, per square inch. 



We then have 



: O.OOISclD + 0.1, 

p D 
2*2' 



5,=- 



628 



Boiler Specifications. 



For the usual diameters we have the following results : 



a — 


4 = 60] 


aounds. 


7 = 105 


pounds. 


10 = 15C 


pounds. 


13 = 175 pounds^ 


D 


5 


S 


8 


S 


8 


S 


8 


S 


24 


.24 


3000 


.35 


3600 


.48 


3750 


.58 


3700 


36 


.31 


3500 


.48 


3900 


.64 


4000 


.80 


4000 


42 


.35 


3600 


.54 


4000 


.73 


4300 


.92 


4000 


72 


.43 


5000 


.85 


4400 


1.18 


4600 


1.50 


4200 



The general character of the material entering into boiler work is well 
described in the specifications of the American Boiler Manufacturers' 
Association, given herewith : 

Uniform American Boiler Specifications 

Adopted by the American Boiler Manufacturers' Association. 

(See Proceedings 1889, pages 49, 50, 66-81, 84-88.) 
(See Proceedings 1897, pages 42-54, 61-77, 207-208.) 
(See Proceedings 1898, pages 49-100.) 



MATERIALS. 

1. Cast-iron.— Should be of soft, gray texture and high degree of duc- 
tility. To be used only for hand-hole plates, crabs, yokes, etc., and man- 
heads. It is a dangerous metal to be used in mud drums, legs, necks, 
headers, man-hole rings, or any part of a boiler subject to tensile strains; 
its use is prohibited for such parts. 

2. Steel.— Homogeneous steel made by the open-hearth or crucible pro- 
cesses, and having the following qualities, is to be used in all boilers. 

Tensile Strength, Elongation, Chemical Tests.— Shell plates not exposed to 
the direct heat of the fire or gases of combustion, as in the external shells 
of internally-fired boilers, may have from 65,000 to 70,000 pounds tensile 
strength ; elongation not less than 24 per cent, in 8 inches ; phosphorus 
not over 0.035 per cent. ; sulphur not over 0.035 per cent. 

Shell plates in any way exposed to the direct heat of the fire or the 
gases of combustion, as in the external shells or heads of externally-fired 
boilers, or plates on which any flanging is to be done, to have from 60,000 
to 65,000 pounds tensile strength ; elongation not less than 27 per cent, in 
8 inches ; phosphorus not over 0.03 per cent. ; sulphur not over 0.025 per 
cent. 

Fire-box plates, or such as are exposed to the direct heat of the fire or 
flanged on the greater portion of their periphery, to have 55,000 to 62,000 
pounds tensile strength ; elongation, 30 per cent, in 8 inches ; phosphorus 
not over 0.03 per cent. ; sulphur not over 0.025 cent. 

For all plates the elastic limit to be at least one-half the ultimate 
strength; percentage of manganese and carbon left to the judgment of 
the steel maker. 

Test Section to be 8 inches long, planed or milled edges : its cross-sectional 
area not less than one-half of 1 square inch, nor width less than the thick- 
ness of the plate. 

Bending Test— Steel up to %-inch thickness must stand bending double 
and being hammered down on itself ; above that thickness it must bend , 
round a mandrel of diameter of 1% times the thickness of plate down to 
180 degrees. All without showing signs of distress. 

Bending test piece to be in length not less than 16 times the thickness of 

Elate, and rough, shear edges milled or filed off. Such pieces to be cut 
oth lengthwise and crosswise of the plate. 



Boiler Specifications. 629 

All tests to be made at the steel mill. Three pulling tests and three 
bending tests to be made from each heat. If one fails the manufacturer 
may furnish and test a fourth piece, but if two fail the entire heat to be 
rejected. 
Mr Certified copies of tests to be furnished each member of A. B. M. A. from 
heats from which his plates are made. 

3. Rivets to be of good charcoal iron or a soft, mild steel, having the 
same physical and chemical properties as the fire-box plates, and must test 
hot and cold by driving down on an anvil with the head in a die, by nick- 
ing and bending, by bending back on themselves cold, without developing 
cracks or flaws. 

4. Boiler Tubes of charcoal iron or mild steel specially made for the 
purpose, and lap-welded or drawn. They should be round, straight, free 
from scales, blisters, and mechanical defects, each tested to 500 pounds 
internal hydrostatic pressure. 

This fact and manufacturer's name to be plainly stencilled on each tube. 
Standard Thicknesses by Birmingham wire gauge to be 

No. 13 for tubes 1 inch, 1% inches, 1% inches, and 1%. inches diameter ; 

No. 12 for tubes 2 inches, 2% inches, and 2% inches diameter ; 

No. 11 for tubes 2% inches, 3 inches, 3% inches, and V/% inches diameter ; 

No. 10 for tubes 2>% inches and 4 inches diameter ; 

No. 9 for tubes 434 inches and 5 inches diameter. 

Tests.— A section cut from 1 tube taken at random from a lot of 150 or 
less must stand hammering down cold vertically without cracking or split- 
ting when down solid. 

Length of test pieces : 

% inch for tubes from 1 inch to 1% inches diameter ; 
1 inch for tubes from 2 inches to 2% inches diameter ; 
Vyi inches for tubes from 2%. inches to 3% inches diameter; 
1>| inches for tubes from sy» inches to 4 inches diameter ; 
1% inches for tubes from 4>| inches to 5 inches diameter. 

All tubes must stand expanding flange over on tube plate and bending 
without flaw, crack, or opening of the weld. 

5. Stay Bolts to be made of iron or mild steel specially manufactured 
for the purpose, and must show on 

Test Section 8 inches long, net : 

For Iron, tensile strength not less than 46,000 pounds ; elastic limit not 
less than 26,000 pounds ; elongation not less than 22 per cent, for bolts of 
less than one (1) square inch area, nor less than 20 per cent, for bolts one 
(1) square inch and more in net area. 

For Steel, tensile strength not less than 55,000 pounds ; elastic limit not 
less than 33,000 pounds ; elongation not less than 25 per cent, for bolts of 
less than one (1) square inch area, nor less than 22 per cent, for bolts one 
(1) square inch and more in net area. 

Tests.— A bar taken from a lot of 1000 pounds or less at random, threaded 
l with a sharp die "V" thread with rounded edges, must bend cold 180° 
around a bar of same diameter without showing any crack or flaws. 

Another piece, similarly chosen and threaded, to be screwed into well- 
| fitting nuts formed of pieces of the plates to be stayed, and riveted over so 
as to form an exact counterpart of the bolt in the finished structure ; to be 
pulled in testing machine and breaking stress noted ; if it fails by pulling 
; apart the tensile stress per square inch of net section is its measure of 
strength ; if it fails by shearing the shear stress per square inch of mean 
section in shear is this measure. The mean section in shear is the product 
of half the thickness of the plate by the circumference at half height of 
thread. 

6. Braces and Stays.— Material to be fully equal to stay-bolt stock, 
and tensile strength to be determined by testing a bar not less than ten 

■ ( F<(10) inches long from each lot of 1000 pounds or less. 

II. WORKMANSHIP AND DIMENSIONS. 

7. Flanging, Bending, and Forming to be done at a heat suited to 
the material, but no bending must be done or blow struck on any plate 



630 Boiler Specifications. 

which no longer shows red by daylight at the working point and at least 
4 inches "beyond it. 

8. Rolling must be done cold by gradual and regular increments from 
the straight plate to the exact circle required, and the whole circurn^ 
ference, including the lap, rolled to a true circle. 

9. Bumped Head uniformly dished to a segment of a sphere should 
have a thickness equal to that of a cylindrical shell of solid plate of same 
material, whose diameter is equal to the radius of curvature of the dished 
head. 

Rivet-holes, man-holes, etc., to be allowed for by proportionate increase 
in the thickness. 

10. Riveting.— Holes made perfectly true and fair by clean-cutting 
punches or drills. Sharp edges and burrs removed by slight counter- 
sinking and burr-reaming before and after sheets are joined together. 

Under side of original rivet head mast be flat, square, and smooth. For 
rivets % inch to Jf inch diameter allow 1% diameters for length of stock 
to form the head, and less for larger rivets. Allow 5 per cent, more stock 
for driven head for button set or snap rivets. Use light regulation riveting 
hammers until rivet is well upset in the hole ; after that, snap and heavy 
mauls. For machine riveting more stock to be left for driven head to 
make it equal to original head, as fixed by experiment. 

Total pressure on the die about 80 tons for 1%-inch to 1%-inch rivets ; 
65 tons for 1-inch rivets ; 57 tons for if-inch rivets ; 35 tons for %-inch 
rivets. 

Make heads of rivets equal in strength to shanks by making head at 
periphery of shank of a height equal to one- third the diameter of shank 
and giving a slight fillet at this point. 

Approximately, make rivet-holes double thickness of thinnest plate; 
pitch, 3 times rivet-hole ; pitch lines of staggered rows % pitch apart; lap 
for single riveting equal to pitch, for double riveting 1% pitch, and % 
pitch more for each additional row of rivets ; exact dimensions determined 
by making resistance to shear of aggregate rivet section at least 10 per 
cent, greater than tensile strength of net or standing metal. 

11. Rivet=holes punched with good, sharp punches and well-fitting 
dies in A. B. M. A. steel up to %-inch thickness ; in thicker plates punch 
and ream with a fluted reamer or drill the holes. 

12. Drift Pin to be used only with light hammers to pull plates into 

Elace and round up the hole, but never to enlarge or gouge holes with 
eavy hammers. 

13. Calking to be done by hand or pneumatic hammer and Conery or 
round-nosed tool. Avoid excessive calking ; the fit must be made in the 
laying of the plates. The square-nosed tool may be used for finishing, 
with great care to avoid nicking lower plate. Calking edges must be 
prepared by bevel planing, shearing, or chipping. 

14. Flat Surfaces.— State the thickness of the plate, t, in sixteenths of 
an inch ; the pitch, p, in inches, and use a constant : 

C= 112 for plates T 7 S inch and under, with screw stays with riveted 

ends. 
C = 120 for plates over T 7 ff inch, with screw stays with riveted ends. 
C= 140 for all plates when, in addition to screw threads in the plates, 

a nut is used inside and outside of each plate. 

When salt, acids, or alkali are contained in the feed water, this latter 
construction is imperative. 

Rule. — Multiply this constant, C, by the square of the thickness of the 
plate expressed in sixteenths of an inch, and divide by the square of the 
pitch expressed in inches ; the quotient is the safe working pressure, P. 




15. Tube=holes, either punched % inch less than required diameter 
and reamed to full size, or drilled, then slightly countersunk on both sides, 
should be ^ inch to ^g inch larger than diameter of tube, according to 



Boiler Specifications. 631 

size of tube ; if copper ferrules are used, the hole to he a neat fit for the 
ferrule. Tube sheet to be annealed after punching and before reaming. 

16. Tube Setting.— Ends of tubes to be annealed (in the tube mill) 
before setting. The tube to extend through the sheet T x 5 inch for every inch 

Kyi diameter. Exr^and until tight in hole and no more. On end exposed to 
direct flame, flange the tube j>artly over on sheet, finishing by beading 
tool, which must not come in contact with the plate ; expand slightly after 
beading. 

Copper ferrules, No. 18 to 14 wire gauge, should be used in fire-tube 
boilers on ends subject to direct heat. 

17. Riveted and Lap=welded Flues, as prescribed in Rule II., Sections 
8, 9, 10, 11, 12, and 13 of Regulations of Board of Supervising Inspectors of 
Steam Vessels, approved February, 1895. 

18. Corrugated Furnace Flues, as prescribed in Sections 14 and 15 of 
the same Rule. 

19. Stay Bolts to be carefully threaded with sharp, clean dies, "V" 
thread, with rounded edges; threading machine equipj>ed with a lead 
screw; holes tapped with tap extending through both sheets to neat, 
smooth fit, so that bolts can be put in by hand-lever or wrench with a 
steady pull ; £ diameter to project for riveting over ; with hollow stay bolts 
use slender drift pin in the bore while riveting, and drive it home to 
expand the bolt after riveting. 

Height of nuts used on screw stays to be at least 50 j)er cent, of diam- 
eter of stay. Largest permissible pitch for screw stays is 10 inches. 

20. Braces and Stays shall be subjected to careful inspection and tests, 
as per Sections 6 and 2. Welding to be avoided where possible, but good, 
clean welds to be allowed a value of 80 per cent, of the solid bar. Rivets 
by which braces are attached, when the pull on them is other than at right 
angles, to be allowed only half the stress permitted for rivets in the seams. 

21. Man=holes should be flanged in, out of the solid plate, on a radius 
not less than 3 times the metal thickness to a straight flange ; when the 
plate is y<i inch or less in thickness a reinforce ring to be shrunk around it. 
Cast-iron reinforce flanges never to be used. 

22. Domes to be avoided when possible; cylindrical portion to be 
flanged down to the shell of the boiler, and this shell flanged up inside 
the dome or reinforced by a collar flanged at the joint, the flanges double- 
riveted. 

23. Drums should be put on with collar flanges of A. B. M. A. steel not 
less than % inch thick, double-riveted to shell and drum and single-riveted 
to the neck or leg, or the flanges may be formed on these legs. 

24. Saddles or Nozzles to be of flanged steel plate or of soft cast^steel, 
never of cast-iron. 

III. FACTORS OF SAFETY. 

25. Rivet Seams, when proportioned as prescribed in Section 10 with 
materials tested as per Sections 2 and 3, shall have 4>^ as factor of safety ; 
when not so tested, but inspection of materials indicates good quality, a 
factor of safety of 5 is to be taken, and at most 55,000 pounds tensile 
strength assumed for the steel plate and 40,000 pounds shear strength for 
the rivets, all figured on the actual net standing metal. 

26. Flat Surfaces, proportioned as per Section 14, have, in the con- 
stants there given, a factor of safety of 5 or a little over. 

27. Bumped Heads, proportioned as per Section 9, to be subject to a 
. factor of safety of 5. 

28. Stay Bolts, proportioned and tested as per Sections 19 and 5, to 
have a factor of safety of 5 applied to the lowest stress found. 

29. Braces and Stays, when tested as per Sections 6 and 2, to be 
allowed a factor of safety of 5 ; when not so tested, but careful inspection 



632 Safety Valves. 



shows good stock, they may be used up to 6500 pounds actual direct pull 
for wrought-iron, and 8000 pounds for mild steel, all per square inch of 
actual net metal. 

IV. HYDROSTATIC PRESSURE. 

30. The hydrostatic test to be made on completed boilers built strictly 
to these specifications is never to exceed working pressure by more than 
one-third of itself, and this excess limited to 100 pounds per square inch. 
The water used for testing to have a temperature of at least 125° F. 



V. HANGING OR SUPPORTING THE BOILER. 

31. The boiler should be supported on points where there is the 
greatest excess of stress. Excessive local stresses from weight of boiler 
and contents must be avoided, and distortion of parts prevented, by using 
long lugs or brackets ; and only half the stress which they may carry in 
the seams to be allowed on rivets. 

The supports must permit rebuilding the furrlace without disturbing 
the proper suspension of the boiler. The boiler should be slightly inclined, 
so that a little less water shows at the gauge cocks than at the opposite 
end. 



SAFETY VALVES. 

Weighted Valves. 

A = area of valve, in square inches ; 

F= distance from centre of valve to fulcrum, in inches ; 

L = length of lever, in inches, from fulcrum to weight; 

W= weight of ball, in pounds ; 

P = blowing-off pressure, in pounds, per square inch. 

Then we have 

p== WL 



Let 



£ = 



TT = 



AF ■ 

AFP 

W ' 

AFP 



If lever is not balanced, its effect, and the effect of valve and spindle, 
must be added to pressure and be taken into account in calculating L and 
W. If w = weight of lever and v = weight of valve and spindle, in 
pounds ; c = distance of centre of gravity of lever from fulcrum ; then, 
if p = pressure per square inch on valve due to weight of lever and valve 
alone, 

w X c v 
P = -AF~ + A- 

In most cases effect of valve and spindle may be neglected. With long, 
heavy levers p will require adding to P to ascertain the blowing-off pressure. 

Various rules are given for the area of safety valves, these usually being 
based on a certain number of square inches of valve area per square foot 
of grate surface, although sometimes the area of the valve is based on the 
heating surface of the boiler. 

The United States Treasury Department, through its Board of Super- 
vising Inspectors of Steam Vessels, has established the following rules : 

"Lever safety valves to be attached to marine boilers shall have an 
area of not less than one square inch to two square feet of grate surface in 



Safety Valves. 633 



the "boiler, and the seats of all such safety valves shall have an angle of 
inclination of 45° to the centre line of their axes. 

"The valves shall be so arranged that each boiler shall have one sepa- 
rate safety valve, unless the arrangement is such as to preclude the possi- 
1 bility of shutting off the communication of any boiler with the safety 
valve or valves employed. This arrangement shall also apply to lock-up 
safety valves when they are employed. 

"Any spring-loaded safety valves constructed so as to give an increased 
lift by the operation of steam after being raised from their seats, or any 
spring-loaded safety valve constructed in any other manner, or so as to 
give an effective area equal to that of the afore-mentioned spring-loaded 
safety valve, may be used in lieu of the common lever-weighted valves on 
all boilers on steam vessels, and all such spring-loaded safety valves shall 
be required to have an area of not less than 1 square inch to 3 square feet 
of grate surface of the boiler, and each spring-loaded safety valve shall be 
supplied with a lever that will raise the valve from its seat a distance of 
not less than that equal to one-eighth the diameter of the valve opening, 
and the seats of all such safety valves shall have an angle of inclination 
to the centre line of their axis of 45°. But in no case shall any spring- 
loaded safety valve be used in lieu of the lever-weighted safety valve 
without having first been approved by the Board of Supervising Inspec- 
tors." 

The Boiler Inspection Department of the city of Philadelphia gives the 
following formula for boilers with natural draft : 



22.5G 
A = - 



8.62 * 

in which A is the area of combined safety valves, in inches ; G is area of 
grate, in square feet; Pis pressure of steam, in pounds, per square inch to 
be carried in the boiler above the atmosphere. 

The following table gives the results of the formula for 1 square foot of 
grate, as applied to boilers used at different pressures. 

Pressure per Square Inch. 

10 20 30 40 50 60 70 80 90 100 110 120 150 175 

1.21 0.79 0.58 0.46 0.38 0.33 0.29 0.25 0.23 0.21 0,19 0.17 0.142 0.123 

Valve area in square inches, corresponding to 1 square foot of grate. 

When forced draft is used, the area of grate for purposes of safety valve 
computation is to be estimated at 1 square foot for each 16 pounds of fuel 
burned per hour. 

Hutton's rule is 

V p 

A = area of valve, in square inches ; 

G = area of grate, in square feet ; 

P = pressure, in pounds, per square inch. 

The area of a safety valve may be determined from the evaporative 
power of the boiler. 
Let 

A = area of safety valve, in square inches ; 

P = steam pressure, in pounds, per square inch ; 

E = evaporative capacity of the boiler, in pounds of water, per hour. 

Then we have 

40i/P 



634 



Safety Valves. 



Minimum Size of Safety Valve Areas Allowed by 
Board of Trade. 





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1.250 


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.559 


89 


.360 


126 


.265 


163 


.210 


16 


1.209 


53 


.551 


90 


.357 


127 


.264 


164 


.209 


17 


1.171 


54 


.543 


91 


.353 


128 


.262 


165 


.208 


18 


1.136 


55 


.535 


92 


.350 


129 


.260 


166 


.207 


19 


1.102 


56 


.528 


93 


.347 


130 


.258 


167 


.206 


20 


1.071 


57 


.520 


94 


.344 


131 


.256 


168 


.204 


21 


1.041 


58 


.513 


95 


.340 


132 


.255 


169 


.203 


22 . 


1.013 


59 


.506 


96 


.337 


133 


.253 


170 


.202 


23 


.986 


60 


.500 


97 


.334 


134 


.251 


171 


.201 


24 


.961 


61 


.493 


98 


.331 


135 


.250 


172 


.200 


25 


.937 


62 


.487 


99 


.328 


136 


.248 


173 


.199 


26 


.914 


63 


.480 


100 


.326 


137 


.246 


174 


.198 


27 


.892 


64 


.474 


101 


.323 


138 


.245 


175 


.197 


28 


.872 


65 


.468 


102 


.320 


139 


.243 


176 


.196 


29 


.852 


66 


.462 


103 


.317 


140 


.241 


177 


.195 


30 


.833 


67 


.457 


104 


.315 


141 


.240 


178 


.194 


31 


.815 


68 


.451 


105 


.312 


142 


.238 


179 


.193 


32 


.797 


69 


.446 


106 


.309 


143 


.237 


180 


.192 


33 


.781 


70 


.441 


107 


.307 


144 


.235 


181 


.191 


34 


.765 


71 


.436 


108 


.304 


145 


.234 


182 


.190 


35 


.750 


72 


.431 


109 


.302 


146 


.232 


183 


.189 


36 


.735 


73 


.426 


110 


.300 


147 


.231 


184 


.188 


37 


.721 


74 


.421 


111 


.297 


148 


.230 


185 


.187 


38 


.707 


75 


.416 


112 


.295 


149 


.228 


186 


.186 


39 


.694 


76 


.412 


113 


.292 


150 


.227 


187 


.185 


40 


.681 


77 


.407 


114 


.290 


151 


.225 


188 


.184 


41 


.669 


78 


.403 


115 


.288 


152 


.224 


189 


.183 


42 


.657 


79 


.398 


116 


.286 


153 


^223 


190 


.182 


43 


.616 


80 


.394 


117 


.284 


154 


.221 


191 


.181 


44 


.635 


81 


.390 


118 


.281 


155 


.220 


192 


.181 


45 


.625 


82 


.386 


119 


.279 


156 


.219 


193 


.180 


46 


.614 


83 


.382 


120 


.277 


157 


.218 


194 


.179 


47 


.604 


84 


.378 


121 


.275 


158 


.216 


195 


.178 


48 


.595 


85 


.375 


122 


.273 


159 


.215 


196 


.177 


49 


.585 


86 


.371 


123 


.271 


160 


.214 


197 


.176 


50 


.576 


87 


.367 


124 


.269 


161 


.213 


198 


.176 


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.568 


88 


.364 


125 


.267 


162 


.211 


200 


.174 



Boiler Incrustation. 635 

Lloyd's Rules for Safety Valves. 

Two safety valves to be fitted to each boiler and loaded to the working 
pressure in the presence of the surveyor. In the case of boilers of greater 
' working pressure than 60 pounds per square inch, the safety valves may 
be loaded to 5 pounds above the working pressure. If common valves are 
used, their combined areas to be at least half a square inch to each square 
foot of grate surface. If improved valves are used, they are to be tested 
under steam in the presence of the surveyor; the accumulation in no case 
to exceed 10 per cent, of the working pressure. 

An approved safety valve also to be fitted to the superheater. 

In winch boilers one safety valve will be allowed, provided its area be 
not less than half a square inch per square foot of grate surface. 

Each valve to be arranged so that no extra load can be added when 
steam is up, and to be fitted with easing gear, which must lift the valve 
itself. All safety valve spindles to extend through the covers and to be 
fitted with sockets and cross handles, allowing them to be lifted and 
turned round in their seats, and their efficiency tested at any time. 

The German rule for safety valves, as given in the " Ingenieurs Taschen- 
buch Hiitte," is 

in which /is the area of valve, in square millimetres, per square metre of 
heating surface in the boiler ; p is the maximum boiler pressure, in atmos- 
pheres; and v is the volume of steam, in litres, per kilogramme at the 
pressure, p, as given in the steam tables. 
We have from this formula 

Areas of Safety Valves, 

in Square Millimetres, per Square Metre of Heating Surface. 
p = press, atm. . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 
v = sp. volume. 896 612 467 378 313 276 244 218 198 181 167 154 144 135 
/= sq. mm. per 

sq. metre.. 449 263 188 147 119 102 89 78 71 63 59 54 50 48 



INCRUSTATION IN BOILERS. 

Whenever possible, pure water should be used for feeding boilers. 
When the water is impure the result is the formation of scale, producing 
diminished efficiency and possible injury to the boiler from overheating. 

The principal impurities in water are calcium carbonate and calcium 
sulphate, together with suspended earth and organic matter. Water of 
condensation from steam engines contains more or less oil from the lubri- 
cant used in the steam, and this may produce a very injurious coating in 
the boiler. 

By far the best plan is to remove or neutralize the impurities of the 
water before it is fed into the boiler, since the scale, when it is once 
formed, is difficult to remove. 

No general rules can be given for the purification of water, since differ- 
ent waters require different treatment. The best plan is to have the water 
analyzed and adopt the course indicated by the nature of the salts found 
in it. 

The following extracts from a paper by Messrs. Hunt and Clapp, in the 
"Transactions of the American Institute of Mining Engineers for 1888," is 
an authoritative statement of the subject : 

"By far the most common commercial analysis of water is made to 
determine its fitness for making steam. Water containing more than 5 
parts per 100,000 of free sulphuric or nitric acid is liable to cause serious 
corrosion, not only of the metal of the boiler itself, but of the pipes, cylin- 
ders, pistons, and valves with which the steam comes in contact. Sul- 
phuric acid is the only one of these acids liable to be present in the water 



636 Boiler Incrustation. 



from natural sources, it being often produced in the water of the coal and 
iron districts by the oxidation of iron pyrites to sulphate of iron, which, 
being soluble, is lixiviated from the earth strata and carried into the 
stream, the presence of organic matter taken up by the water in its after- 
course reducing the iron and lining the bottom of the stream with red 
oxide of iron, leaving a considerable proportion of the sulphuric acid 
free in the water. This is a troublesome feature with the water necessarily 
used in many of the iron districts of this country. The sulphuric acid 
may come from other natural chemical reactions than the one described 
above. Muriatic and nitric acids, as well as sulphuric acid, may be con- 
veyed into water through the refuse of various kinds of manufacturing 
establishments being discharged into it. 

"The large total residue in water used for making steam causes the 
interior linings of the boilers to become coated, clogs their action, and 
often produces a dangerous hard scale, which prevents the cooling action 
of the water from protecting the metal against burning. 

" Lime and magnesia bicarbonates in water lose their excess of carbonic 
acid on boiling, and often, especially when the water contains sulphuric 
acid, produce, with the other solid residues constantly being formed by the 
evaporation, a very hard and insoluble scale. 

11 A larger amount than 100 parts per 100,000 of total solid residue will 
ordinarily cause troublesome scale, and should condemn the water for use 
in steam boilers, unless a better supply cannot be obtained. 

" The following is a tabulated form of the causes of trouble with water 
for steam purposes, and the proposed remedies, given by Professor L. M. 
Norton in his lecture on * Industrial Chemistry.' 



1 Causes of Incrustation. 






11 1. Deposition of suspended matter. 

11 2. Deposition of dissolved salts from concentration. 

"3. Deposition of carbonates of lime and magnesia by boiling off car- 
bonic acid, which holds them in solution. 

"4. Deposition of sulphates of lime, because sulphate of lime is but 
slightly soluble in cold water, less soluble in hot water, insoluble above 
140° C. (284° F.). 

"5. Deposition of magnesia, because magnesium salts decompose at 
high temperature. 

"6. Deposition of lime soap, iron soap, etc., formed by saponification 
of grease. 



Various Means of Preventing Incrustation. 



"1. Filtration. 
"2. Blowing off. 

"3. Use of internal collecting apparatus or devices for directing the 
circulation. 

1 ' 4. Heating feed water. 

"5. Chemical or other treatment of water in boiler. 

"6. Introduction of zinc into boiler. 

"7. Chemical treatment of water outside of boiler." 



Prevention and Cure of Boiler Troubles Due to Water. 

Sediment, mud, clay, . Filtration . 






Incrustation. 



suimciit, iuuu, ua L , Filtration> 

eic \ Blowing off. 

Readily soluble salts Blowing off. 

Bicarbonate of magne- f ^{P?^ and Precipitating. 4 
sia, lime, and iron... \ Lime 

I Magnesia. 
Rulnhatp of limp \ Carbonate of soda. 

i Barium chloride. 



Boiler Incrustation. 



637 



Corrosion . . 



Priming . 



Organic matter. 



Grease . 



( Precipitate with alum and filter. 
. < Precipitate with ferric chloride 
( and filter. 
( Slaked lime and filter. 
' \ Carbonate of soda and filter. 
Chloride or sulphate of 

magnesia Carbonate of soda. 

Acid Alkali. 

Dissolved carbonic ( Slaked lime, 
acid and oxygen — < Caustic soda. 
( Heating. 

f Precipitate with alum or ferric 
' \ chloride and filter. 
Carbonate of soda in 
^ large quantities Barium chloride. 

The following table shows the solubility of various scale-making 
materials in steam boilers, showing in the last column the temperatures 
at which they become insoluble. Although sulphate of lime does not 
become entirely insoluble until a temperature of nearly 400° F., corre- 
sponding to a pressure of about 225 pounds, a large proportion of it is pre- 
cipitated at about 310° F., or about 65 pounds pressure. It will be seen, 
therefore, that most of these impurities may be precipitated by using a 
feed-water heater of sufficient size to permit the precipitated impurities to 
settle and be blown off before passing into the boiler. 



Solubilities of Scale= making Minerals. 



Substance. 


Soluble in 

parts of 

pure water 

at 30° F. 


Soluble in parts 

of carbonic 

acid, water 

cold. 


Soluble in 

parts of 

pure water 

at 212° F. 


Insoluble 

in water 

at 


Carbonate of lime 

Sulphate of lime 

Carbonate of magnesia. 
Phosphate of lime 


62500 

500 

5500 


150 


62500 

460 

9600 


302° F. 
392° F. 


150 
1333 




212° F. 


Oxide of iron 






212° F 


Silica 




Undetermined. 




212° F. 











Analyses of Boiler Scale. 

(Chandler.) 



Sulphate 
of lime. 


Magnesia. 


Silica. 


Peroxide of 
iron. 


Water. 


Carbonate of 
lime. 


74.07 
71.37 


9.19 


.65 
1.76 
2.60 
4.79 
5.32 
7.75 
2.07 

.65 
2.92 
8.24 


.08 


1.14 


14.78 


62.86 
53.05 


18.95 


.92 


1.28 


12.62 


46.83 










30.80 
4.95 

.88 
4.81 


31.17 
2.61 
2.84 


1.08 

1.03 

.36 


2.44 
.63 
.15 


26.93 
86.25 
93.19 


30.07 



















638 



Steam Engine. 



Analysis, in Parts per 100,000, of Water Giving Bad 
Results in Steam Boilers. 

(A. E. Hunt.) 





© 


















a 




a o 


a -a 
















d 


Waters. 




«2 ® 
O ."S 

S o 


©* 


© 

■ a 


5 

'o 

03 






a 

o 

"S 




73 

s 

o 




icarbon 
deposite 
ing. 


III 

III 


o 


c3 

a 
3 

o 


"9 


© 

a 
2 


o 


c3 
G 

a 

J3 


© 
| 

2 




ffl 


CQ 


H 


H 


02 


o 


M 


o 


«*1 


o 


Coal-mine water 


110 


25 


119.0 


39.00 


890 


590.0 


780 


30 


640 




Salt well 


151 


38 


19 


48.00 


360 


990 


38 


?! 


30 


13 1 


Soring 


75 


89 


95.0 


120.00 


310 


210 


75 


10 


80 


36 


Monongahela River . . 
Monongahela River . . 
Monongahela River . . 
Allegheny River, 


130 


21 


161.0 


33.00 


910 


38.0 


70 








80 


70 


94.0 


81.00 


91 q 


210.0 


90 








32 


82 


61.0 


1.04 


98 


1.9 


38 




























near oil- works 


30 


50 


41,0 


68.00 


890 


42.0 


23 

















THE STEAM ENGINE. 
Horse=power. 

The measure of the power of steam engines is the Horse=power, origi- 
nally selected by Watt as a basis on which to sell his engines. Tests of a 
number of powerful draught horses showed an effort corresponding to 
22,000 foot-pounds per minute, and Watt increased this by 50 per cent., in 
order to assure his customers that he was furnishing ample power ; this 
being the origin of the well-known value of 33,000 foot-pounds per minute, 
or 550 foot-pounds per second, as a commercial horse-power. 

In the metric system the cheval-vapeur is taken as 75 kilogrammetres per 
second, this corresponding to 32,548 foot-pounds per minute, the metric 
horse-power thus being 0.9863 times the British horse-power. The latter 
will always be understood, unless otherwise stated. 

In France it has been suggested to use a new unit, equal to 100 kilo- 
grammetres per second, this being called the Poncelei, and being practically 
equivalent to the kilowatt. 

Since 1 B. T. U. = 778 foot-pounds, it requires the expenditure of 42.416 
B. T. U. per minute to produce 1 horse-power, if all the heat is converted 
into mechanical energy. 

In the steam engine the power is usually developed by the pressure of 
the expansive force of the steam upon the piston in the cylinder. Since 
the speed of the piston is not uniform, varying from zero to a maximum 
twice for every revolution of the crank, it is necessary to take the total 
distance travelled in one minute as the average or mean speed. 

The pressure of the steam upon the piston is also variable, and hence it 
is necessary to determine the mean effective pressure, in order that the 
horse-power may be computed. For a completed engine the mean effec- 
tive pressure may be determined by use of the indicator, but for a proposed 
design it is computed in accordance with the laws of the expansion of 
steam. 

According to the law of Mariotte, considering steam as a gas, the prod- 
uct of the pressure and the volume is constant, or 

pv = C. 



Steam Engine. 639 



When the steam in a cylinder be permitted to follow a portion of the 
stroke at full boiler pressure, and is then cut off and allowed to expand 
for the remainder of the stroke, the expansion curve may be considered as 
an equilateral hyperbola, the pressure at any point being inversely as the 
| volume. When the volume has been doubled, the pressure will fall to 
one-half the initial ; when it becomes three times what it was at the point 
of cut-off, the pressure will be one-third the initial pressure, and so on. 
In this way it is quite possible to construct a theoretical diagram for any 
degree of cut-off or any expansion ratio, and measure the mean pressure 
throughout the stroke. 

Instead of performing this work, however, the mean effective pressure 
may be computed immediately by means of a table of hyperbolic loga- 
rithms. 

Let P == initial pressure, absolute, — i.e., above vacuum ; 
p = mean effective pressure, including vacuum ; 
r = expansion ratio == total stroke divided by length up to point 
of cut-off. 



Then 



p 1 + hyp, log, r 



Hence, by taking the hyperbolic logarithm of the expansion ratio and 
adding 1, and dividing by the expansion ratio, we have a number which, 
multiplied by the initial pressure, will give the mean effective pressure. 

Thus, if the steam is admitted at 100 pounds gauge pressure, or 114.7 
pounds absolute pressure, and cut off at % the stroke, we have 

r = 4, 
and p = 1117 1 + h yP- 1 Qg-4 > 

The hyperbolic logarithm of 4 is 1.3863, and hence we have 

„. i-fl.3863 
p = 114,7 — — 

= 114.7 X 0.5966 

= 68.43 pounds absolute 

= 53.73 pounds above atmosphere. 

There is always a loss of pressure in practice due to cylinder considera- 
tion, etc., and in practice about 70 per cent, of the theoretical mean effec- 
tive pressure is attained. 

In the above computations care must be taken always to use the absolute 
pressure, — i.e., the pressure above vacuum, — the resulting mean effective 
pressure being that existing above vacuum. For a high-pressure engine, 
therefore, atmospheric pressure must be deducted. 



640 




Hyperbolic 


Logarithms. 






N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


1.01 


.009 9503 


1.65 


.500 7752 


2.29 


.828 5518 


2.93 


1.075 0024 


1.02 


.019 8026 


1.66 


.506 8175 


2.30 


.832 9091 


2.94 


1.078 4095 


1.03 


.029 5588 


1.67 


.512 8236 


2.31 


.837 2475 


2.95 


1.081 8051 


1.04 


.039 2207 


1.68 


.518 7937 


2.32 


.841 5671 


2.96 


1.085 1892 


1.05 


.048 7902 


1.69 


.524 7285 


2.33 


.845 8682 


2.97 


1.088 5619 


1.06 


.058 2689 


1.70 


.530 6282 


2.34 


.850 1509 


2.98 


1.091 9233 


1.07 


.067 6586 


1.71 


.536 4933 


2.35 


.854 4153 


2.99 


1.095 2733 


1.08 


.076 9610 


1.72 


.542 3242 


2.36 


.858 6616 


3.00 


1.098 6123 


1.09 


.086 1777 


1.73 


.548 1214 


2.37 


.862 8899 


3.01 


1.101 9400 


1.10 


.095 3102 


1.74 


.553 8851 


2.38 


.867 1004 


3.02 


1.105 2568 


1.11 


.104 3600 


1.75 


.559 6157 


2.39 


.871 2933 


3.03 


1.108 5626 


1.12 


.113 3287 


1.76 


.565 3138 


2.40 


.875 4687 


3.04 


1.111 8575 


1.13 


.122 2176 


1.77 


.570 9795 


2.41 


.879 6267 


3.05 


1.115 1415 


1.14 


.131 0283 


1.78 


.576 6133 


2.42 


.883 7675 


3.06 


1.118 4149 


1.15 


.139 7619 


1.79 


.582 2156 


2.43 


.887 8912 


3.07 


1.121 6775 


1.16 


.148 4200 


1.80 


.587 7866 


2.44 


.891 9980 


3.08 


1.124 9295 


1.17 


.157 0037 


1.81 


.593 3268 


2.45 


.896 0880 


3.09 


1.128 1710 


1.18 


.165 5144 


1.82 


.598 8365 


2.46 


.900 1613 


3.10 


1.131 4021 


1.19 


.173 9533 


1.83 


.604 3159 


2.47 


.904 2181 


3.11 


1.134 6227 


1.20 


.182 3215 


1.84 


.609 7655 


2.48 


.908 2585 


3.12 


1.137 8330 


1.21 


.190 6203 


1.85 


.615 1856 


2.49 


.912 2826 


3.13 


1.141 0330 


1.22 


.198 8508 


1.86 


.620 5764 


2.50 


.916 2907 


3.14 


1.144 2227 


1.23 


.207 0141 


1.87 


.625 9384 


2.51 


.920 2827 


3.15 


1.147 4024 


1.24 


.215 1113 


1.88 


.631 2717 


2.52 


.924 2589 


3.16 


1.150 5720 


1.25 


.223 1435 


1.89 


.636 5768 


2.53 


.928 2193 


3.17 


1.153 7315 


1.26 


.231 1117 


1.90 


.641 8538 


2.54 


.932 1640 


3.18 


1.156 8811 


1.27 


.239 0169 


1.91 


.647 1032 


2.55 


.936 0933 


3.19 


1160 0209 


1.28 


.246 8600 


1.92 


.652 3251 


2.56 


.940 0072 


3.20 


1.163 1508 


1.29 


.254 6422 


1.93 


.657 5200 


2.57 


.943 9058 


3.21 


1.166 2709 


1.30 


.262 3642 


1.94 


.662 6879 


2.58 


.947 7893 


3.22 


1.169 3813 


1.31 


.270 0271 


1.95 


.667 8293 


2.59 


.951 6578 


3.23 


1.172 4821 


1.32 


.277 6317 


1.96 


.672 9444 


2.60 


.955 5114 


3.24 


1.175 5733 


1.33 


.285 1789 


1.97 


.678 0335 


2.61 


.959 3502 


3.25 


1.178 6549 


1.34 


.292 6696 


1.98 


.683 0968 


2.62 


.963 1743 


3.26 


1.181 7271 


1.35 


.300 1045 


1.99 


.688 1346 


2.63 


.966 9838 


3.27 


1.184 7899 


1.36 


.307 4846 


2.00 


.693 1472 


2.64 


.970 7789 


3.28 


1.187 8434 


1.37 


.314 8107 


2.01 


.698 1347 


2.65 


.974 5596 


3.29 


1.190 8875 


1.38 


.322 0834 


2.02 


.703 0974 


2.66 


.978 3261 


3.30 


1.193 9224 


1.39 


.329 3037 


2.03 


.7080357 


2.67 


.982 0784 


3.31 


1.196 9481 


1.40 


.336 4722 


2.04 


.712 9497 


2.68 


.985 8167 


3.32 


1.199 9647 


1.41 


.343 5897 


2.05 


.717 8397 


2.69 


.989 5411 


3.33 


1.202 9722 


1.42 


.350 6568 


2.06 


.722 7059 


2.70 


.993 2517 


3.34 


1.205 9707 


1.43 


.357 6744 


2.07 


.727 5485 


2.71 


.996 9486 


3.35 


1.208 9603 


1.44 


.364 6431 


2.08 


.732 3678 


2.72 


1.000 6318 


3.36 


1.211 9409 


1.45 


.371 5635 


2.09 


.737 1640 


2.73 


1.004 3015 


3.37 


1.214 9127 


1.46 


.378 4364 


2.10 


.741 9373 


2.74 


1.007 9579 


3.38 


1.217 8757 


1.47 


.385 2624 


2.11 


.746 6879 


2.75 


1.011 6008 


3.39 


1.220 8299 


1.48 


.392 0420 


2.12 


.751 4160 


2.76 


1.015 2306 


3.40 


1.223 7754 


1.49 


.398 7761 


2.13 


.756 1219 


2.77 


1.018 8473 


3.41 


1.226 7122 


1.50 


.405 4651 


2.14 


.760 8058 


2.78 


1.022 4509 


3.42 


1.229 6405 


1.51 


.412 1096 


2.15 


.765 4678 


2.79 


1.026 0415 


3.43 


1.232 5605 


1.52 


.418 7103 


2.16 


.770 1082 


2.80 


1.029 6194 


3.44 


1.235 4714 


1.53 


.425 2677 


2.17 


.774 7271 


2.81 


1.033 1844 


3.45 


1.238 3742 


1.54 


.431 7824 


2.18 


.779 3248 


2.82 


1.036 7368 


3.46 


1.241 2685 


1.55 


.438 2549 


2.19 


.783 9015 


2.83 


1.040 2766 


3.47 


1.244 1545 


1.56 


.444 6858 


2.20 


.788 4573 


2 84 


1.043 8040 


3.48 


1.247 0322 


1.57 


.451 0756 


2.21 


.792 9925 


2.85 


1.047 3189 


3.49 


1.249 9017 


1.58 


.457 4248 


2.22 


.797 5071 


2.86 


1.050 8216 


3.50 


1.252 7629 


1.59 


.463 7340 


2.23 


.802 0015 


2.87 


1.054 3120 


3.51 


1.255 6160 


1.60 


.470 0036 


2.24 


.806 4758 


2.88 


1.057 7902 


3.52 


1.258 4609 


1.61 


.476 2341 


2.25 


.810 9302 


2.89 


1.061 2564 


3.53 


1.261 2978 


1.62 


.482 4261 


2.26 


.815 3648 


2.90 


1.064 7107 


3.54 


1.264 1266 


1.63 


.488 5800 


2.27 


.819 7798 


2.91 


1.068 1530 


3.55 


1.266 9475 


1.64 


.494 6962 


2.28 


.824 1754 


2.92 


1.071 5836 


3.56 


1.269 7605 







H 


YPEKBOLIC 


Logarithms. 




641 


N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


3.57 


1.272 5655 


4.21 


1.437 4626 


4.85 


1.578 9787 


5.49 


1.702 9282 


"3.58 


1.275 3627 


4.22 


1.439 8351 


4.86 


1.581 0384 


5.50 


1.704 7481 


3.59 


1.278 1521 


4.23 


1.442 2020 


4.87 


1.583 0939 


5.51 


1.706 5646 


3.60 


1.280 9338 


4.24 


1.444 5632 


4.88 


1.585 1452 


5.52 


1.708 3778 


3.61 


1.283 7077 


4.25 


1.446 9189 


4.89 


1.587 1923 


5.53 


1.710 1878 


3.62 


1.286 4740 


4.26 


1.449 2691 


4.90 


1.589 2352 


5.54 


1.711 9944 


3.63 


1.289 2326 


4.27 


1.451 6138 


4.91 


1.591 2739 


5.55 


1.713 7979 


3.64 


1.291 9836 


4.28 


1.453 9530 


4.92 


1.593 3085 


5.56 


1.715 5981 


3.65 


1.294 7271 


4.29 


1.456 2867 


4.93 


1.595 3389 


5.57 


1.717 3950 


3.66 


1.297 4631 


4.30 


1.458 6149 


4.94 


1.597 3653 


5.58 


1.719 1887 


3.67 


1.300 1916 


4.31 


1.460 9379 


4.95 


1.599 3875 


5.59 


1.720 9792 


3.68 


1.302 9127 


4.32 


1.463 2553 


4.96 


1.601 4057 


5.60 


1.722 7666 


3.69 


1.305 6264 


4.33 


1.465 5675 


4.97 


1.603 4198 


5.61 


1.724 5507 


3.70 


1.308 3328 


4.34 


1.467 8743 


4.98 


1.605 4298 


5.62 


1.726 3316 


3.71 


1.311 0318 


4.35 


1.470 1758 


4.99 


1.607 4358 


5.63 


1.728 1094 


3.72 


1.313 7236 


4.36 


1.472 4720 


5.00 


1.609 4379 


5.64 


1.729 8840 ' 


3.73 


1.316 4082 


4.37 


1.474 7630 


5.01 


1.611 4359 


5.65 


1.731 6555 


3.74 


1.319 0856 


4.38 


1.477 0487 


5.02 


1.613 4300 


5.66 


1.733 4238 


3.75 


1.321 7558 


4.39 


1.479 3292 


5.03 


1.615 4200 


5.67 


1.735 1891 


3.76 


1.324 4189 


4.40 


1.481 6045 


5.04 


1.617 4060 


5.68 


1.736 9512 


3.77 


1.327 0749 


4.41 


1.483 8746 


5.05 


1.619 3882 


5.69 


1.738 7102 


3.78 


1.329 7240 


4.42 


1.486 1396 


5.06 


1.621 3664 


5.70 


1.740 4661 


3.79 


1.332 3660 


4.43 


1.488 3995 


5.07 


1.623 3408 


5.71 


1.742 2189 


3.80 


1.335 0010 


4.44 


1.490 6543 


5.08 


1.625 3112 


5.72 


1.743 9687 


3.81 


1.337 6291 


4.45 


1.492 9040 


5.09 


1.627 2778 


5.73 


1.745 7155 


3.82 


1.340 2504 


4.46 


1.495 1487 


5.10 


1.629 2405 


5.74 


1.747 4591 


3.83 


1.342 8648 


4.47 


1.497 3883 


5.11 


1.631 1994 


5.75 


1.749 1998 


3.84 


1.345 4723 


4.48 


1.499 6230 


5.12 


1.633 1544 


5.76 


1.750 9374 


3.85 


1.348 0731 


4.49 


1.501 8527 


5.13 


1.635 1056 


5.77 


1.752 6720 


3.86 


1.350 6671 


4.50 


1.504 0774 


5.14 


1.637 0530 


5.78 


1.754 4036 


3.87 


1.353 2544 


4.51 


1.506 2971 


5.15 


1.638 9967 


5.79 


1.756 1323 


3.88 


1.355 8351 


4.52 


1.508 5119 


5.16 


1.640 9365 


5.80 


1.757 8579 


3.89 


1.358 4091 


4.53 


1.510 7219 


5.17 


1.642 8726 


5.81 


1.759 5805 


3.90 


1.360 9765 


4.54 


1.512 9269 


5.18 


1.644 8050 


5.82 


1.761 3002 


3.91 


1.363 5373 


4.55 


1.515 1272 


5.19 


1.646 7336 


5.83 


1.763 0170 


3.92 


1.366 0916 


4.56 


1.517 3226 


5.20 


1.648 6586 


5.84 


1.764 7308 


3.93 


1.368 6394 


4.57 


1.519 5132 


5.21 


1.650 5798 


5.85 


1.766 4416 


3.94 


1.371 1807 


4.58 


1.521 6990 


5.22 


1.652 4974 


5.86 


1.768 1496 


3.95 


1.373 7156 


4.59 


1.523 8800 


5.23 


1.654 4112 


5.87 


1.769 8546 


3.96 


1.376 2440 


4.60 


1.526 0563 


5.24 


1.656 3214 


5.88 


1.771 5567 


3.97 


1.378 7661 


4.61 


1.528 2278 


5.25 


1.658 2280 


5.89 


1.773 2559 


3.98 


1.381 2818 


4.62 


1.530 3947 


5.26 


1.660 1310 


5.90 


1.774 9523 


3.99 


1.383 7912 


4.63 


1.532 5568 


5.27 


1.662 0303 


5.91 


1.776 6458 


4.00 


1.386 2943 


4.64 


1.534 7143 


5.28 


1.663 9260 


5.92 


1.778 3364 


4.01 


1.388 7912 


4.65 


1.536 8672 


5.29 


1.665 8182 


5.93 


1.780 0242 


4.02 


1.391 2818 


4.66 


1.539 0154 


5.30 


1.667 7068 


5.94 


1.781 7091 


4.03 


1.393 7663 


4.67 


1.541 1590 


5.31 


1.669 5918 


5.95 


1.783 3912 


4.04 


1.396 2446 


4.68 


1.543 2981 


5.32 


1.671 4733 


5.96 


1.785 0704 


4.05 


1.398 7168 


4.69 


1.545 4325 


5.33 


1.673 3512 


5.97 


1.786 7469 


4.06 


1.401 1829 


4.70 


1.547 5625 


5.34 


1.675 2256 


5.98 


1.788 4205 


4.07 


1.403 6429 


4.71 


1.549 6879 


5.35 


1.677 0965 


5.99 


1.790 0914 


4.08 


1.406 0969 


4.72 


1.551 8087 


5.36 


1.678 9639 


6.00 


1.791 7594 


4.09 


1.408 5449 


4.73 


1.553 9252 


5.37 


1.680 8278 


6.01 


1.793 4247 


4.10 


1.410 9869 


4.74 


1.556 0371 


5.38 


1.682 6882 


6.02 


1.795 0872 


4.11 


1.413 4230 


4.75 


1.558 1446 


5.39 


1.684 5453 


6.03 


1.796 7470 


4.12 


1.415 8531 


4.76 


1.560 2476 


5.40 


1.686 3989 


6.04 


1.798 4040 


4.13 


1.418 2774 


4.77 


1.562 3462 


5.41 


1.688 2491 


6.05 


1.800 0582 


4.14 


1.420 6957 


4.78 


1.564 4405 


5.42 


1.690 0958 


6.06 


1.801 7098 


4.15 


1.423 1083 


4.79 


1.566 5304 


5.43 


1.691 9391 


6.07 


1.803 3586 


4.16 


1.425 5150 


4.80 


1.568 6159 


5.44 


1.693 7790 


6.08 


1.805 0047 


4.17 


1.427 9160 


4.81 


1.570 6971 


5.45 


1.695 6155 


6.09 


1.806 6481 


4.18 


1.430 3112 


4.82 


1.572 7739 


5.46 


1.697 4487 


6.10 


1.808 2887 


4.19 


1.432 7007 


4.83 


1.574 8464 


5.47 


1.699 2786 


6.11 


1.809 9267 


4.20 


1.435 0845 


4.84 


1.576 9147 


5.48 


1.701 1051 


6.12 


1.811 5621 



41 



642 




H 


YPEKBOLIC 


Logarithms. 






N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


6.13 


1.813 1947 


6.77 


1.912 5011 


7.41 


2.002 8305 


8.05 


2.085 6720 


6.14 


1.814 8247 


6.78 


1.913 9771 


7.42 


2.004 1790 


8.06 


2.086 9135* 


6.15 


1.816 4520 


6.79 


1.915 4509 


7.43 


2.005 5258 


8.07 


2.088 1534 


6.16 


1.818 0767 


6.80 


1.916 9226 


7.44 


2.006 8708 


8.08 


2.089 3918 


6.17 


1.819 6988 


6.81 


1.918 3921 


7.45 


2.008 2140 


8.09 


2.090 6287 


6.18 


1.821 3182 


6.82 


1.919 8594 


7.46 


2.009 5553 


8.10 


2.091 8640 


6.19 


1.822 9351 


6.83 


1.921 3247 


7.47 


2.010 8949 


8.11 


2.093 0984 


6.20 


1.824 5493 


6.84 


1.922 7877 


7.48 


2.012 2327 


8.12 


2.094 3306 


6.21 


1.826 1608 


6.85 


1.924 2486 


7.49 


2.013 5687 


8.13 


2.095 5613 


6.22 


1.827 7699 


6.86 


1.925 7074 


7.50 


2.014 9030 


8.14 


2.096 7905 


6.23 


1.829 3763 


6.87 


1.927 1641 


7.51 


2.016 2354 


8.15 


2.098 0182 


6.24 


1.830 9801 


6.88 


1.928 6186 


7.52 


2.017 5661 


8.16 


2.099 2444 


6.25 


1.832 5814 


6.89 


1.930 0710 


7.53 


2.018 8950 


8.17 


2.100 4691 


6.26 


1.834 1801 • 


6.90 


1.931 5214 


7.54 


2.020 2221 


8.18 


2.101 6923 


6.27 


1.835 7763 


6.91 


1.932 9696 


7.55 


2.021 5475 


8.19 


2.102 9140 


6.28 


1.837 3699 


6.92 


1.934 4157 


7.56 


2.022 8711 


8.20 


2.104 1341 


6.29 


1.838 9610 


6.93 


1.935 8598 


7.57 


2.024 1929 


8.21 


2.105 3529 


6.30 


1.840 5496 


6.94 


1.937 3017 


7.58 


2.025 5131 


8.22 


2.106 5702 


6.31 


1.842 1356 


6.95 


1.938 7416 


7.59 


2.026 8315 


8.23 


2.107 7861 


6.32 


1.843 7191 


6.96 


1.940 1794 


7.60 


2.028 1482 


8.24 


2.108 9998 


6.33 


1.845 3002 


6.97 


1.941 6152 


7.61 


2.029 4631 


8.25 


2.110 2128 


6.34 


1.846 8787 


6.98 


1.943 0489 


7.62 


2.030 7763 


8.26 


2.111 4243 


6.35 


1.848 4547 


6.99 


1.944 4805 


7.63 


2.032 0878 


8.27 


2.112 6343 


6.36 


1.850 0283 


7.00 


1.945 9101 


7.64 


2.033 3976 


8.28 


2.113 8428 


6.37 


1.851 5994 


7.01 


1.947 3376 


7.65 


2.034 7056 


8.29 


2.115 0499 


6.38 


1.853 1680 


7.02 


1.948 7632 


7.66 


2.036 0119 


8.30 


2.116 2555 


6.39 


1.854 7342 


7.03 


1.950 1866 


7.67 


2.037 3166 


8.31 


2.117 4596 


6.40 


1.856 2979 


7.04 


1.951 6080 


7.68 


2.038 6195 


8.32 


2.118 6622 


6.41 


1.857 8592 


7.05 


1.953 0275 


7.69 


2.039 9207 


8.33 


2.119 8634 


6.42 


1.859 4181 


7.06 


1.954 4449 


7.70 


2.041 2203 


8.34 


2.121 0632 


6.43 


1.860 9745 


7.07 


1.955 8604 


7.71 


2.042 5181 


8.35 


2.122 2615 


6.44 


1.862 5285 


7.08 


1.957 2739 


7.72 


2.043 8143 


8.36 


2.123 4584 


6.45 


1.864 0801 


7.09 


1.958 6853 


7.73 


2.045 1088 


8.37 


2.124 6539 


6.46 


1.865 6293 


7.10 


1.960 0947 


7.74 


2.046 4016 


8.38 


2.125 8479 


6.47 


1.867 1761 


7.11 


1.961 5022 


7.75 


2.047 6928 


8.39 


2.127 0405 


6.48 


1.868 7205 


7.12 


1.962 9077 


7.76 


2.048 9823 


8.40 


2.128 2317 


6.49 


1.870 2625 


7.13 


• 1.964 3112 


7.77 


2.050 2701 


8.41 


2.129 4214 


6.50 


1.871 8021 


7.14 


1.965 7127 


7.78 


2.051 5563 


8.42 


2.130 6098 


6.51 


1.873 3394 


7.15 


1.967 1123 


7.79 


2.052 8408 


8.43 


2.131 7967 


6.52 


1.874 8743 


7.16 


1.968 5099 


7.80 


2.054 1237 


8.44 


2.132 9822 


6.53 


1.876 4069 


7.17 


1.969 9056 


7.81 


2.055 4049 


8.45 


2.134 1664 


6.54 


1.877 9371 


7.18 


1.971 2993 


7.82 


2.056 6845 


8.46 


2.135 3491 


6.55 


1.879 4650 


7.19 


1.972 6911 


7.83 


2.057 9624 


8.47 


2.136 5304 


6.56 


1.880 9906 


7.20 


1.974 0810 


7.84 


2.059 2388 


8.48 


2.137 7104 


6.57 


1.882 5138 


7.21 


1.975 4689 


7.85 


2.060 5135 


8.49 


2.138 8889 


6.58 


1.884 0317 


7.22 


1.976 8549 


7.86 


2.061 7866 


8.50 


2.140 0661 


6.59 


1.885 5533 


7.23 


1.978 2390 


7.87 


2.063 0580 


8.51 


2.141 2419 


6.60 


1.887 0696 


7.24 


1.979 6212 


7.S8 


2.064 3278 


8.52 


2.142 4163 


6.61 


1.888 5837 


7.25 


1.981 0014 


7.89 


2.065 5961 


8.53 


2.143 5893 


6.62 


1.890 0954 


7.26 


1.982 3798 


7.90 


2.066 8627 


8.54 


2.144 7609 


6.63 


1.891 6048 


7.27 


1.983 7562 


7.91 


2.068 1277 


8.55 


2.145 9312 


6.64 


1.893 1119 


7.28 


1.985 1308 


7.92 


2.069 3911 


8.56 


2.147 1001 


6.65 


1.894 6168 


7.29 


1.986 5035 


7.93 


2.070 6530 


8.57 


2.148 2676 


6.66 


1.896 1194 


7.30 


1.987 8743 


7.94 


2.071 9132 


8.58 


2.149 4339 


6.67 


1.897 6198 


7.31 


1.989 2432 


7.95 


2.073 1719 


8.59 


2.150 5987 


6.68 


1.899 1179 


7.32 


1.990 6103 


7.96 


2.074 4290 


8.60 


2.151 7622 


6.69 


1.900 6138 


7.33 


1.991 9754 


7.97 


2.075 6845 


8.61 


2.152 9243 


6.70 


1.902 1075 


7.34 


1.993 3387 


7.98 


2.076 9384 


8.62 


2.154 0851 


6.71 


1.903 5989 


7.:',.") 


1.994 7002 


7.99 


2.078 1907 


8.63 


2.155 2445 


6.72 


1.905 0881 


7.36 


1.996 0599 


8.00 


2.079 4415 


8.64 


2.156 4026 * 


6.73 


1.906 5751 


7.37 


1.997 4177 


8.01 


2.080 6907 


8.65 


2.157 5593 


6.74 


1.908 OC.OO 


7.38 


1.998 773,6 


8.02 


2.081 9384 


8.66 


2.158 7147 


6.75 


1.909 5425 


7.39 


2.000 1278 


8.03 


2.0S3 1845 


8.67 


2.159 8687 


6.76 


1.911 0228 


7.40 


2.001 4800 


8.04 


2.084 4290 


8.68 


2.161 0215 







Hyperbolic 


Logarithms. 




643 


N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


8.69 


2.162 1729 


9.33 


2.233 2350 


9.97 


2.299 5806 


71 


4.262 6799 


£.70 


2.163 3230 


9.34 


2.234 3062 


9.98 


2.300 5831 


72 


4.276 6661 


8.71 


2.164 4718 


9.35 


2.235 3763 


9.99 


2.301 5846 


73 


4.290 4594 


8.72 


2.165 6192 


9.36 


2.236 4452 


10.0 


2.302 5851 


74 


4.304 0651 


8.73 


2.166 7653 


9.37 


2.237 5130 


11.0 


2.397 8953 


75 


4.317 4881 


8.74 


2.167 9101 


9.38 


2.238 5797 


12.0 


2.484 9067 


76 • 


4.330 7333 


8.75 


2.169 0536 


9.39 


2.239 6452 


13.0 


2.564 9494 


77 


4.343 8054 


8.76 


2.170 1959 


9.40 


2.240 7096 


14.0 


2.639 0573 


78 


4.356 7088 


8.77 


2.171 3367 


9.41 


2.241 7729 


15.0 


2.708 0502 


79 


4.369 4479 


8.78 


2.172 4763 


9.42 


2.242 8350 


16.0 


2.772 5887 


80 


4.382 0266 


8.79 


2.173 6116 


9.43 


2.243 8960 


17.0 


2.833 2133 


81 


4.394 4492 


8.80 


2.174 7517 


9.44 


2.244 9559 


18 


2.890 3718 


82 


4.406 7193 


8.81 


2.175 8874 


9.45 


2.246 0147 


19.0 


2.944 4390 


83 


4.418 8406 


8.82 


2,177 0218 


9.46 


2.247 0723 


20.0 


2.995 7323 


84 


4.430 8168 


8.83 


2.178 1550 


9.47 


2.248 1288 


21.0 


3.044 5224 


85 


4.442 6513 


8.84 


2.179 2868 


9.48 


2.249 1843 


22.0 


3.091 0425 


86 


4.454 3473 


8.85 


2.180 4174 


9.49 


2.250 2386 


23.0 


3.135 4942 


87 


4.465 9081 


8.86 


2.181 5467 


9.50 


2.251 2917 


24.0 


3.178 0538 


88 


4.477 3368 


8.87 


2.182 6747 


9.51 


2.252 3438 


25.0 


3.218 8758 


89 


4.488 6364 


8.88 


2.183 8015 


9.52 


2.253 3948 


26.0 


3.258 0965 


90 


4.499 8097 


8.89 


2.184 9270 


9.53 


2.254 4446 


27.0 


3.295 8369 


91 


4.510 8595 


8.90 


2.186 0512 


9.54 


2.255 4934 


28.0 


3.332 2045 


92 


4.521 7886 


8.91 


2.187 1742 


9.55 


2.256 5411 


29.0 


3.367 2958 


93 


4.532 5995 


8.92 


2.188 2959 


9.56 


2.257 5877 


30.0 


3.401 1974 


94 


4.543 2948 


8.93 


2.189 4163 


9.57 


2.258 6332 


31.0 


3.433 9872 


95 


4.553 8769 


8.94 


2.190 5355 


9.58 


2.259 6776 


32.0 


3.465 7359 


96 


4.564 3482 


8.95 


2.191 6535 


9.59 


2.260 7209 


33.0 


3.496 5076 


97 


4.574 7110 


8.96 


2.192 7702 


9.60 


2.261 7631 


34.0 


3.526 3605 


98 


4.584 9675 


8.97 


2.193 8856 


9.61 


2.262 8042 


35.0 


3.555 3481 


99 


4.595 1199 


8.98 


2.194 9998 


9.62 


2.263 8442 


36.0 


3.583 5189 


100 


4.605 1702 


8.99 


2.196 1128 


9.63 


2.264 8832 


37.0 


3.610 9179 


101 


4.615 1205 


9.00 


2.197 2245 


9.64 


2.265 9211 


38.0 


3.637 5862 


102 


4.624 9728 


9.01 


2.198 3350 


9.65 


2.266 9579 


39.0 


3.663 5617 


103 


4.634 7290 


9.02 


2.] 99 4443 


9.66 


2.267 9936 


40.0 


3.688 8795 


104 


4.644 3909 


9.03 


2.200 5523 


9.67 


2.269 0282 


41.0 


3.713 5721 


105 


4.653 9604 


9.04 


2.201 6591 


9.68 


2.270 0618 


42.0 


3.737 6696 


106 


4.663 4391 


9.05 


2.202 7647 


9.69 


2.271 0944 


43.0 


3.761 2001 


107 


4.672 8288 


9.06 


2.203 8691 


9.70 


2.272 1258 


44.0 


3.784 1896 


108 


4.682 1312 


9.07 


2.204 9722 


9.71 


2.273 1562 


45.0 


3.806 6525 


109 


4.691 3479 


9.08 


2.206 0741 


9.72 


2.274 1856 


46.0 


3.828 6414 


110 


4.700 4804 


9.09 


2.207 1748 


9.73 


2.275 2138 


47.0 


3.850 1476 


111 


4.709 5302 


9.10 


2.208 2744 


9.74 


2.276 2411 


48.0 


3.871 2010 


112 


4.718 4989 


9.11 


2.209 3727 


9.75 


2.277 2673 


49.0 


3.891 8203 


113 


4.727 3878 


9.12 


2,210 4697 


9.76 


2.278 2924 


50.0 


3.912 0230 


114 


4.736 1985 


9.13 


2.211 5656 


9.77 


2.279 3165 


51.0 


3.931 8256 


115 


4.744 9321 


9.14 


2.212 6603 


9.78 


2.280 3395 


52.0 


3.951 2437 


116 


4.753 5902 


9.15 


2.213 7538 


9.79 


2.281 3614 


53.0 


3.970 2919 


117 


4.762 1739 


9.16 


2.214 8461 


9.80 


2.282 3823 


54.0 


3.988 9841 


118 


4.770 6846 


9.17 


2.215 9372 


9.81 


2.283 4022 


55.0 


4.007 3332 


119 


4.779 1235 


9.18 


2.217 0272 


9.82 


2.284 4211 


56.0 


4.025 3517 


120 


4.787 4917 


9.19 


2.218 1160 


9.83 


2.285 4389 


57.0 


4.043 0513 


121 


4.795 7906 


9.20 


2.219 2034 


9.84 


2.286 4556 


58.0 


4.060 4430 


122 


4.804 0210 


9.21 


2.220 2898 


9.85 


2.287 4714 


59.0 


4.077 5374 


123 


4.812 1844 


9.22 


2.221 3750 


9.86 


2.288 4861 


60.0 


4.094 3446 


124 


4.820 2816 


9.23 


2.222 4590 


9.87 


2.289 4998 


61.0 


4.110 8739 


125 


4.828 3137 


9.24 


2.223 5418 


9.88 


2.290 5124 


62,0 


4.127 1344 


126 


4.836 2819 


9.25 


2.224 6235 


9.89 


2.291 5241 


63.0 


4.143 1347 


127 


4.844 1871 


9.26 


2.225 7040 


9.90 


2.292 5347 


64.0 


4.158 8839 


128 


4.852 0303 


9.27 


2.226 7833 


9.91 


2.293 5443 


65.0 


4.174 3873 


129 


4.859 8124 


9.2S 


2.227 8615 


9.92 


2.294 5529 


66.0 


4.189 6547 


130 


4.867 5345 


19.29 


2.228 9385 


9.93 


2.295 5604 


67.0 


4.204 6926 


131 


4.875 1973 


9.30 


2.230 0144 


9.94 


2.296 5670 


68.0 


4.219 5077 


132 


4.882 8019 


9.31 


2.231 0890 


9.95 


2.297 5725 


69.0 


4.234 1065 


133 


4.890 3491 


9.32 


2.232 1626 


9.96 


2.298 5770 


70.0 


4.248 4952 


134 


4.897 8398 



644 




Expansion of 


Steam. 








Mean Pressure Above Vacuum of Expanding 








Steam, 








! 


















» ., 










Expansion ratio. 








Absolute 


1.333 


1.5 


1.6 


2 


2.666 


3 


4 


8 


steam 


















pressure, 


















P. 






Steam cut-off, fraction of stroke. 








% 


% 


% 


K 


% 


Vs 


K 


Vs 


25 


24.130 


23.481 


22.938 


21.164 


18.567 


17.488 


19.913 


9.6232 


30 


28.956 


28.100 


27.524 


25.396 


22.280 


20.986 


17.897 


11.548 


35 


33.782 


32.874 


32.110 


29.630 


25.992 


24.484 


20.880 


13.472 


40 


38.608 


37.468 


36.700 


33.862 


28.964 


27.982 


23.862 


15.396 


45 


43.474 


42.151 


41.287 


38.095 


32.677 


31.479 


26.845 


17.320 


50 


48.262 


46.835 


45.875 


42.328 


37.133 


34.977 


29.828 


19.246 


55 


53.088 


51.518 


50.462 


46.561 


40.846 


38.474 


32.811 


21.170 


60 


57.914 


56.202 


55.050 


50.794 


44.559 


41.972 


35.794 


23.095 


65 


62.740 


60.885 


59.637 


55.027 


48.273 


45.470 


38.777 


25.020 


70 


67.566 


65.569 


64.225 


59.260 


51.986 


48.967 


41.760 


26.944 ; 


75 


72.393 


70.252 


68.812 


63.493 


55.700 


52.465 


44.743 


28.869 


80 


77.216 


74.936 


73.400 


67.726 


59.413 


55.963 


47.726 


30.794 


85 


82.042 


79.619 


77.987 


71.959 


63.126 


59.461 


50.709 


32.718 


90 


86.866 


85.303 


82.574 


76.192 


66.840 


62.958 


53.692 


34.643 ! 


95 


91.699 


89.986 


87.163 


80.425 


70.553 


66.456 


56.675 


36.568 


100 


96.524 


93.670 


91.750 


84.657 


74.267 


69.954 


59.657 


38.493 


105 


101.35 


98.353 


96.337 


88.890 


77.981 


73.451 


62.640 


40.417 


110 


106.17 


103.04 


100.92 


93.123 


81.694 


76.949 


65.622 


42.342 


115 


111.00 


107.72 


105.51 


97.356 


85.407 


80.447 


68.606 


44.267 ! 


120 


115.83 


112.40 


110.10 


101.59 


89.121 


83.944 


71.589 


46.191 


125 


120.65 


117.08 


114.68 


105.82 


92.834 


87.442 


74.572 


48.116 


130 


125.48 


121.77 


119.27 


110.05 


96.548 


90.940 


77.555 


50.041 


135 


130.30 


L26.46 


123.86 


114.28 


100.26 


94.437 


80.538 


51.966 


140 


L35.13 


L31.13 


128.45 


118.52 


103.97 


97.935 


83.520 


53.890 


145 


139.96 


135.82 


133.03 


122.75 


107.68 


101.43 


86.502 


55.815 


150 


144.78 


140.50 


137.62 


126.98 


111.40 


104.93 


89.485 


57.739 


165 


1 19.60 


145.18 


1 42.20 


131.22 


115.11 


108.42 


92.468 


59.663 


160 


L54.43 


1 19.87 


11(1.79 


135.45 


118.82 


111.92 


95.451 


61.588 - 


I.;:, 


L59.26 


L54.55 


151.38 


139.68 


122.54 


115.42 


98.434 


63.513 


170 


L64.08 


L59.23 


155.97 


143.92 


126.25 


118.92 


101.41 


65.437 


17", 


L68.91 


L63.92 


160.55 


148.15 


129.96 


122.42 


104.40 


67.362 


L80 


L73.73 


L68.60 


165.14 


152.38 


133.68 


125.91 


107.38 


69.287 


185 


L78.56 


L73.28 


Kilt. 73 


156.61 


137.39 


129.41 


110.36 


71.212 


190 


L83.39 


177.97 


L74.32 


L60.85 


141.10 


132 91 


113.35 


73.136 


L96 


188.2] 




178.90 


165.08 


144.82 


136.41 


116.33 


75.061 


•j, i, | 


L93.04 


187.84 


183.50 


169.31 


148.53 


139.91 


119.31 


76.986 


■J |l) 




L96.7] 


L92.68 


177.78 


155.96 


146.90 


125.27 


80.835 


220 


212.34 




201.85 


186.25 


163.39 


153.90 


131.24 


84.684 i 




221.99 


215.45 


211.03 


L94.71 


170.82 


160.89 


137.20 


88.534 : 


2 111 


281.65 


224.83 


220.20 


203.18 


178.23 


167.89 


143.17 


92.383 




241.80 






211.64 


185.67 


174.88 


149.13 


96.232** 


260 


250.96 






220.U 


193.18 


181.88 


155.11 


100.08 


270 


260.61 


252.91 


2 17.73 


228.67 


200.52 


L88.87 


161.07 


103.93 




270.26 


262.28 




237.04 


207.95 


195.87 


167.04 


107.78 


300 






275.24 




222.80 


209.86 


178.97 


115.48 



Expansion of Steam. 



645 



Mean Pressure for High=pressure Engines Above 
Atmosphere. 





Expansion ratio. 


Pressure 
above 


1.333 


1.5 


1.6 


2 


2.666 


3 


4 


8 


atmos- 


















phere, P. 






Steam 


3ut-off, fraction of stroke. 








% 


% 


% 


% 


% 


Vs 


M 


Ys 


25 


23.908 


22.768 


22.000 


19.162 


14.264 


13.282 


9.162 


.696 


30 


28.774 


27.451 


26.587 


23.395 


17.977 


16.779 


12.145 


2.620 


35 


33.562 < 


32.135 


31.175 


27.628 


22.433 


20.277 


15.128 


4.546 


40 


38.388 


36.818 


35.762 


31.861 


26.146 


23.774 


18.111 


6.470 


45 


43.214 


41.502 


40.350 


36.094 


29.859 


27.272 


21.094 


8.395 


50 


48.040 


46.185 


44.937 


40.327 


33.573 


30.770 


24.077 


10.320 


55 


52.866 


50.869 


49.625 


44.560 


37.286 


34.267 


27.060 


12.244 


60 


57.693 


55.552 


54.112 


48.793 


41.000 


37.765 


30.043 


14.169 


65 


62.516 


60.236 


58.700 


53.026 


44.713 


41.263 


33.026 


16.094 


70 


67.342 


64.919 


63.287 


57.259 


48.426 


44.761 


36.009 


18.018 


75 


72.166 


70.603 


67.874 


61.492 


52.140 


48.258 


38.992 


19.943 


80 


76.999 


75.286 


72.463 


65.725 


55.853 


51.756 


41.975 


21.868 


85 


81.824 


78.970 


77.050 


69.957 


59.567 


55.254 


44.957 


23.793 


90 


86.65 


83.653 


81.637 


74.190 


63.281 


58.751 


47.940 


25.717 


95 


91.47 


88.34 


86.22 


78.423 


66.994 


62.249 


50.922 


27.642 


100 


96.30 


93.02 


90.81 


82.656 


70.707 


65.747 


53.906 


29.567 


105 


101.13 


97.70 


95.40 


86.89 


74.421 


69.244 


56.889 


31.491 


110 


105.95 


102.38 


99.98 


91.12 


78.134 


72.742 


59.872 


33.416 


115 


110.78 


107.07 


104.57 


95.35 


81.848 


76.240 


62.855 


35.341 


120 


115.60 


111.75 


109.16 


99.58 


85.56 


79.737 


65.838 


37.266 


125 


120.43 


116.43 


113.75 


103.82 


89.27 


83.235 


68.820 


39.190 


130 


125.26 


121.12 


118.33 


108.05 


92.98 


86.73 


71.802 


41.115 


135 


130.08 


125.80 


122.92 


112.28 


96.70 


90.23 


74.785 


43.039 


140 


134.90 


130.48 


127.50 


116.52 


100.41 


93.72 


77.768 


44.963 


145 


139.73 


135.17 


132.09 


120.75 


104.12 


97.22 


80.751 


46.888 


150 


144.56 


139.85 


136.68 


124.98 


107.84 


100.72 


83.734 


48.813 


155 


149.38 


144.83 


141.27 


129.22 


111.85 


104.22 


86.71 


50.737 


160 


154.21 


149.22 


145.85 


133.45 


115.26 


107.72 


89.70 


52.662 


165 


159.03 


153.90 


150.44 


137.68 


118.98 


111.21 


92.68 


54.587 


170 


163.86 


158.58 


155.03 


141.91 


122.69 


114.71 


95.66 


56.812 


175 


168.69 


163.27 


159.62 


146.15 


126.40 


118.21 


98.65 


58.436 


180 


173.51 


167.95 


164.20 


150.38 


130.12 


121.71 


101.63 


60.361 


185 


178.34 


172.64 


168.80 


154.81 


133.83 


125.21 


104.61 


62.286 


190 


183.16 


177.32 


173.39 


158.81 


137.54 


128.71 


107.59 


64.210 


195 


187.99 


182.01 


177.98 


163.08 


141.26 


132.20 


110.57 


66.135 


200 


192.81 


186.69 


182.58 


167.31 


144.97 


135.70 


113.55 


68.060 


210 


202 46 


195.06 


191.74 


175.78 


152.40 


142.70 


119.52 


71.908 


220 


212.11 


205.43 


200.93 


184.24 


159.83 


149.69 


125.48 


75.758 


230 


221.77 


214.79 


210.10 


192.71 


167.24 


156.69 


131.39 


79.603 


240 


231.42 


224.16 


219.27 


201.17 


174.68 


163.68 


137.41 


83.456 


250 


241.08 


233.57 


228.45 


209.64 


182.19 


170.68 


143.39 


87.30 


260 


250.73 


242.89 


237.62 


218.10 


189.53 


177.67 


149.35 


91.15 


270 


260.38 


252.26 


246.79 


226.57 


196.96 


184.67 


155.32 


95.00 


280 


270.04 


261.62 


255.94 


235.03 


204.39 


191.66 


161.29 


98.86 


300 


289.34 


280.35 


264.30 


251.95 


219.24 


205.56 


173.22 


106.55 



646 Expansion of Steam. 

In the preceding computations and tables it has been assumed that 

there was no clearance or waste space between the piston and the cylinder 

head at the end of the stroke. In practice, the clearance amounts to from 

2 to 7 per cent, of the cylinder volume. This may be taken into account 

by adding the clearance to both the length of the stroke and the length qj 

the admission portion in determining the expansion ratio, r. Thus, if the 

stroke is 24 inches and the steam is cut off at 6 inches, the expansion ratio 

will be 2 f = 4, if clearance is neglected. If, however, there is a space of 

Y 2 inch between the piston and the cylinder head at the end of the stroke, 

we have 

24.5 nm 
r = = 3f77 . 

6.0 
and this is the ratio to be used in computation. 

Most Economical Point of Cut=off. 

(W. D. Marks.) 

To find the most economical point of cut-off,— that is, its inverse, that number 
of expansions which will result in the greatest economy of steam from the boiler, 
per horse-power, per hour. 

Notation. 

e = the true point of cut-off = the reciprocal of the true number of 

expansions ; 
B = the absolute back pressure during exhaust, in pounds, per square 

inch ; 
P b = the absolute pressure at cut-off ; 
s = the stroke of piston, in feet ; 
d = the diameter of cylinder, in feet ; 

62.5 
A = ~^> 
S = the specific volume of steam at cut-off ; 

N 
T b = the temperature of the steam at cut-off (Fahr.) ; 
T e = the temperature of the steam during exhaust ; 
N = the number of strokes per minute = twice the revolutions of 

crank ; 
C = the constant of condensation = 0.018 pounds of steam for about 
82 pounds gauge pressure. 



B , / 1 . 0.194N Dd 

e = ~K + It + -r)-Zd + D nat log - 

Example. Let 



e 



\W have 

A 
D= 0.0274, 



P b = 100 pounds absolute ; 
B = 15 pounds absolute ; 

s = 4tV<t ; 

d L.Sfeet; 
N = 150 per minute. 



._()!-, | ] . °- 194 \ 0- 0274 X 1.5 X 2.3026 . 1 

e ~°- ( l f To" j -0T 233 X 1.5 + 0.0274 COm - l0g - 7' 



c = 0.1-"> 0.0952 log. --. 

c 



Expansion of Steam. 



647 



We must solve this transcendental equation tentatively, trying values 
until the two members balance. 

Assume e = | of stroke plus clearance. We have 

0.20 = 0.15 + 0.066 = 0.216. 

This error of 0.016 is closer work than can be realized in practice, and 
we can take 5 expansions as the best number. 

Between £ and % would have been near enough for all practical pur- 
poses. 

To find the proper ratio of stroke to diameter under the given conditions, 
assuming 5 expansions and diameter = l%feet. 
Inverting the above equation, we have 

d 



(*«)l 



nat. log. - 



-0.194 



• = 8.56, 



1.5 



8.56X1.5 + 1) (^y^)- 0.194 



= 6.4 feet, nearly. 



With slow-moving engines it will be found that long stroke is most 
economical, while on the other hand high-speed engines require short 
stroke for greatest economy. If we double the speed of this engine, mak- 
ing N = 300, the stroke s = 2.4 feet, for greatest economy. 

In order to construct the curve representing the expansion of steam in 
a cylinder, under the assumption that the expansion is isothermal,— i.e., 
that pv = constant,— the following method may be used : 



T D 




XA 



Isothermal Curve. 



Draw the line, AC, to represent the position of zero pressure, or perfect 
vacuum, making the length, AC, represent the stroke of the piston. Make 
AX equal to the clearance, expressed in terms of the stroke, — that is, 



AX=AC 



clearance volume 
volume swept through by piston' 



Erect the perpendicular, DF, to represent the admission pressure on any 
convenient scale, and draw the horizontal line, YDF. Mark the point, E, 
so that DE represents the length of the stroke during which steam is ad- 
mitted, — i.e., if the expansion ratio is 6, DE will be one-sixth of AC,— and 
draw BE. Take any points, 1, 2, 3, 4, and join them to X, and also drop 



G48 



Expansion of Steam. 



perpendiculars from 1, 2, 3, 4. Draw horizontal lines from the intersections, 
Xi Xo A" 3f X A , on EB, and where these horizontal lines intersect the cor- 
responding verticals will be points on the curve, as at r 1} Y 2 , Y 3 , F 4 . 

The hvperbolic curve represents isothermal expansion, it being assumed 
that the temperature is kept constant. Other curves have been considered^ 
in connection with the expansion of steam, and according to the investi- 
gations of Rankine, Zeuner, and others, these may be expressed by the 
general equation : 

p V m — constant, 



the 




Polytropic Curve. 



! exponent, ra, being varied according to the curve under consideration. 

For dry saturated steam, according to Rankine, m = if = 1.0625 ; while, 
according to Zeuner, m = 1.0646. For adiabatic expansion, in which the 
expanding steam neither receives nor gives out heat, Rankine gives m = 

JLQ — l.lH. 

Any of these curves may be constructed by the computation of any 
desired number of ordinates, using logarithms, or more conveniently by 

the so-called "polytropic" 
Y diagram. 

p'.i'o Draw the rectangular 

°\e axes, YOX, and make i> 

equal the portion of the 
stroke during which the in- 
itial pressure, p , is main- 
tained. Draw OA, making 
the angle, a, any convenient 
value, and also draw OB, 
making the angle, /3, so that 

1 4- tan |3 = (1-|- tan a)™, 

choosing ra according to the 
curve to be drawn, as above. 
Then, starting from C, draw 
Cc at 45° from OC and back 
at right angles to CO, zig- 
zagging back and forth, as 
shown; also drop the verti- 
cal, ED, prolonged to e, and construct a similar zigzag between OA and 
OX, making alternate angles of 45° and 90° with OX. The intersections 
of the normals to OF and OX, when prolonged, will then give points in 
the curve, as at 1, 2, 3, etc. 

In practice, it has been found that the isothermal curve represents 
practical working conditions as closely as any which can be drawn. 

The principal source of loss in steam engines is the initial condensation 
of the steam, which takes place when it first enters the cylinder. The 
difference in temperature between steam at 100 pounds and steam atatmos- 
pheric pressure is about 125° F., and as the cylinder walls absorb and part 
with heat readily the incoming steam meets the walls which have just 
been fooled t<> the temperature of the previous exhaust. For this reason it 
has been found wasteful to attempt to realize the high economy due to 
large expansion ratios in a single cylinder. For high-pressure engines the 
best results are obtained with 4 or 6 expansions, — i.e., cutting off the steam 
at J , t<> | of the stroke,— while for condensing engines from 8 to 10 expan- 
sions is about the highest that can be used to advantage. 

In onhr t<> use higher expansion ratios successfully, the expansion 
is performed in two or more cylinders, giving compound or multiple- 
expansion engines. A number of rules and methods have been given 
for determining the best relative areas of cylinders for compound and 
multiple-expansion engines, depending upon'the desire of the designer. 
In many cases, it is wished to make the work performed in the various 
cylinders approximately equal; in others, it is desired to equalize the ini- 
tial stresses; and in others, to equalize the drop in temperature. 

For compound engines various empirical rules have been given, gener- 
ally based upon the Initial pressure or upon the expansion ratio. 

Thus, if r he tin exp insion ratio, the cylinder ratio is often made equal 
to \ r. In marine practice, the cylinder ratio usually ranges from 1 to 4 
for loo pounds pressure i<> i to ;> for L20 pounds pressure. 



Multiple-expansion Engines. 



649 



L 



For triple-expansion engines the ratios found in practice, according to 
Whitham, are about as follows : 

Cylinder Ratios for TripIe=expansion Engines. 



Initial pressure. 


High pressure. 


Intermediate. 


Low pressure. 


130 
140 
150 
160 


1 
1 
1 
1 


2.25 
2.40 
2.55 
2.70 


5.00 
5.85 
6.90 
7.25 



For quadruple-expansion engines, operating at pressures of 160 pounds 
and over, the following proportions are found : 

Cylinder Ratios for Quadruple=expansion Engines. 



Initial 
pressure. 


High 
pressure. 


First 
intermediate. 


Second 
intermediate. 


Low- 
pressure. 


160 


1 


2.00 


4.00 


8 


180 


1 


2.10 


4.20 


9 


200 


1 


2.15 


4.60 


10 


220 


1 


2.20 


4.80 


11 



The subject is best studied by drawing a single diagram for the initial 
pressure and expansion ratio given, this being then divided up according 
to the distribution of power desired among two, three, or four cylinders, as 
the proposed design may be for a compound, triple, or quadruple engine. 

For this purpose the isothermal curve will be sufficiently accurate. 

Thus, the diagram may be drawn as for a single engine, as shown here- 
with, and divided into 
three portions of equal 
area, Z>, D', and D" ; this 
being best done tenta- 
tively, the areas being 
measured by the plani- 
meter. If the total area 
is first measured and 
divided by three, the por- 
tion D can be laid off very 
closely after one or two 
trials, and the same for 
B' and D". The areas of 
the several parts will then 
be proportional to the 
volumes of the various 
cylinders, and, since they 
are all made of the same 
stroke in practice, the 
cylinder ratios will be 
proportional to the 
lengths, s, s', and s". Any 
other subdivision of the total expansion may be considered in the same 
manner. ■ 

The thermal efficiency of any heat motor is limited by the range of 
temperature through which the impelling fluid acts. This efficiency is the 




Triple-expansion Diagram. 



650 Indicator Diagrams. 



ratio obtained by dividing the heat converted into work by the total heat 
taken in. This ratio must always be less than unity, and its maximum 
value for any range of temperature is found from the ratio 



Ti— T 2 



in which 7\ is the absolute temperature of reception, = temperature F. -f 
461, = temperature C. + 273 ; while T 2 is the absolute temperature of rejec- 
tion. Considering all temperatures as absolute,— that is, as measured from 
the absolute zero, we have 

„ ~ . temperature of reception — temperature of rejection 

Maximum efficiency == - t £ — r i — r- i • 

temperature of reception 

Thus, in the case of an engine in which the steam enters at a tempera- 
ture of 341° F., or 802° absolute, corresponding to an absolute pressure of 
120 pounds per square inch, and is rejected in the condenser at a tempera- 
ture of 60° F., or 521° absolute, we have 

T x — n _ 802° — 521° _ 
~~%~ ~ 802^ ~ °- d5 ' 

so that, if all the heat in the steam were converted into mechanical 
energy, the efficiency could not exceed 35 per cent. 

In actual practice the thermal efficiency rarely attains 12 per cent., the 
highest recorded efficiency being that of the Reynolds pumping engine 
at Boston, Massachusetts. This engine has the record of a performance 
of 187.8 B. T. U. per indicated horse-power, corresponding to a thermal 
efficiency of 22% per cent. 



Indicator Diagrams. 

The steam-engine indicator is a form of recording pressure gauge, 
arranged to be attached to the cylinder of a steam engine so as to draw a 
curve representing the pressure within the cylinder at every point in the 
stroke. Originally invented by Watt, and greatly improved by McNaught, 
Richards, Thompson, and others, it is now a standard instrument of the 
engineer. The details of construction of the various styles of instruments 
on the market are fully given in the hand-books issued by the manufac- 
turer, and hence the diagrams themselves will only be discussed here. 



Cut Off* 




gComp tessloa 
Atmospheric line 

Typical Indicator Diagram. 

In the typical diagram, given herewith, the general form obtained from 
a angle-cylinder engine In good condition is shown. If the area of the 
diagram (best measured by a planimeter) is divided bv the length and this 
multiplied by the Bcale oi the spring, the mean effective pressure in the 
cylinder is obtained, rhia mean effective pressure multiplied by the area 
pi the piston, Insquare Inches, gives the total force acting upon the piston, 
In pounds, and by multiplying this force by the number of feet of piston 



Indicator Diagrams. 651 

travel per minute, the power, in foot-pounds, per minute is obtained. From 
this the horse-power is found by dividing by 33,000. 
Thus, if 

p = mean effective pressure, in pounds, per square inch ; 
a = area of piston, in square inches ; 
5 = piston speed, in feet, per minute. 

= aXpXs 
* 33000 * 

If a number of computations are to be made upon a given engine, the 
area of the piston may conveniently be divided by 33,000 to obtain a con- 
stant factor, corresponding to the horse-power developed by 1 pound mean 
effective pressure at 1 foot piston speed. This constant need then only be 
multiplied by the actual speed and pressure to give the power in each case. 

It must be remembered that the indicator is only a recording pressure 
gauge, and that it merely shows the pressure at every point in the stroke. 
The interpretation given to the record is a matter in which the judgment 
of the observer must in great measure supply. 

In general, the indicator diagram shows the action of the valve gear, 
including the points of cut-off, release, and compression ; also, the freedom 
of the exhaust and the equality of action in both the forward and back- 
ward strokes. To this extent the indicator is of great assistance in adjust- 
ing the valves and in maintaining a correct adjustment. 

The indicator diagram may also be used to determine the steam con- 
sumption of the engine,— at least the theoretical consumption may thus be 
determined, and by comparison with actual measurements the proportion 
of steam accounted for by the indicator may be computed. 

The steam consumption is usually stated in terms of the equivalent 
weight of water. Several methods may be used in computing the rate of 
water consumption. The following, due to Mr. Jesse Warrington, is con- 
venient in that it does not require any data concerning the dimensions or 
speed of the engine, being determined solely from the indicator diagram. 

Divide the constant number 859,375 by the volume of steam at the termi- 
nal pressure and by the mean effective pressure. The quotient will be the 
desired rate. 

This constant is the number of pounds of water that would be used in 1 
hour by an engine developing 1 horse-power, if run by water (instead of 
steam), at 1 pound pressure per square inch. Then, with pressure of more 
than 1 pound, the amount required would be as many times less as the 
pressure was greater than 1 pound, and when steam is used the amount 
would be as much less as the volume of the steam at the pressure at which 
it is released is greater than that of an equal weight of water ; hence, the 
above rule. The constant is found as follows : The standard horse-power 
being 33,000 foot-pounds, or 33,000 pounds lifted 1 foot per minute, would 
be equivalent to 33,000 X 12 = 396,000 pounds lifted 1 inch per minute; 
hence, an engine whose piston displacement was 396,000 cubic inches per 
minute would develop 1 horse-power with 1 pound mean effective pressure 
on the piston. This for 1 hour would be 396,000 X 60 minutes = 23,760,000 
cubic inches per hour. Then suppose the engine to be run by water at 1 
pound pressure per square inch, instead of steam, and taking 27.648 as the 
number of cubic inches of water per pound, 23,760,000 h- 27.648 = 859,375, 
which is the desired constant. 

The water consumption thus determined is not corrected for clearance 
or for compression, but this may be done from the diagram, as follows : 
Prolong the expansion curve beyond the point of release until it reaches 
the end of the diagram, this giving the terminal point of the curve as it 
would have been had the exhaust valve not been opened. Draw a hori- 
zontal line from this terminal point through the compression curve to the 
other end of the diagram. The ratio of the length from terminal to com- 
pression curve, divided by the total length of the diagram, will give a 
factor which, when multiplied by the previously-computed water con- 
sumption, will give the result corrected for clearance and compression. 
These methods are naturally dependent upon the tightness of the valves 
for their accuracy. 

In order to simplify the work of computation, the following table has 
been made. 



652 



Water Consumption. 



Water Consumption Table. 



p 


W 


P 


W 


P 


W 


P 


W 


P 
71 


W 


P 

88 


W 


P 


W 


3 


39.10 


20 


34.99 


37 


33.72 


54 


32.98 


32.46 


32.07 


105 


31.73 


4 


38.47 


21 


34.89 


38 


33.67 


55 


32.94 


72 


32.43 


89 


32.05 


106 


31.71 


5 


37.95 


22 


34.79 


39 


33.62 


56 


32.91 


73 


32.40 


90 


32.03 


107 


31.69 


6 


37.54 


23 


34.70 


40 


33.57 


57 


32.88 


74 


32.38 


91 


32.00 


108 


31.67 


7 


37.22 


24 


34.61 


41 


33.52 


58 


32.85 


75 


32.36 


92 


31.98 


109 


31.65 


8 


36.93 


25 


34.53 


42 


33.47 


59 


32.82 


76 


32.34 


93 


31.96 


110 


31.63 


9 


36.67 


26 


34.45 


43 


33.42 


60 


32.79 


77 


32.32 


94 


31.94 


111 


31.61 


10 


36.44 


27 


34.37 


44 


33.38 


61 


32.76 


78 


32.30 


95 


31.92 


112 


31.59 


11 


36.24 


28 


34.29 


45 


33.34 


62 


32.73 


79 


32.27 


96 


31.90 


113 


31.57 


12 


36.06 


29 


34.22 


46 


33.30 


63 


32.70 


80 


32.25 


97 


31.88 


114 


31.55 


13 


35.89 


30 


34.15 


47 


33.26 


64 


32.67 


81 


32.23 


98 


31.86 


115 


31.54 


14 


35.73 


31 


34.08 


48 


33.22 


65 


32.64 


82 


32.20 


99 


31.84 


116 


31.53 


15 


35.59 


32 


34.01 


49 


33.18 


66 


32.61 


83 


32.18 


100 


31.82 


117 


31.52 


16 


35.46 


33 


33.95 


50 


33.14 


67 


32.58 


84 


32.16 


101 


31.80 


118 


31.51 


17 


35.34 


34 


33.89 


51 


33.10 


68 


32.55 


85 


32.14 


102 


31.78 


119 


31.50 


18 


35.22 


35 


33.83 


52 


33.06 


69 


32.52 


86 


32.12 


103 


31.77 


120 


31.49 


19 


35.10 


36 


33.77 


53 


33.02 


70 


32 49 


87 


32.09 


104 


31.75 


121 


31.48 



Under P is found the absolute terminal pressure. Under W, opposite 
the terminal pressure, is found a factor which, when multiplied by the 
absolute terminal pressure and divided by the mean effective pressure, will 
give the theoretical water consumption. From this it will be seen that the 
best economy is attained by a low terminal pressure combined with a high 
mean effective pressure, conditions which are incompatible either with 
underloading or overloading. 

The relation between the actual and the computed water consumption 
of simple engines, both condensing and non-condensing, for various points 
of cut-off is given in the following table from the practice of the Buckeye 
Engine Company. 



Steam-engine Performances. 



653 



Table of Standard Engine Performance. 





T L cut-off. 


© 


Mean effective 




Kates, in pounds of water per indicated 


© 


pressure, in 
pounds. 


a 

1 

© 


horse-power per 


hour. 




3 






Actual. 


Theoretical. 




'3 


Non- 


Con- 








+i 


M 










g 




con- 
densing. 


densing. 




Non- 
con- 
densing. 


Con- 
densing. 


Non- 
con- 
densing. 


Con- 
densing. 




40 


3.65 


13.65 


6.41 


72.0 


38,0 


51.4 


16.4 


146 


45 


5.42 


15.42 


7.00 


58.5 


35.0 


38.5 


16.0 


120 


50 


7.19 


17.19 


7.59 


49.0 


33.0 


31.9 


15.6 


93 


55 


8.96 


18.96 


8.17 


43.5 


31.5 


28.1 


15.2 


80 


60 


10.73 


20.73 


8.76 


39.0 


30.0 


25.3 


14.9 


70 


65 


12.50 


22.50 


9.35 


35.7 


28.6 


23.3 


14.6 


62 


70 


14.27 


24.27 


9.93 


33.0 


27.7 


21.8 


14.4 


55 


75 


16.04 


26.04 


10.52 


31.0 


26.7 


20.6 


14.2 


50 


80 


17.81 


27.81 


11.11 


29.0 


26.0 


19.7 


14.0 


46 


85 


19.58 


29.58 


11.70 


27.5 


25.3 


19.0 


13.8 


43 


90 


21.36 


31.36 


12.28 


26.0 


24.5 


18.4 


13.6 


40 


95 


23.13 


33.13 


12.87 


25.0 


23.7 


17.9 


13.5 


37 


100 


24.9 


34.9 


13.46 


24.0 


23.0 


17.5 


13.4 


35 




T V 5 o cut-off. 


40 


9.05 


19.05 


9.07 


54.0 


30.0 


31.3 


16.8 


64 


45 


11.32 


21.32 


9.87 


47.0 


28.5 


27.7 


16.4 


56 


50 


13.59 


23.59 


10.72 


42.0 


27.0 


25.3 


16.1 


51 


55 


15.86 


25.86 


11.55 


38.0 


26.0 


23.4 


15.8 


'47 


60 


18.12 


28.12 


12.38 


34.5 


25.0 


22.1 


15.6 


43 


65 


20.39 


30.39 


13.20 


32.0 


24.0 


21.1 


15.4 


40 


70 


22.66 


32.66 


14.03 


30.0 


23.0 


20.3 


15.2 


38 


75 


24.92 


34.92 


14.86 


28.0 


22.2 


19.5 


15.0 


36 


80 


27.19 


37.19 


15.69 


26.0 


21.3 


18.8 


14.8 


35 


85 


29.46 


39.46 


16.51 


24.5 


20.4 


18.4 


14.6 


34 


90 


31.72 


41.72 


17.34 


23.0 


19.5 


18.0 


14.5 


33 


95 


33.93 


43.93 


18.17 


22.0 


18.7 


17.6 


14.4 


32 


100 


36.26 


46.26 


19.0 


21.0 


18.0 


17.3 


14.3 


32 




| cut=off. 


40 


13.46 


23.46 


11.79 


45.0 


24.0 


27.9 


17.7 


51 


45 


16.15 


26.15 


12.87 


41.5 


23.3 


25.7 


17.3 


45 


50 


18.85 


28.85 


13.94 


37.0 


22.5 


24.0 


16.9 


40 


55 


21.54 


31.54 


15.00 


33.6 


21.7 


22.7 


16.6 


38 


60 


24.24 


34.24 


16.08 


31.0 


21.0 


21.7 


16.4 


36 


65 


26.93 


36.93 


17.15 


29.0 


20.3 


20.9 


16.2 


35 


70 


29.63 


39.63 


18.23 


27.5 


19.6 


20.2 


16.0 


34 


75 


32.32 


42.32 


19.31 


26.0 


19.0 


19.6 


15.8 


33 


80 


35.02 


45.02 


20.39 


24.5 


18.4 


19.1 


15.7 


33 


85 


37.71 


47.71 


21.46 


23.3 


18.0 


18.7 


15.6 


32 


90 


40.41 


50.41 


22.54 


22.0 


17.4 


18.4 


15.5 


32 


95 


43.1 


53.1 


23.62 


21.0 


16.9 


18.1 


15.4 


31 


100 


45.8 


55.8 


24.7 


20.0 


16.4 


17.8 


15.3 


31 



654 



Steam-engine Performances. 



Table of Standard Engine Performance.— continued. 









25 
TOO 


or \ cut=off. 










Mean effective 




Rates, in pounds of water per indicated 


5q 


pressure, in 




horse-power per 


hour. 




I 


pounds. 


DO 
















P 
















g 


Actual. 


Theoretical. 




'3 


Non- 


Con- 










-*j 
















con- 
densing. 


densing. 




Non- 
con- 
densing. 


Con- 
densing. 


Non- 

con- 

densing. 


Con- 
densing. 




40 


17.34 


27.34 


14.49 


39.0 


22.0 


27.2 


18.5 


46 


45 


20.39 


30.39 


15.81 


36.0 


21.5 


25.3 


18.2 


44 


50 


23.45 


33.45 


17.13 


33.5 


21.0 


24.0 


17.9 


42 


55 


26.50 


36.50 


18.45 


31.2 


20.5 


22.9 


17.6 


40 


60 


29.56 


39.56 


19.77 


29.0 


20.0 


22.0 


17.4 


39 


65 


32.61 


42.61 


21.09 


27.6 


19.5 


21.3 


17.2 


38 


70 


35.67 


45.67 


22.41 


26.4 


19.0 


20.8 


17.0 


37 


75 


38.72 


48.72 


23.73 


25.3 


18.5 


20.4 


16.8 


36 


80 


41.78 


51.78 


25.05 


24.0 


18.0 


20.0 


16.6 


35 


85 


44.83 


54.83 


26.37 


23.0 


17.7 


19.6 


16.5 


34 


90 


47.89 


57.89 


27.69 


22.0 


17.4 


19.3 


16.4 


33 


95 


50.94 


60.94 


29.01 


21.2 


17.2 


19.0 


16.3 


32 


100 


54.0 


64.0 


30.33 


20.4 


17.0 


18.7 


16.2 


31 




t 3 q cut=off. 


40 


20.75 


30.75 


17.11 


38.0 


22.5 


27.0 


19.4 


43 


45 


24.13 


34.13 


18.67 


35.0 


22.0 


25.5 


19.1 


41 


50 


27.50 


37.50 


20.24 


33.0 


21.6 


24.3 


18.8 


40 


55 


30.87 


40.87 


21.80 


31.2 


21.2 


23.3 


18.5 


39 


60 


34.24 


44.24 


23.37 


29.5 


20.7 


22.5 


18.3 


38 


65 


37.61 


47.61 


24.94 


28.2 


20.3 


21.9 


18.1 


37 


70 


40.98 


50.98 


26.51 


27.0 


19.9 


21.4 


17.9 


36 


75 


44.35 


54.35 


28.07 


26.0 


19.5 


21.0 


17.7 


35 


80 


47.72 


57.72 


29.64 


25.0 


19.0 


20.6 


17.5 


34 


85 


51.09 


61.09 


31.20 


24.0 


18.8 


20.2 


17.3 


33 


90 


54.46 


64.46 


32.77 


23.0 


18.5 


19.9 


17.2 


32 


95 


67.83 


67.83 


34.33 


22.2 


18.3 


19.6 


17.1 


31 


100 


61.2 


71.2 


35.9 


21.5 


18.0 


19.4 


17.0 


30 




T %\ cut-off. 


40 


23.70 


33.70 


19.80 


37.0 


24.0 


27.5 


20.4 


41 


45 


27.32 


37.32 


21.61 


35.2 


23.5 


26.3 


20.0 


40 


50 


30.94 


40.94 


23.42 


33.7 


23.0 


25.3 


19.7 


39 


55 


34.56 


44.56 


25.23 


32.0 


22.5 


24.4 


19.5 


38 


60 




48.18 


27.04 


30.4 


22.0 


23.6 


19.3 


37 


6o 


41.80 


51.80 


28.85 


29.3 


21.7 


22.9 


19.1 


36 


70 


45.42 


56. 12 


30. ti<; 


28.0 


21.2 


22.3 


18.9 


35 


75 


49.05 


68.05 


82.47 


27.0 


20.8 


21.8 


18.7 


34 


80 


62.68 


62.68 


34.28 


26.0 


20.5 


21.4 


18.5 


33 


85 


56.31 


66.81 


36.09 


25.4 


20.2 


21.1 


18.4 


32 


90 


59.9-4 


69.94 


87.90 


24.0 


20.0 


20.8 


18.3 


31 


95 


68.67 


7: , ..r,7 


89.71 


23.2 


19.7 


20.6 


18.2 


30 


100 


f.7.1'0 


77.20 


n.^-2 


22.8 


19.4 


20.4 


18.1 


30 



Steam-engine Performances. 



655 



Table of Standard Engine Performance. - 



- Continued. 





T 4 o cut-off. 


9 


Mean effective 




Rates, 


in pound 


3 of water per indicated 


1 

u 
Pn 


pressure, in 
pounds. 


0Q 
P 




horse-power per 


hour. 




3 






Actual. 


Theoretical. 




'3 


Non- 


Con- 












+5 


H 










g 




con- 
densing. 


densing. 




Non- 
con- 
densing. 


Con- 
densing. 


Non- 
con- 
densing. 


Con- 
densing. 




40 


26.22 


36.22 


22.44 


38.0 


25.0 


28.3 


21.4 


40 


45 


30.08 


40.08 


24.49 


36.3 


24.6 


26.9 


21.1 


39 


50 


33.95 


43.95 


26.55 


34.5 


24.3 


25.8 


20.8 


38 


53 


37.81 


47.81 


28.60 


33.0 


23.8 


25.0 


20.5 


37 


60 


41.68 


51.68 


30.66 


31.5 


23.5 


24.4 


20.2 


36 


65 


45.54 


55.54 


32.71 


30.0 


23.0 


23.9 


20.0 


35 


70 


49.41 


59.41 


34.77 


29.0 


22.7 


23.4 


19.8 


34 


75 


53.27 


63.27 


36.82 


28.0 


22.3 


23.0 


19.6 


33 


80 


57.14 


67.14 


38.88 


27.0 


22.0 


22.6 


19.4 


32 


85 


61.00 


71.00 


40.93 


26.0 


21.7 


22.2 


19.3 


31 


90 


64.87 


74.87 


42.99 


25.0 


21.5 


21.9 


19.2 


30 


95 


68.73 


78.73 


45.04 


24.2 


21.2 


21.6 


19.1 


29 


100 


72.6 


82.6 


47.1 


23.4 


21.0 


21.4 


19.0 


29 




J cut=off. 


40 


30.50 


40.50 


27.78 


41.0 


29.5 


28.5 


23.4 


39 


45 


34.75 


44.75 


30.33 


39.0 


28.8 


27.6 


23.1 


38 


50 


39.00 


49.00 


32.88 


37.0 


28.3 


26.9 


22.8 


37 


55 


43.25 


53.25 


35.43 


35.5 


27.9 


26.3 


22.5 


36 


60 


47.50 


57.50 


37.98 


34.0 


27.5 


25.8 


22.2 


35 


65 


51.75 


61.75 


40.52 


32.5 


27.1 


25.3 


22.0 


34 


70 


56.00 


66.00 


43.07 


31.0 


26.7 


24.9 


21.8 


33 


75 


60.25 


70.25 


45.61 


30.0 


26.3 


24.5 


21.6 


32 


80 


64.50 


74.50 


48.16 


29.0 


25.8 


24.2 


21.5 


31 


85 


68.75 


78.75 


50.70 


28.0 


25.4 


23.9 


21.4 


30 


90 


73.00 


83.00 


53.25 


27.0 


24.9 


23.7 


21.3 


30 


95 


77.25 


87.25 


55.79 


26.0 


24.5 


23.5 


21.2 


29 


100 


81.5 


91.5 


58.34 


25.0 


24.0 


23.3 


21.1 


29 



The water rates given under the heading "Throt." in the tables show 
the number of pounds of water per indicated horse-power per hour used by 
throttling engines, at same (non-condensing) mean effective pressure and 
initial pressure as on same line. 

Standard Engine Tests. 

The final report of the Committee of the American Society of Mechani- 
cal Engineers upon the standardizing of steam-engine tests (1902) contains 
a large amount of valuable information upon the whole subject of engine 
performance, and an abridgement of it is here given. 

The Committee recommends the heat-unit basis, believing it to be the 
only fundamental basis for the determination of engine performance. 

The expressions of engine economy which meet all the requirements 
noted are the number of heat units consumed per hour, both per indicated 
and per brake horse-power, and these are recommended as the desired 



656 Steam-engine Testing. 

standards of comparison. The heat-unit standard does not interfere in 
any way with the common terms of expressing economy of engines. The 
hourly weights of coal, gas, oil, or other fuel, or weight of steam consumed 
per horse-power, heretofore commonly employed, are additional forms of 
stating economy, and are none the less useful within their limitations. *- 
They should by no means be abandoned. In the scheme now presented 
these additional or subsidiary forms of stating economy, as applied to 
particular classes of engines, are suitably provided for. 

The heat consumption of a steam-engine plant required for the standard 
test is ascertained by measuring the quantity of steam consumed by the 
plant, calculating the total heat of the entire quantity, and crediting this 
total with that portion of the heat rejected by the plant, which is utilized 
and returned to the boiler. The term " engine plant," as here used, should 
include the entire equipment of the steam plant which is concerned in 
the production of the power, embracing the main cylinder or cylinders ; 
the jackets and reheaters; the air, circulating, and boiler feed pumps, if 
steam driven ; and any other steam-driven mechanism or auxiliaries neces- 
sary to the working of the engine. 

The indicated horse-power for the proposed standard is that determined 
by the use of steam-engine indicators. It should be confined to the power 
developed in the main cylinder or cylinders, and should not include that 
developed in the cylinders of auxiliaries. 

One of the important subsidiary forms of expressing efficiency is that 
based on a so-called "standard coal" unit. The assumption is made that 
the heat consumed by the engine is generated from coal of a fixed heat 
value, as implied by the term "standard coal." 

The term "standard coal" refers to a coal which imparts to the steam 
10,000 B. T. U. for each pound of the dry coal consumed. It is coal having 
a calorific value of 12,500 B. T. U., used in what may be termed a " standard 
boiler," which gives an efficiency of 80 per cent, (referred to the coal). 
Although chosen arbitrarily, these figures, as a matter of fact, apply closely 
to the average coals of the United States. 

In treating of the subject of engine testing as relating primarily to the 
determination of matters of economy, it must not be forgotten that capac- 
ity is often of even greater importance than economy. In that large class 
of steam engines which are required to run at a certain limited and con- 
stant speed there should be a considerable reserve of capacity beyond the 
rated power. It is recommended that when a steam engine is operating at 
its rated power at a given pressure there should be a sufficient reserve to 
allow a drop of at least 15 per cent, in the gauge pressure without sensible 
reduction in the working speed of the engine, and allow an overload at 
the stated pressure amounting to at least 25 per cent. 



Rules for Conducting Steam=engine Tests. 

Code of 1902. 
American Society of Mechanical Engineers. 

I. Object of Test.— Ascertain at the outset the specific object of the 
test, whether it be to determine the fulfilment of a contract guarantee, to 
ascertain the highest economy obtainable, to find the working economy 
and delects under conditions as they exist, to ascertain the performance 
under special conditions, to determine the effect of changes in the condi- 
tions, or to find the performance of the entire boiler and engine plant, and 
prepare for the test accordingly. 

II. General Condition of the Plant.— Examine the engine and the 
entire plant concerned in the test; note its general condition and any 
points oi design, construction, or operation which bear on the objects in 
view. Make a special examination of the valves and pistons for leakage 
by applying the working pressures with the engine at rest, and observe the 
quantity ol steam, n any, blowing through per hour. 

If the trial has for an object the determination of the highest efficiency 
obtainable, the valves and pistons must first be made tight, and all parts 
ot the engine and its auxiliaries, and allother parts of the plant concerned, 
should be put in the best possible working condition. 



Steam-exgine Testing. 657 

III. Dimensions, etc.— Measure or check the dimensions of the cylin- 
\ ders in any case, this being done when they are hot. If they are much 

worn the average diameter should be determined. Measure also the clear- 
ance, which should be done, if possible, by filling the spaces with water 
to »previously measured, the piston being placed at the end of the stroke. If 
the clearance cannot be measured directly, it can be determined approxi- 
mately from the working drawings of the cylinder. 

Measure also the dimensions of auxiliaries and accessories ; also those of 
the boilers, so far as concerned in attaining the objects. It is well to sup- 
plement these determinations with a sketch or sketches showing the gen- 
eral features and arrangement of the different parts of the plant. 

IV. Coal.— When the trial involves the complete plant, embracing 
boilers as well as engine, determine the character of coal to be used. The 
class, name of the mine, size, moisture, and quality of the coal should be 
stated in the report. It is desirable, for purposes of comparison, that the 
coal should be of some recognized standard quality for the locality where 
the plant is situated. 

V. Calibration of Instruments.— All instruments and apparatus 
should be calibrated and their reliability and accuracy verified by com- 
parison with recognized standards. Such apparatus as is liable to change 
or become broken during a test, as gauges, indicator springs, and ther- 
mometers, should be calibrated before and after the test. The accuracy of 
scales should be verified by standard weights. When a water-meter is 
used, special attention should be given to its calibration, verifying it both 
before and after the trial, and, if possible, during its progress, the condi- 
tions in regard to water pressure and rate of flow being made the same in 
the calibrations as exist throughout the trial. 

VI. Leakages of Steam, Water, etc.— In all tests except those of a 
complete plant made under conditions as they exist, the boiler and its 
connections, both steam and feed, as also the steam piping leading to the 
engine and its connections, should, so far as possible, be made tight. If 
absolute tightness cannot be obtained (in point of fact it rarely can be), 
proper allowance should be made for such leakage in determining the 
steam actually consumed by the engine. This, however, is not required 

i where a surface condenser is used and the water consumption is deter- 
mined by measuring the discharge of the air pump. In such cases it is 
necessary to make sure that the condenser is tight, both before and after 
the test, against the entrance of circulating water, or, if such occurs, to 
make proper correction for it, determining it under the working difference 
of pressure. When the steam consumption is determined by measuring 
the discharge of the air pump, any leakage about the valve or piston-rods 
of the engine should be carefully guarded against. 

Make sure that there is no leakage at any of the connections with the 
apparatus provided for measuring and supplying the feed water which 
could affect the results. All connections should, so far as possible, be 
visible and be blanked off, and where this cannot be done satisfactory 
assurance should be obtained that there is no leakage either in or out. 

VII. Duration of Test.— The duration of a test should depend largely 
upon its character and the objects in view. The standard heat test of an 
engine, and, likewise, a test for the simple determination of the feed-water 
consumption, should be continued for at least five hours, unless the class 
of service precludes a continuous run of so long duration. It is desirable 
to prolong the test the number of hours stated to obtain a number of 
consecutive hourly records as a guide in analyzing the reliability of the 
whole. 

Where the water discharged from a surface condenser is measured for 
successive short intervals of time, and the rate is found to be uniform, the 
test may be of a much shorter duration than where the feed water is meas- 
ured to the boi]er. The longer the test with a given set of conditions the 
more accurate the work, and no test should be so short that it cannot be 
divided into several intervals which will give results agreeing substantially 
with each other. 

The commercial test of a complete plant, embracing boilers as well as 
engine, should continue at least one full day of twenty-four hours, whether 
the engine is in motion during the entire time or not. A continuous coal 

42 



658 Steam-engine Testing. 

test of a boiler and engine should be of at least ten hours' duration, or the 
nearest multiple of the interval between times of cleaning fires. 

VIII. Starting and Stopping a Test.— (a) Standard Heat Test and 
Feed-water Test of Engine: The engine having been brought to the normaV . 
condition of running, and operated a sufficient length of time to be thor- 
oughly heated in all its parts, and the measuring apparatus having been 
adjusted and set to work, the height of water in the gauge glasses of the 
boilers is observed, the depth of water in the reservoir from which the feed 
water is supplied is noted, the exact time of day is observed, and the test 
held to commence. Thereafter the measurements determined upon for 
the test are begun and carried forward until its close. If practicable, the 
test may be commenced at some even hour or minute, but it is of the first 
importance to begin at such time as reliable observations of the water 
heights are obtained, whatever the exact time happens to be when these 
are satisfactorily determined. When the time for the close of the test 
arrives, the water should, if possible, be brought to the same height in the 
glasses and to the same depth in the feed-wat r reservoir as at the begin- 
ning, delaying the conclusion of the test, if necessary, to bring about this 
similarity of conditions. If differences occur, the proper corrections must 
be made. 

(b) Complete Engine and Boiler Test: For a continuous running test of 
combined engine or engines, and boiler or boilers, the same directions 
apply for beginning and ending the feed-water measurements as that just 
referred to under Section a. The time of beginning and ending such a 
test should be the regular time of cleaning the fires, and the exact time of 
beginning and ending should be the time when the fires are fully cleaned, 
just preparatory to putting on fresh coal. In cases where there are a num- 
ber of boilers, and it is inconvenient or undesirable to clean all fires at 
once, the time of beginning the test should be deferred until they are all 
cleaned and in a satisfactory state, all the fires being then burned down to 
a uniformly thin condition, the thickness and condition being estimated 
and the test begun just before firing the new coal previously weighed. 
The ending of the test is likewise deferred until the fires are all satisfac- 
torily cleaned, being again burned down to the same uniformly thin 
condition as before, and the time of closing being taken just before 
replenishing the fires with new coal. 

For a commercial test of a combined engine and boiler, whether the 
engine runs continuously for the full twenty-four hours of the day or only 
a portion of the time, the fires in the boilers being banked during the time 
when the engine is not in motion, the beginning and ending of the test 
should occur at the regular time of cleaning the fires, the method followed 
being that already given. In cases where the engine is not in continuous 
motion, as, for example, in textile mills, where the working time is ten or 
eleven hours out of the twenty-four, and the fires are cleaned and banked 
at the close of the day's work, the best time for starting and stopping a test 
is the time just before banking, when the fires are well burned down and 
the thickness and condition can be most satisfactorily judged. In these, 
as in all other cases noted, the test should be begun by observing the exact 
time, the thickness and condition of the fires on the grates, the height of 
water in the gauge glasses of the boilers, the depth of the water in the 
reservoir from which the feed water is supplied, and other conditions 
relating to the trial, the same observations being again taken at the end of 
the test, and the conditions in all respects being made as nearly as possible 
the same as at the beginning. 

IX. Measurement of Heat Units Consumed by the Engine.— The 

measurement of the heat consumption requires the measurement of each 
supply of feed water to the boiler,— that is, the water supplied by the main 
feed pump, that supplied by auxiliary pumps, such as jacket water, water 
from separators, drips, etc., and water supplied by gravity or other means ; 
also, the determination of the temperature of the water supplied from 
each source, together with the pressure and quality of the steam. ^g 

The temperatures at the various points should be those applying to the 
working conditions. The temperature of the feed water should be taken 
near the boiler. This causes the engine to suffer a disadvantage from the 
heat lost by radiation from the pipes which carry the water to the boiler, 
but it is, nevertheless, advisable on the score of simplicity. Such pipes 






Steam-engine Testing. 659 

would, therefore, be considered a portion of the engine plant. This con- 
i forms with the rule already recommended for the tests of pumping engines 
where the dutv per million heat units is computed from the temperature 
of the feed water taken near the boiler. It frequently happens that the 
,* measurement of the water requires a change in the usual temperature of 
I .upply. For example, where the main supply is ordinarily drawn from a 
hot- well in which the temperature is, say, 100° F., it may be necessary, 
owing to the low level of the well, to take the supply from some source 
under a pressure or head sufficient to fill the weighing tanks used, and this 
supply may have a temperature much below that of the hot-well ; possibly 
as low as 40° F. The temperature to be used is not the temperature of the 
water as weighed in this case, but that of the working temperature of the 
hot- well. The working temperature in cases like this must be determined 
by a special test and included in the log sheets. 

The heat to be determined is that used by the entire engine equipment, 
embracing the main cylinders and all auxiliary cylinders and mechanism 
concerned in the operation of the engine, including the air pump, circu- 
lating pump, and feed pumps, also the jacket and reheater, when these 
are used. No deduction is to be made for steam used by auxiliaries, unless 
these are shown by test to be unduly wasteful. In this matter an excep- 
tion should be made in cases of guarantee tests where the engine contractor 
furnishes all the auxiliaries referred to. He should, in that case, be respon- 
sible for the whole, and no allowance should be made for inferior economy, 
if such exists. Should a deduction be made on account of the auxiliaries 
being unduly wasteful, the method of waste and its extent, as compared 
with the wastes of the main engine or other standard of known value, 
shall be reported definitely. 

The steam pressure and the quality of the steam are to be taken at some 
point conveniently near the throttle valve. The quantity of steam used 
by the calorimeter must be determined and properly allowed for. (See 
Article XVI., on " Quality of Steam.") 

X. Measurement of Feed Water or Steam Consumption of 
Engine, etc.— The method of determining the steam consumption appli- 
cable to all plants is to measure all the feed water supplied to the boilers, 
and deduct therefrom the water discharged by separators and drips, as 
also the water and steam which escapes on account of leakage of the 
boiler and its pipe connections and leakage of the steam main and branches 
connecting the boiler and the engine. In plants where the engine ex- 
hausts into a surface condenser, the steam consumption can be measured 
by determining the quantity of water discharged by the air pump, cor- 
rected for any leakage of the condenser, and adding thereto the steam 
used by jackets, reheaters, and auxiliaries, as determined independently. 
If the leakage of the condenser is too large to satisfactorily allow for it, 
the condenser should, of course, be repaired and the leakage again deter- 
mined before making the test. 

In measuring the water it is best to carry it through a tank or tanks 
resting on platform weighing scales suitably arranged for the purpose, the 
water being afterwards emptied into a reservoir beneath, from which the 
pump is supplied. 

Where extremely large quantities of water must be measured, or in 
some places relatively small quantities, the orifice method of measuring is 
one that can be applied with satisfactory results. In this case the average 
head of water on the orifice must be determined, and, furthermore, it is 
important that means should be at hand for calibrating the discharge of 
the orifice under the conditions of use. 

The corrections or deductions to be made for leakage above referred to 
should be applied only to the standard heat-unit test and tests for deter- 
mining simply the steam or feed-water consumption, and not to coal tests 
of combined engine and boiler equipment. In the latter, no correction 
should be made except for leakage of valves connecting to other engines 
and boilers, or for steam used for purposes other than the operation of the 
. plant under test. Losses of heat due to imperfections of the plant should 
*? j**"be charged to the plant, and only such losses as are concerned in the work- 
ing of the engine alone should be charged to the engine. 

In measuring jacket water or any supply under pressure which has a 
temperature exceeding 212° F., the water should first be cooled, as may be 
done by discharging it into a tank of cold water previously weighed, or by 



660 Steam-engine Testing. 

passing it through a coil of pipe submerged in running and colder water, 
preventing thereby the loss of evaporation which occurs when such hot 
water is discharged into the open air. 

XI. Measurement of Steam Used by Auxiliaries.— Although tho, 
steam used by the auxiliaries— embracing the air pump, circulating pump, 
feed pump, and any other apparatus of this nature, supposing them to be 
steam-driven, also the steam jackets, reheaters, etc., which consume steam 
required for the operation of the engine— is all included in the measure- 
ment of the steam consumption, as pointed out in Article X., yet it is 
highly desirable that the quantity of steam used by the auxiliaries, and in 
many cases that used by each auxiliary, should be determined exactly, so 
that the net consumption of the main engine cylinders may be ascertained 
and a complete analysis made of the entire work of the engine plant. 
Where the auxiliary cylinders are non-condensing, the steam consumption 
can often be measured by carrying the exhaust for the purpose into a tank 
of cold water resting on scales or through a coil of pipe surrounded by 
cold running water. Another method is to run the auxiliaries as a whole, 
or one by one, from a spare boiler (preferably a small vertical one), and 
measure the feed water supplied to this boiler. The steam used by the air 
and circulating pumps may be measured by running them under, as near as 
possible, the working conditions and speed, the main engine and other 
auxiliaries being stopped, and testing the consumption by the measuring 
apparatus used on the main trial. For a short trial, to obtain approximate 
results, measurement can be made by the water gauge-glass method, the 
feed supply being shut off. When the engine has a surface condenser, the 
quantity of steam used by the auxiliaries may be ascertained by allowing 
the engine alone to exhaust into the condenser, measuring the feed water 
supplied to the boiler and the water discharged by the air pump, and sub- 
tracting one from the other, after allowing for losses by leakage. 

XII. Coal Measurement. — (a) Commercial Tests: In commercial tests 
of the combined engine and boiler equipment, or those made under ordi- 
nary conditions of commercial service, the test should, as pointed out in 
Article VII., extend over the entire period of the day,— that is, twenty-four 
hours,— or a number of days of that duration. Consequently, the coal 
consumption should be determined for the entire time. If the engine runs 
but a part of the time, and during the remaining portion the fires are 
banked, the measurement of coal should include that used for banking. 
It is well, however, in such cases, to determine separately the amount con- 
sumed during the time the engine is in operation and that consumed during 
the period while the fires are banked, so as to have complete data for pur- 
poses of analysis and comparison, using suitable precautions to obtain 
reliable measurements. The measurement of coal begins with the first 
firing, after cleaning the furnaces and burning down at the beginning of 
the test, as pointed out in Article VIII., and ends with the last firing, at 
the expiration of the allotted time. 

(b) Continuous Running Tests: In continuous running tests which, as 
pointed out in Article VII., cover one or more periods which elapse between 
the cleaning of the fires, the same principle applies as that mentioned 
under the above heading (a), —viz., the coal measurement begins with the 
first firing, after cleaning and burning down, and the measurement ends 
with the last firing, before cleaning and burning down at the close of the 
trial. 

(c) Coal Tests in General: When not otherwise specially understood, a 
coal test of a combined engine and boiler plant is held to refer to the com- 
mercial test above noted, and the measurement of coal should conform 
thereto. 

In connection with coal measurements, whatever the class of tests, it is 
important to ascertain the percentage of moisture in the coal, the weight 
of ashes and refuse, and, where possible, the approximate and ultimate 
analysis of the coal, following all the methods and details advocated in the 
latest report of the Boiler Test Committee of the Society. (See " Transac- * 
tions of the American Society of Mechanical Engineers," Volume XXL," 
page 34.) 

(d) Other Fuels than Coal: For all other solid fuels than coal the same 
directions in regard to measurement should be followed as those given for 
coal. If the boilers are run with oil or gas, the measurements relating to 



Steam-engine Testing. 661 

stopping and starting are mnch simplified, because the fuel is burned as 
fast as supplied and there is no body of fuel constantly in the furnace, as 
in the case of using solid fuel. When oil is used it should be weighed, 
and when gas is used it should be measured in a calibrated gas-meter or a 
gasometer. 

XIII. Indicated Horse=power.— The indicated horse-power should be 
determined from the average mean effective pressure of diagrams taken at 
intervals of twenty minutes, and at more frequent intervals if the nature 

; of the test makes this necessary, for each end of each cylinder. With 
variable loads, such as those of engines driving generators for electric 
railroad work, and of rubber-grinding and rolling-mill engines, the dia- 
grams cannot be taken too often. In cases like the latter, one method of 
obtaining suitable averages is to take a series of diagrams on the same 
blank card without unhooking the driving cord, and apply the pencil at 
^successive intervals of ten seconds until two minutes' time or more has 
elapsed, thereby obtaining a dozen or more indications in the time covered. 
This tends to insure the determination of a fair average for that period. 
In taking diagrams for variable loads, as, indeed, for any load, the pencil 
should be applied long enough to cover several successive revolutions, 
so that the variations produced by the action of the governor may be 
'properly recorded. To determine whether the governor is subject to what 
lis called "racing'' or "hunting, " a "variation diagram" should be obtained; 
— that is, one in which the pencil is applied a sufficient time to cover a 
'complete cycle of variations. When the governor is found to be working 
in this manner the defect should be remedied before proceeding with the 
■test. 

It is seldom necessary, as far as average power measurements are con- 
cerned, to obtain diagrams at precisely the same instant ac the two ends of 
the cylinder, or at the same instant on all the cylinders, when there are 
more than one. All that is required is to take the diagrams at regular in- 
tervals. Should the diagrams vary so much among themselves that the 
average may not be a fair one, it signifies that they should be taken more 
'frequently, and not that special care should be employed to obtain the 
- diagrams of each set at precisely the same time. When diagrams are taken 
during the time when the engine is working up to speed at the start, or 
when a study of valve setting and steam distribution is being made, they 
should be taken at as nearly the same time as practicable. In cases where 
the diagrams are to be taken simultaneously, the best plan is to have an 
■operator stationed at each indicator. This is desirable, even where an 
electric or other device is employed to operate all the instruments at once, 
for, unless there are enough operators, it is necessary to open the indicator- 
cocks some time before taking the diagrams and run the risk of clogging 
the pistons and heating the high-pressure springs above the ordinary work- 
ing temperature. 

The most satisfactory driving rig for indicating seems to be some form 
of well-made pantagraph, with driving cord of fine annealed wire leading 
to the indicator. The reducing motion, whatever it may be, and the con- 
nections to the indicator should be so perfect as to produce diagrams of 
equal lengths when the same indicator is attached to either end of the 
cylinder, and produce a proportionate reduction of the motion of the piston 
at every point of the stroke, as proved by test. 

The use of a three-way cock and a single indicator connected to the two 
ends of the cylinder is not advised, except in cases where it is impracticable 
to use an indicator close to each end. If a three-way cock is used the 
1 error produced should be determined and allowed for. 

To determine the average power developed in cases where the engine 
starts from rest during the progress of the trial, as in a commercial test of 
a plant where the engine runs only a portion of the twenty-four hours, a 
-number of diagrams should be taken during the period of getting up speed 
and applying the working load, the corresponding speed for each set of 
diagrams being counted. The power shown by these diagrams for ihe pro- 
portionate time should be included in the average for the whole run, and 
ithe duration should be the time the throttle valve is open. 

XIV. Testing Indicator Springs.— To make a perfectly satisfactory 
comparison of indicator springs with standards, the calibration should be 

, made, if this were practical, under the same conditions as those pertaining 



662 Steam-engine Testing. 



to their ordinary use. Owing to the fact that the pressure of the steam in 
the indicator cylinder and the corresponding temperature are undergoing 
continual changes, it becomes almost impossible to compare the springs 
with any standard under such conditions. There must be a constant press- 
ure during the time that the comparison is being made. Although thr-j 
best that can be done is not altogether satisfactory, it seems that we must ' 
be content with it. To bring the conditions as nearly as possible to those 
of the working indicator, the steam should be admitted to the indicator as 
short a time as practicable for each of the pressures tried, and then the 
indicator cock should be closed and the steam exhausted therefrom before 
another pressure is tried. By this means the parts are heated and cooled 
somewhat the same as under the working conditions. We recommend, 
therefore that for each required pressure the first step be to open and close 
the indicator cock a number of times in quick succession, then to quickly 
draw the line on the paper for the desired record, observing the gauge or 
other standard at the instant when the line is drawn. A corresponding 
atmospheric line is taken immediately after obtaining the line at the given 
pressure so as to eliminate any difference in the temperature of the parts 
of the indicator. This appears to be a better method (although less readily 
carried on and requiring more care) than the one heretofore more com- 
monly used, where the indicator cock is kept continually open and the 
pressure is gradually rising or falling through the range of comparison 

The calibration should be made for at least five points, two of these 
being for the pressures corresponding as near as may be to the initial and 
back pressures, and three for intermediate points equally distant. 

For pressures above the atmosphere the proper standard recommended 
is the dead-weight testing apparatus or a reliable mercury column or an 
accurate steam gauge proved correct, or of known error, by either of these 
standards. For pressures below the atmosphere the best standard to use is 
a mercury column. 

The correct scale of spring to be used for working out the mean effective 
pressure of the diagrams should be the average based on the calibration. 

XV Brake Horse=power.— This term applies to the power delivered 
from the fly-wheel shaft of the engine. It is the power absorbed by a 
friction brake applied to the rim of the wheel or to the shaft. A form of 
brake is preferred that is self-adjusting to a certain extent, so that it will, 
of itself tend to maintain a constant resistance at the rim of the wheel. 
One of the simplest brakes for comparatively small engines which may be 
made to embody this principle consists of a cotton or hemp rope or a 
number of ropes, encircling the wheel, arranged with weighing scales or 
other means for showing the strain. An ordinary band brake may also be 
constructed so as to embody the principle. The wheel should be provided 
with interior flanges for holding water used for keeping the rim cool. 

The water-friction brake is considered most satisfactory, not only tor 
small powers but for large powers. It is especially adapted for high speeds, 
and has the advantage of being self-cooling. 

XVI Quality of Steam.— When ordinary saturated steam is used its 
quality should be obtained by the use of a throttling calorimeter attached 
to the main steam pipe near the throttle valve. When the steam is super- 
heated the amount of superheating should be found by the use of a 
thermometer placed in a thermometer-well filled with mercury, inserted 
in the pipe. The sampling pipe for the calorimeter should, if possible, be 
attached to a section of the main pipe having a vertical direction, with the 
steam preferably passing upward, and the sampling nozzle should be made 
of a half-inch pipe having at least twenty %-inch holes m its perforated 
surface. The readings of the calorimeter should be corrected for radiation 
of the instrument, or they should be referred to a normal reading. It 
the steam is superheated, the amount of superheating should be ob- 
tained by referring the reading of the thermometer to that ot the same 
thermometer when the steam within the pipe is saturated, and not by 
taking the difference between the reading of* the thermometer and the 
temperature of saturated steam at the observed pressure as given in aq 
steam table. 

XVII Speed.— There are several reliable methods of ascertaining the 
speed or the number of revolutions of the engine crank-shaft per minute. 
The simplest is the familiar method of counting the number of turns for a 



Steam-engine Testing. 663 

period of one minute, with the eye fixed on the second-hand of a time- 
piece. Another is the use of a counter held for a minute or a number of 
minutes against the end of the main shaft. Another is the use of a reliable 
calibrated tachometer held likewise against the end of the shaft. The 
^most reliable method, and the one we recommend, is the use of a con- 
tinuous recording engine register or counter, taking the total reading each 
time that the general test data are recorded, and computing the revolutions 
per minute corresponding to the difference in the readings of the instru- 
ment. When the speed is above 250 revolutions per minute it is almost 
impossible to make a satisfactory counting of the revolutions without the 
use of some form of mechanical counter. 

The determination of variation of speed during a single revolution, or 
the effect of the fluctuation due to sudden changes of the load, is also 
desirable, especially in engines driving electric generators used for lighting 
purposes. There is at present no recognized standard method of making 
such determinations, and, if such are desired, the method employed may 
be devised by the person making the test and described in detail in the 
report. 

XVIII. Recording the Data.— Take note of every event connected 
with the progress of the trial, whether it seems at the time to be important 
or unimportant. Record the time of every event and time of taking 
every weight and every observation. Observe the pressures, temperatures, 
water heights, speeds, etc., every twenty or thirty minutes when the con- 
ditions are practically uniform, and at much more frequent intervals if the 
conditions vary. Observations which concern the feed-water measurement 
should be made with special care at the expiration of each hour of the 
trial, so as to divide the tests into hourly periods and show the uniformity 
of the conditions and results as the test goes forward. Where the water 
discharged from a surface condenser is weighed, it may be advisable to 
divide the test by this means into periods of less than one hour. 

The data and observations of the test should be kept on properly-pre- 
pared blanks or in note-books containing columns suitably arranged for a 
clear record. As different observers have their own individual ideas as to 
how such records should be kept, no special form of log sheet is given as a 
necessary part of the code. 

XIX. Uniformity of Conditions. — In a test having for an object the 
determination of the maximum economy obtainable from an engine, or 
where it is desired to ascertain with special accuracy the effect of pre- 
determined conditions of operation, it is important that all the conditions 
under which the engine is operated should be maintained uniformly 
constant. This requirement applies especially to the pressure, the speed, 
the load, the rate of feeding the various supplies of water, the height of 
water in the gauge glasses, and the depth of water in the feed-water 
reservoir. 

XX. Analysis of Indicator Diagrams. — (a) Steam Accounted for by the 
Indicator : The simplest method of computing the steam accounted for by 
the indicator is the use of the formula, 

M= M 18 ™p. [{C+E)X Wc-(H+E)X Wh], 

which gives the weight, in pounds, per indicated horse-power per hour. 
In this formula the symbol " M. E. P." refers to the mean effective pressure. 
In multiple-expansion engines this is the combined mean effective pressure 
referred to the cylinder in question. The symbol C refers to the proportion 
of the stroke completed at points on the expansion line of the diagram 
near the actual cut-off or release, the symbol H to the proportion of com- 
pression, and the symbol E to the proportion of clearance, all of which 
are determined from the indicator diagram. The symbol Wc refers to the 
weight of 1 cubic foot of steam at the cut-off or release pressure, and the 
symbol Wh to the weight of 1 cubic foot of steam at the compression press- 
ure, these weights being taken from steam tables of recognized accuracy. 
The points near the cut-off and release on the expansion line, and the 
point on the compression line, are located as shown on the sample dia- 
gram. They are the points in the case of the expansion and compression 
lines of the diagram which mark the complete closure of the valve. The 



664 



Steam-engine Testing. 



point near the cut-off, for example, lies where the curve of expansion 
begins after the rounding of the diagram due to the wire-drawing, which 
occurs while the valve is closing. This cut-off may be located by finding 
the point where the curve is tangent to a hyperbolic curve. 




Belease, 



Compression. 



Atmospheric line ^^ -^^ TegM 
Showing Points where "Steam Accounted for by Indicator" is Computed. 



Should the point in the compression curve be at the same height as the 
point in the expansion curve, then Wc = Wh, and the formula becomes 



13750 
M. E. P. 



x (c-h) : 



Wc, 



in which (C — H) represents the distance between the two points divided 
by the length of the diagram. 

When the load and all other conditions are substantially uniform, it is 
unnecessary to work up the steam accounted for by the indicator from all 
the diagrams taken. Five or more sample diagrams may be selected and 
the computations based on the samples instead of on the whole. 

(b) Sample Indicator Diagrams: In order that the report of a test may 
afford complete information regarding the conditions of the test, sample 
indicator diagrams should be selected from those taken and copies ap- 
pended to the tables of results. In cases where the engine is of the multi- 
ple-expansion type, these sample diagrams may also be arranged in the 
form of a "combined" diagram. 

(c) The Point of Cut-off: The term " cut-off," as applied to steam engines, 
although somewhat indefinite, is usually considered to be at an earlier 
point in the stroke than the beginning of the real expansion line. That 
the cut-off point may be defined in exact terms for commercial purposes, 
as used in steam-engine specifications and contracts, the Committee recom- 
mends that, unless otherwise specified, the commercial cut-off, which seems 




HG F 

Four-valve Engine, Slow Speed. 

Commercial Cut-off : 




Single-valve Engine, High Speed. 
BC 



to be an appropriate expression for this term, be ascertained as follows : 
through a point showing the maximum pressure during admission draw a 
line parallel to the atmospheric line. Through the point on the expansion 
line, near the actual cut-off, referred to in Section XX. (a), draw a hyper- 
bolic curve. The point where these two lines intersect is to be considered 
the commercial cut-off point. The percentage is then found by dividing the 



Steam-engine Testing. 



665 




Ideal Ratio of Expansion 



length of the diagram measured to this point by the total length of the 
diagram, and multiplying the result by 100. 

The principle involved in locating the commercial cut-off is shown in 
the preceding diagrams, the first of which represents a diagram from a 
m slow-speed Corliss engine, and the second a diagram from a single-valve, 
high-speed engine. In the latter case, where, owing to the fling of the 
pencil, the steam line vibrates, the maximum pressure is found by taking 
a mean of the vibrations at the highest point. 

The commercial cut-off, as thus determined, is situated at an earlier point 
of the stroke than the actual cut-off used in computing the "steam ac- 
counted for" by the indicator and referred to in Section XX. (a). 

(d) Ratio of Expansion: The " commercial" ratio of expansion is the 
quotient obtained by dividing the volume corresponding to the piston 
displacement, including clearance, 

by the volume of the steam at the IRC. S . N 

Commercial CUt-off, including Clear- li ^-^ V ^ Boiler Pressure 

ance. In a multiple-expansion 
engine the volumes are those per- 
taining to the low-pressure cylinder 
and high-pressure cylinder, respec- 
tively. 

The " ideal" ratio of expansion is 
the quotient obtained by dividing 
the volume of the piston displace- 
ment by the volume of the steam at 
the cut-off (the latter being referred 
to the throttle-valve pressure), less 
the volume equivalent to that re- 
tained at compression. In a multiple-expansion engine the volumes to be 
used are those pertaining to the low-pressure cylinder and high-pressure 
cylinder, respectively. 

(e) Diagram Factor: The diagram factor is the proportion borne by the 

I actual mean effective pressure measured from the indicator diagram to that 
' of a diagram in which the various operations, of admission, expansion, 
release, and compression are carried on under assumed conditions. The 
factor recommended refers to an ideal diagram which represents the 
maximum power obtainable from the steam accounted for by the indicator 
diagrams at the point of cut-off, assuming, first, that the engine has no 
clearance ; second, that there are no losses through wire-drawing the 
! steam either during the admission or the release ; third, that the expan- 
sion line is a hyperbolic curve ; and fourth, that the initial pressure is that 
of the boiler and the back pressure that of the atmosphere for a non- 
! condensing engine, and of the condenser for a condensing engine. 

The diagram factor is useful for comparing the steam distribution losses 
in different engines, and is of special use to the engine designer, for by 
multiplying the mean effective pressure obtained from the assumed theo- 
retical diagrams by it he will obtain the actual mean effective pressure 
, that should be developed in an engine of the type considered. The expan- 
sion and compression curves are taken as hyperbolas, because such curves 
are ordinarily used by engine builders in their work, and a diagram based 
on such curves will be more useful to them than one where the curves are 
, constructed according to a more exact law. 

In cases where there is a considerable loss of pressure between the boiler 
, and the engine, as where steam is transmitted from a central plant to a 
1 number of consumers, the pressure of the steam in the supply main should 
| be used in place of the boiler pressure in constructing the diagrams. 

XXI. Standards of Economy and Efficiency.— The hourly consump- 

: tion of heat, determined by employing the actual temperature of the feed 

water to the boiler, as pointed out in Article IX. of the Code, divided by 

the indicated and brake horse-power,— that is, the number of heat units 

consumed per indicated and per brake horse-power per hour are the stand- 

j , ards of engine efficiency recommended by the Committee. The consump- 

i tion per hour is chosen rather than the consumption per minute, so as to 

] conform with the designation of time applied to the more familiar units of 

coal and water measurement, which have heretofore been used. The 

British standard, where the temperature of the feed water is taken as that 

I corresponding to the temperature of the back-pressure steam, allowance 



666 



Steam-engine Testing. 



being made for any drips from jackets or reh eaters, is also included in the 
tables. 

It is useful in this connection to express the efficiency in its more scien- 
tific form, or what is called the "thermal efficiency ratio." The thermal 
efficiency ratio is the proportion which the heat equivalent of the power r 
developed bears to the total amount of heat actually consumed, as deter- 
mined by test. The heat converted into work, represented by 1 horse- 
power, is 1,980,000 foot-pounds per hour, and this, divided by 778, equals 
2545 B. T. U. Consequently, the thermal efficiency ratio is expressed by the 
fraction __,._ 

2545 

B. T. U. per horse-power per hour* 

XXII. Heat Analysis.— For certain scientific investigations it is useful 

to make a heat analysis of 
the diagram to show the 
interchange of heat from 
steam to cylinder walls, etc., 
which is going on within 
the cylinder. This is un- 
necessary for commercial 
tests. 

XXIII. Temperature- 
entropy Diagram. — The 

study of the neat analysis 
is facilitated by the use of 
the temperature-entropy di- 
agram, in which areas rep- 
resent quantities of heat, 
the coordinates being the 
absolute temperature and 
entropy. Such a diagram 
is here given. When the 
quantity given in the steam 
tables is plotted, two curves, 
AA and BB, are obtained 
which may be termed the 
water line and the steam 
. line, AA being the logarith- 
** mic curve if the specific 
heat of the water is taken 
as constant. The diagram 
refers to a unit weight of 
the agent, and the heat 
necessary to raise a pound 
of water from the tempera- 
ture, ma, to the tempera- 
ture, pa', and evaporate it 
at that temperature, is repre- 
sented by the area, aa'b'qm. 
If the steam be now ex- 
panded adiabatically, the 
temperature will fall to qs, 

will 




and x per cent. = 



ab 



P Q 

Temperature-entropy Diagram. 



to pump a pound of water into the boiler 
evidently 

h + L\ — xL 2 — w 
h + Li 



remain as steam, the rest 
being liquefied. If the 
steam is now rejected, it 
carries away with it the 
heat, sqma, the work area 
being a'b'sa, from which 
must be deducted the work, 
w (expressed in heat units), 
The efficiency of this cycle is 



Steam-engine Testing. 



667 



in which 



ar -f a'b' 
ab 



To 



By the action of the walls a portion of the steam is liquefied prior to the 
expansion, which, therefore, begins at e ; and since the cooling action of 
the walls continues, the expansion line falls off to ef, from which point a 
reverse action takes place and the expansion line bends over to g. Finally, 
since the release takes place before the condenser temperature is reached, 
the heat rejection starts at g, following a line of equal volume until the 
exhaust-port temperature is reached at j. If enough heat is added dur- 
ing expansion to keep the steam theoretically saturated, — as, for example, 
by a water jacket,— such additional heat is represented by the area, b'bnq, 
and the additional work obtained by the triangle, b'bs. If the steam is 
superheated sufficiently to 
give by expansion theoreti- 
cally dry steam at the end, 
such additional heat is rep- 
resented by the area, b'vnq, 
and the additional work by 
b'vbs. Neither of these extra 
amounts of work are realized 
in practice, and it is evident 
from the diagram that the 
heat thus applied is in both 
cases less efficient than in 
the principal cycle. Never- 
theless, the action in each 
case is to bring the point, e, & : 
nearer the point, b', and to 
effect a notable net economy. 

The Carnot cycle would 
be obtained if in the Ran- 
kine cycle the rejection of 
heat were stopped at r and 
the temperature of the mix- 
ture raised to a' by com- 
pression. This cannot be 
practically accomplished, 
but a system of feed-water 
heaters* has been suggested 
and exemplified in the 
Nordberg engine, which is 
theoretically a close equiva- 
lent to it. Where steam is 
expanded in, say, three cyl- 
inders, the feed water may 
be successively heated from 
the receiver intermediate 
between each pair, the 



h 




Temperature-entropy Diagram. 



effect of which is illustrated in the above diagram. The expansion line 
follows the heavy line, being carried over to y by the first feed- water heater 
and to y' by the second feed-water heater. With an infinite number of 
such feed-water heaters the line, yy f , would be parallel to aa', and the 
cycle equivalent to that of Carnot. 

XXIV. Ratio of Economy of an Engine to that of an Ideal 

Engine.— The ideal engine recommended for obtaining this ratio is that 
which was adopted by the Committee appointed by the Civil Engineers of 
London to consider and report a standard thermal efficiency for steam 
engines. This engine is one which follows the Rankine cycle, where steam 
at a constant pressure is admitted into the cylinder with no clearance, and, 
after the point of cut-off, is expanded adiabatically to the back pressure. 
In obtaining the economy of this engine the feed water is assumed to be 



668 



Steam-engine Testing. 



returned to the boiler at the exhaust temperature. Such a cycle is prefer- 
able to the Carnot for the purpose at hand, because the Carnot cycle is 
theoretically impossible for an engine using superheated steam produced 




250 Values of t a 300 350~ 

Coefficient for superheat correction 



400° 




Curves showing British Thermal Units Expended per Minute per Indicated 
Horse-power by the Ideal Steam Engine, forming part of the Itankine Cycle. 
(From the Minutes of Proceedings of the Civil Engineers of London.) Tempera- 
tures are expressed in degrees Fahrenheit. The upper and lower portions of the 
upper diagram are to different scales. This is in order that the lower and more 
important part may be read more easily, and accounts for the cusps in the curves. 



at a constant pressure, and the gain in efficiency for superheated steam 
corresponding to the Carnot efficiency will be much greater than that 
possible for the actual cycle. 



Steam-engine Testing. 669 

The economy of the ideal engine recommended can be readily obtained 
from the accompanying chart, which has been copied from the report 
already mentioned of the Committee appointed by the Civil Engineers of 
London. 

In the chart, t a represents the temperature of saturated steam at the 
boiler pressure, in degrees Fahrenheit ; tas, that of the steam furnished to 
the engine, should there be superheating; t e , that of the exhaust. The 
British thermal units consumed per minute per indicated horse-power by 
the ideal engine can be read off directly from the curves given in the 
upper portion of the diagram. Thus, if the temperature of the exhaust, t e , 
is 212° F., and the temperature of the steam at boiler pressure is 350° F., 
the heat consumption is 265 B. T. U. per indicated horse-power per minute. 
If the steam is superheated, the figure obtained as just described is cor- 
rected by employing the factor obtained from the lower part of the dia- 
gram. Opposite the temperature of saturation, corresponding to the 
pressure in the boiler, and on the curve corresponding to the temperature 
of superheated steam, tas, is found a coefficient. This coefficient, multi- 
plied by the exhaust temperature and by the heat consumption per minute 
obtained, — should there be no superheating,— gives the deduction to be 
made on account of the superheating. Thus, if the temperature of the 
superheated steam is 500° F. in the case already considered for saturated 
steam, we find, opposite 350 degrees for t a and on the curve for t a s = 500 
degrees, the coefficient, 0.00015. This gives the correction, 0.00015 X 265 = 
8.5 B. T. U ; and the heat consumption of the engine, when furnished with 
superheated steam, will be 265 — 8.5 = 256.5 B. T. U. per indicated horse- 
power per minute. 

The ratio of the economy of an engine to that of the ideal engine is 
obtained by dividing the heat consumption per indicated horse-power per 
minute for the ideal engine by that of the actual engine. 

XXV. Miscellaneous.— In the case of tests of combined engine and 
boiler plants, where the full data of the boiler performance is to be deter- 
mined, reference should be made to the directions given by the Boiler-test 
Committee of the Society, Code of 1899. (See "Transactions of the Ameri- 
can Society of Mechanical Engineers," Volume XXI., page 34.) 

In tests made for scientific research, and in those made on special forms 
of engines, the line of procedure must be varied according to the special 
objects in view ; and it has been deemed unnecessary to go into particulars 
applying to such tests. 

In testing steam pumping engines and locomotives, in accordance with 
the standard methods of conducting such tests recommended by the com- 
mittees of the Society, reference should be made to the reports of those 
committees in the "Transactions," Volume XII., page 530, and in Volume 
XIV., page 1312. 

XXVI. Report of Test.— The data and results of the test should be 
reported in the manner and in the order outlined in one of the following 
tables, the first of which gives, it is hoped, a complete summary of all the 
data and results as applied not only to the* standard heat-unit test, but also 
to tests of combined engine and boiler for determining all questions of 
performance, whatever the class of service ; the second refers to a short 
form of report giving the necessary data and results for the standard heat 
test ; and the third to a short form of report for a feed- water test. It is the 
intention that the tables should be full enough to apply to any type of 
engine, but where not so, or where special data and results are determined, 
additional results may be inserted under the appropriate headings. Al- 
though these forms are arranged so as to be used for expressing the princi- 
pal data and results of tests of pumping engines and locomotives, as well 
as for all other classes of steam engines, it is not the intention that they 
shall supplant the forms recommended by the committees on Duty Trials 
and Locomotives in cases where the full report of a test of such engines is 
desired. 



(570 Steam-engine Testing. 



Data and Results of Standard Heat Test of Steam 
Engine. 

Arranged according to the Short Form advised by the Engine Test Com- 
mittee of the American Society of Mechanical Engineers. Code of 1902. 

1. Made by of 

on engine located at , 

to determine 



2. Date of trial • • 

3. Type and class of engine ; also of condenser. 



4. Dimensions of main engine : 

(a) Diameter of cylinder, in inches — 

(b) Stroke of piston, in feet 

(c) Diameter of piston-rod, in inches. . 

\d) Average clearance, in per cent 

(e) Ratio of volume of cylinder to 

high-pressure cylinder 

(/) Horse-power constant for 1 pound 
mean effective pressure and 1 
revolution per minute 

5. Dimensions and type of auxiliaries 



1st Cyl. 2d Cyl. 3d Cyl. 



Total Quantities, Time, Etc. 

6. Duration of test • •• • hours. 

7 Total water fed to boilers from mam source of supply, lbs. 
8. Total water fed from auxiliary supplies : 



(a). 



lbs. 
lbs. 



ft-:;::::::;:::::::::::::::... n>, 

9 Total water fed to boilers from all sources lbs. 

10 Moisture in steam or superheating near throttle per cent, or deg. 

11*. Factor of correction for quality of steam 

12. Total dry steam consumed for all purposes ids. 

Hourly Quantities. 

13. Water fed from main source of supply lbs. 

14. Water fed from auxiliary supplies : 



$• 



lbs. 
lbs. 



;;;;;;;;;; ids. 

15. Total water fed to boilers per hour lbs. 

16. Total dry steam consumed per hour IDs. 

17. Loss of steam and water per hour due to drips from 

main steam pipes and to leakage of plant. . . . IDs. 

18. Net dry steam consumed per hour by engine and 

auxiliaries lDS - 

Pressures and Temperatures (Corrected). 

19 Pressure in steam pipe near throttle, by gauge lbs. per sq. in. 

20! Barometric pressure of atmosphere, in inches of mer- 

cury ins. 

21. Pressure in receivers, by gauge los. per sq. in. 

22 Vacuum in condenser, in inches of mercury ms. 

23 Pressure in jackets and reheaters, by gauge lbs. per sq. m. 

24*. Temperature of main supply of feed water deg. if anr. 

25. Temperature of auxiliary supplies of feed water : ^^ 

ft] ::. \v. :::;: :;.'::.'.'.'.'.' .'.'^.'.' deg.'Fahr: 

(c) I"'.'.:'.'. '.".'. ".'.'. deg. Fahr. 



Steam-engine Testing. 671 

26. Ideal feed- water temperature, corresponding to press- 

ure of steam in the exhaust pipe, allowance 
being made for heat derived from jacket or 
reheater drips deg. Fahr. 

Data Relating to Heat Measurement. 

27. Heat units per pound of feed water, main supply B. T. U. 

28. Heat units per pound of feed water, auxiliary supplies : 

(a) B.T.U. 

(&) B.T.U. 

(c) B.T.U. 

29. Heat units consumed per hour, main supply B. T. U. 

30. Heat units consumed per hour, auxiliary supplies : 

(a) B. T. U. 

(&) B.T.U. 

(c) B.T.U. 

31. Total heat units consumed per hour for all purposes . . B. T. U. 

32. Loss of heat per hour due to leakage of plant, drips, 

etc B. T. U. 

33. Net heat units consumed per hour : 

(a) By engine alone B.T.U. 

(&) By auxiliaries B. T. U. 

34. Heat units consumed per hour by engine alone, reck- 

oned from temperature given in line 26 B. T. U. 

Indicator Diagrams. 

IstCyl. 2dCyl. 3d Cyl. 

35. Commercial cut-off, in per cent, of stroke. 

36. Initial pressure, in pounds, per square inch 

above atmosphere 

! 37. Back pressure at mid-stroke, above or 
below atmosphere, in pounds, per 
square inch 

38. Mean effective pressure, in pounds, per 

square inch 

39. Equivalent mean effective pressure, in 

pounds, per square inch : 

(a) Referred to first cylinder 

lb) Referred to second cylinder 

(c) Referred to third cylinder 

; 40. Pressures and percentages used in com- 
puting the steam accounted for 
by the indicator diagrams, meas- 
ured to points on the expansion 

and compression curves 

Pressure above zero, in pounds, per square 
inch : 

(a) Near cut-off 

(&) Near release 

(c) Near beginning of compression 

Percentage of stroke at points where 
pressures are measured : 

(a) Near cut-off 

(b) Near release 

(c) Near beginning of compression 

41. Steam accounted for by indicator, in 

pounds, per indicated horse- 
power per hour : 

(a) Near cut-off 

(b) Near release 

42. Ratio of expansion : 

(a) Commercial 

(&) Ideal 

Speed. 

43. Revolutions per minute rev. 



672 Steam-engine Testing. 



Power. 

44. Indicated horse-power developed by main engine cylinders : 

First cylinder H. P. 

Second cylinder H. P. 

Third cylinder H. P. 

Total H. P. 

45. Brake horse-power developed by engine H. P. 



Standard Efficiency and other Results.* 

46. Heat units consumed by engine and auxiliaries per hour : 

(a) Per indicated horse-power B. T. U. 

(b) Per brake horse-power B. T. U. 

47. Equivalent standard coal, in pounds, per hour : 

(a) Per indicated horse-power lbs. 

(b) Per brake horse-power lbs. 

48. Heat units consumed by main engine per hour, corresponding 

to ideal maximum temperature of feed water given 

in line 26 : 

(a) Per indicated horse-power B. T. U. 

(&) Per brake horse-power B. T. U. 

49. Dry steam consumed per indicated horse-power per hour : 

(a) Main cylinders, including jackets lbs. 

(b) Auxiliary cylinders lbs, 

(c) Engine and auxiliaries lbs. 

50. Dry steam consumed per brake horse-power per hour : . 

(a) Main cylinders, including jackets lbs. 

(b) Auxiliary cylinders lbs. 

(c) Engine and auxiliaries lbs. 

51. Percentage of steam used by main engine cylinders accounted 

for by indicator diagrams, near cut-off of high- 
pressure cylinder per cent. 



Additional Data. 

Add any additional data bearing on the particular objects of the test or 
relating to the special class of service for which the engine is used. Also 
give copies of indicator diagrams nearest the mean and the corresponding 
scales. 



Data and Results of Feed=water Test of Steam 
Engine. 

Arranged according to the Short Form advised by the Engine Test Com- 
mittee of the American Society of Mechanical Engineers. Code of 1902. 

1. Made by of 

on engine located at 

to determine 

2. Date of trial 

3. Type of engine (simple, compound, or other multiple-expansion ; con- 

densing or non-condensing) 

4. Class of engine (mill, marine, locomotive, pumping, electric, or other). 

5. Rated power of engine 

6. Name of builders 

7. Number and arrangement of cylinders of engine ; how lagged ; typej| 

of valves and of condensers 



* The horse-power referred to above (items 4G-50) is that of the main engine, 
exclusive of auxiliaries. 



Steam-engine Testing. 673 

1st Cyl. 2d Cyl. 3d Cyl. 

8. Dimensions of engine 

(a) Single or double acting 

(b) Cylinder dimensions : 

k Bore, in inches 

Stroke, in feet 

Diameter of piston-rod, in inches. 
Diameter of tail-rod, in inches. . . 

(c) Clearance, in per cent, of volume, 

displaced by piston per stroke : 

Head end 

Crank end 

Average 

(d) Ratio of volume of each cylinder 

to volume of high-pressure cyl- 
inder 

(e) Horse-power constant for 1 pound 

mean effective pressure and 1 
revolution per minute 

Total Quantities, Time, Etc. 

9. Duration of test hours. 

I 10. Water fed to boilers from main source of supply lbs. 

11. Water fed from auxiliary supplies : 

(a) lbs. 

(&) lbs. 

(c) lbs. 

12. Total water fed from all sources lbs. 

; 13. Moisture in steam or superheating hear throttle* per cent, or deg. 

I 14. Factor of correction for quality of steam 

| 15. Total dry steam consumed for all purposes lbs. 

Hourly Quantities. 

16. Water fed from main source of supply lbs. 

17. Water fed from auxiliary supplies : 

(a) lbs. 

(b) lbs. 

( c) lbs. 

18. Total water fed to boilers per hour lbs. 

19. Total dry steam consumed per hour lbs. 

20. Loss of steam and water per hour due to leakage of 

plant, drips, etc lbs. 

21. Net dry steam consumed per hour by engine and aux- 

iliaries lbs. 

22. Dry steam consumed per hour : 

(a) Main cylinders lbs. 

(o) Jackets and reheaters lbs. 

Pressures and Temperatures (Corrected). 

23. Steam-pipe pressure near throttle, by gauge lbs. per sq. in. 

24. Barometric pressure of atmosphere, in inches of mer- 

cury ins. 

25. Pressure in first receiver, by gauge lbs. per sq. in. 

26. Pressure in second receiver, by gauge lbs. per sq. in. 

27. Vacuum in condenser : 

(a) In inches of mercury ins. 

(&) Corresponding total pressure lbs. per sq. in. 

28. Pressure in steam jackets, by gauge lbs. per sq. in. 

1 29. Pressure in reheater, by gauge lbs. per sq. in. 

30. Superheating of steam in first receiver deg. Fahr. 

31. Superheating of steam in second receiver deg. Fahr. 



* In case of superheated steam engines determine, if practicable, the tempera- 
ture of the steam in each cylinder. 

43 



674 Steam-engine Testing. 

Indicator Diagrams. 

1st Cyl. 2d Cyl. 3d Cyl. 

32. Commercial cut-off, in per cent., of stroke. 

33. Initial pressure, in pounds, per square inch 

above atmosphere a 

34. Back pressure at mid-stroke above or be- 

low atmosphere, in pounds, per 
square inch 

35. Mean effective pressure, in pounds, per 

square inch 

36. Equivalent mean effective pressure, in 

pounds, per square inch per indi- 
cated horse-power 

(a) Referred to first cylinder. 

lb) Referred to second cylinder. 

(c) Referred to third cylinder. 

37. Pressures and percentages used in com- 

puting the steam accounted for 
by the indicator diagrams, meas- 
ured to points on the expansion 

and compression curves 

Pressures above zero, in pounds, per square 
inch: 

(a) Near cut-off 

lb) Near release 

(c) Near beginning of compression 

Percentage of stroke at points where press- 
ures are measured : 

(a) Near cut-off 

(5) Near release 

(c) Near beginning of compression 

38. Aggregate mean effective pressure, in 

pounds, per square inch referred 
to each cylinder given in heading 

39. Mean back pressure above zero 

40. Steam accounted foi, in pounds, per indi- 

cated horse-power per hour : 
(a) Near cut-off 

(6) Near release 

41. Ratio of expansion : 

(a) Commercial 

(6) Ideal 

Speed. 

42. Revolutions per minute rev. 

43. Piston speed per minute ft. 

Power. 

44. Indicated horse-power developed by main engine cylinders : 

First cylinder H. P. 

Second cylinder H. P. 

Third cylinder H. P. 

Total H. P. 

Efficiency Results. 

45. Dry steam consumed per indicated horse-power per hour : 

(a) Main cylinder, including jackets lbs. 

lb) Auxiliary cylinders, etc lbs. 

(c) Engine and auxiliaries lbs. 

46. Percentage of steam used by main engine cylinders accounted 

for by indicator diagrams : 

1st Cyl. 2d Cyl. 3d Cyl. 

(a) Near cut-off 

(6) Near release * 

Sample Diagrams. 

Copies of indicator diagrams nearest the mean, with corresponding 
scales, should be given in connection with table. 



Steam-engine Performance. 675 

Practical Engine Performances. 

(J. B. Stanwood.) 

NON=CONDENSING ENGINES. 

™ Slide= valve Engine.— 75 to 80 pounds boiler pressure; stroke, long; 
mean effective pressure, 33 to 38 pounds per square inch ; 25 to 100 horse- 
power ; cut-off, % stroke ; performance, about 40 pounds of steam per indi- 
cated horse-power per hour. When valves and piston are tight this has 
been reduced to 33 pounds of dry steam per indicated horse-power per 
hour by careful test. 

Automatic High-speed Engines with Single Valves.— 75 to 80 
pounds boiler pressure ; stroke, about equal to piston diameter ; mean 
effective pressure, 40 pounds per square inch ; 50 to 150 horse-power ; cut- 
off, 34 stroke ; performance, about 40 pounds of steam per horse-power per 
hour. When valves and piston are tight this has been reduced to 32 pounds 
of dry steam per indicated horse-power per hour. Valves difficult to keep 
tight. 

Automatic High-speed Engines with Double Valves.— 75 to 80 
pounds boiler pressure ; stroke, V/% to 2 times piston diameter ; mean effec- 
tive pressure, 40 pounds per square inch ; 50 to 150 horse-power ; cut-off, % 
stroke ; performance, about 35 pounds of steam per indicated horse-power 
per hour. When valves and piston are tight this has been reduced to 30 
pounds of dry steam per indicated horse-power per hour by careful test. 

Automatic Cut=off Engines of the Corliss Type. — Stroke, 2 to 3 
times diameter of piston ; 75 to 90 pounds boiler pressure ; mean effective 
pressure, 40 pounds per square inch ; cut-off, \ to 34 stroke ; performance, 
under 200 horse-power, 29 to 30 pounds of steam per indicated horse-power 
per hour, over 200 horse-power, 27 pounds of steam per indicated horse- 
power per hour. When valves and piston are tight this has been reduced 
to 23% pounds of dry steam per indicated horse-power per hour. 

Compound Engines.— High speed; automatic cut-off; short stroke; 
110 to 120 pounds boiler pressure ; mean effective pressure, 25 to 27 pounds 
per square inch ; 6 expansions ; 100 to 250 horse-power ; performance, 27 
pounds of steam per indicated horse-power per hour. 

CONDENSING ENGINES. 

Automatic Cut=off Engines of the Corliss Type.— Stroke, 2 to 3 
times piston diameter ; 70 to 80 pounds boiler pressure ; mean effective 
pressure, 40 pounds per square inch ; over 200 horse-power ; cut-off, | stroke ; 
about 19 to 20 pounds of steam per indicated horse-power per hour. 

Compound Engines.— High speed; automatic cut-off; short stroke; 
110 to 120 pounds boiler pressure ; mean effective pressure, 27 to 30 pounds 
per square inch ; 9 expansions ; 200 to 500 horse-power ; 17 to 19 pounds 
of steam per indicated horse-power per hour. 

Compound Automatic Cut-off Engines of the Corliss Type. — 
Stroke, on high-pressure cylinder, 2 to 3 times piston diameter ; 110 to 135 
pounds boiler pressure ; mean effective pressure, 14 to 24 pounds per square 
inch ; over 400 horse-power ; 16 to 20 expansions ; 14 to 17 pounds of steam 
per indicated horse-power per hour. In one or two special cases, 13% 
pounds of steam per indicated horse-power per hour has been obtained. 

Steam-engine Proportions. 

The dimensions of many of the parts of a steam engine may be deter- 
mined according to the general methods given in the section on Machine 
Design, pages 416-481, but some additional data will be given here. 

The following proportions are those recommended by James B. Stan- 
wood, M.E., and are based on an extensive practical experience. 

ENGINE PROPORTIONS. 
Pressures on Wearing Surfaces. 

] Main bearings : 140 to 160 pounds per square inch of area, obtained by mul- 
tiplying length by diameter of journal. 
Crank pins : 1000 to 1200 pounds per square inch of area, obtained by multi- 
plying length by diameter of pin. 



676 Steam-engine Proportions. 



Cross-head pins : 1200 to 1600 pounds per square inch of area, obtained by 

multiplying length by diameter of pin. 
Cross-head surface : 35 to 40 pounds per square inch of area. 

Non-condensing engines are usually designed for 100 pounds pressure 
per square inch of piston. 

Sizes of Engine Parts, in Relation to Piston. 

Diameter of piston. 

Main shaft, diameter 0.42 to 0.50 

Main bearing, length 0.85 to 1.00 

Crank pin, diameter 0.22 to 0.27 

Crank pin, length 0.25 to 0.30 

Cross-head pin, diameter 0.18 to 0.20 

Cross-head pin, length 0.25 to 0.30 

Piston-rod, diameter 0.14 to 0.17 

Area of steam ports : Area of piston. 

Slide-valve engine 0.08 to 0.09 

High-speed automatic engine 0.10 to 0.12 

Corliss engine 0.07 to 0.80 

Area of exhaust ports : 

Slide-valve engine 0.15 to 0.20 

High-speed automatic engine 0.18 to 0.22 

Corliss engine 0.10 to 0.12 

Diameter of steam pipes : 

Slide-valve engine, 34 diameter of piston to % diameter of piston + % 

inch. 
Automatic high-speed engine, % diameter of piston. 
Corliss engine, ■& diameter of piston. 
Diameter of exhaust pipes : 

Slide-valve engine, % diameter of piston. 
Automatic high-speed engine, % diameter of piston. 
Corliss engine, % to % diameter of piston. 

Displacement of piston 
Clearance spaces : in one stroke. 

Slide-valve engine 0.06 to 0.08 

Automatic high-speed engine, single valve 0.08 to 0.15 

Automatic high-speed engine, double valve 0.03 to 0.05 

Automatic cut-off engine, Corliss type, long stroke 0.02 to 0.04 

Weights of engines per rated horse-power : 

Slide-valve engine 125 to 135 pounds 

Automatic high-speed engine 90 to 120 pounds 

Corliss engine 220 to 250 pounds 

Fly-wheels, weight per rated horse-power : 

Slide-valve engine 33 pounds 

Automatic high-speed engine (according to size and 

speed) 25 to 33 pounds 

Corliss engine (according to size and speed) 80 to 120 pounds 

Rules for Fly-wheel Weights, Single=cylinder Engines. 

Let d = diameter of cylinder, in inches ; 

S = stroke of cylinder, in inches ; 
D = diameter of fly-wheel, in feet ; 
R = revolutions per minute ; 
W= weight of fly-wheel, in pounds. 

d-S 
For slide-valve engines, ordinary duty, W= 350 000 -f^—^ ; 

For slide-valve engines, electric lighting, W = 700 000 Tlt< p^r ; 

For aul omatic high-speed engines, W = 1 000 000 ; 

For Corliss engines, ordinary duty, W ' = 700 000 ; 

JJ-Hr 

For Corliss engines, electric lighting, W= 1 000 000 ~ OTV> ; 



Steam-engine Propoetions. 



677 



Steam Passages. 

The dimensions of steam passages should be proportioned, when possi- 
L ble, so that the velocity of flow is not greater than 6000 feet per minute, 
^»but this is not always practicable. The following table will enable the 

diameters of steam pipes and the areas of steam ports to be computed for 

various velocities. 

Steam=pipe Diameters and Port Areas. 









Velocity of steam, 


in feet, 


per minute. 






o 


4000 


6000 


8000 


10000 


12000 


1 


a « • 


^ 


a a • 


i—i 


a r. • 


A 


a - . 


^ 


a d • 


rH 




3§r; 


II 


M^ 


II 


-3 £ i—i 


II 


* s ^ 


II 


!§■; 


II 




ft:H o 


f-* £ 


*;g || 

ft ^ © 


£ o3 

P 

of % 


^ to || 

ft^£ 


P rt 


*.« ii 

ft»~ 

I*! 


"S3 o3 


^2 'I 

a'pU 

'ft£ © 


£ o3 
03~£ 

2 3 


c 2 


BcB 


3 o 


BuB 


08 S 


B Cs 


So 


a 2~b 


c3 o 


a ^a 


c3 O 


°*S 


& S $ 


£.S 


*££ 




d o £ 




oS © ,o3 




$&£ 


"E.S 


.5 a 


B «T3 


o ft 


g <D ~J 


O 'p, 


£vv 


o'a 


-2 © >r 3 


O 'ft 


£%% 


O ft 


p- 


HI 


ft 


CO 


ft 


m 


ft 


CO 


ft 


m 


ft 


100 


.158 


.025 


.129 


.017 


.112 


.013 


.100 


.010 


.091 


.008 


125 


.177 


.031 


.144 


.021 


.125 


.016 


.112 


.013 


.102 


.010 


150 


.194 


.037 


.158 


.025 


.137 


.019 


.123 


.015 


.112 


.013 


175 


.209 


.044 


.171 


.029 


.148 


.022 


.132 


.018 


.121 


.015 


200 


.224 


.050 


.183 


.033 


.158 


.025 


.141 


.020 


.129 


.017 


225 


.237 


.056 


.194 


.038 


.168 


.028 


.150 


.023 


.137 


.019 


250 


.250 


.063 


.204 


' .042 


.177 


.031 


.158 


.025 


.144 


.021 


275 


.262 


.069 


.214 


.046 


.185 


.034 


.166 


.028 


.151 


.023 


300 


.274 


.075 


.224 


.050 


.193 


.038 


.173 


.030 


.157 


.025 


325 


.285 


.081 


.233 


.054 


.201 


.041 


.180 


.033 


.164 


.027 


350 


.296 


.088 


.242 


.058 


.209 


.044 


.187 


.035 


.171 


.029 


375 


.306 


.094 


.250 


.063 


.217 


.047 


.194 


.038 


.177 


.031 


400 


.316 


.100 


.258 


.067 


.224 


.050 


.200 


.040 


.183 


.033 


425 


.326 


.106 


.266 


.071 


.231 


.053 


.206 


.043 


.188 


.035 


450 


.335 


.113 


.274 


.075 


.238 


.056 


.212 


.045 


.193 


.038 


475 


.344 


.119 


.281 


.079 


.244 


.059 


.218 


.048 


.199 


.040 


500 


.353 


.125 


.288 


.083 


.250 


.063 


.224 


.050 


.204 


.042 


525 


.362 


.131 


.295 


.088 


.256 


.066 


.229 


.053 


.209 


.044 


550 


.371 


.138 


.302 


.092 


.262 


.069 


.235 


.055 


.214 


.046 


575 


.380 


.144 


.309 


.096 


.268 


.072 


.240 


.058 


.219 


.048 


600 


.388 


.150 


.316 


.100 


.274 


.075 


.245 


.060 


.224 


.050 


625 


.395 


.156 


.323 


.104 


.279 


.078 


.250 


.063 


.228 


.052 


650 


.403 


.163 


.329 


.108 


.285 


.081 


.255 


.065 


.232 


.054 


675 


.411 


.169 


.335 


.113 


.290 


.084 


.260 


.068 


.237 


.056 


700 


.418 


.175 


.341 


.117 


.296 


.088 


.265 


.070 


.241 


.058 


725 


.426 


.181 


.347 


.121 


.301 


.091 


.269 


.073 


.246 


.060 


750 


.433 


.188 


.353 


.125 


.306 


.094 


.274 


.075 


.250 


.063 


775 


.440 


.194 


.359 


.129 


.311 


.097 


.278 


.078 


.254 


.065 


800 


.447 


.200 


.365 


.133 


.316 


.100 


.283 


.080 


.259 


.067 


825 


.454 


.206 


.371 


.137 


.321 


.103 


.287 


.083 


.262 


.069 


850 


.461 


.213 


.376 


.141 


.326 


.106 


.292 


.085 


.266 


.071 


875 


.468 


.219 


.382 


.145 


.331 


.109 


.296 


.088 


.270 


.073 


900 


.474 


.225 


.388 


.150 


.336 


.113 


.300 


.090 


.274 


.075 


925 


.481 


.231 


.393 


.154 


.340 


.116 


.304 


.093 


.277 


.077 


950 


.487 


.238 


.398 


.158 


.344 


.119 


.308 


.095 


.281 


.079 


975 


.494 


.244 


.403 


.162 


.349 


.122 


.312 


.098 


.285 


.081 


1000 


.500 


.250 


.408 


.166 


.353 


.125 


.316 


.100 


.289 


.083 


1025 


.506 


.256 


.413 


.170 


.357 


.128 


.320 


.103 


.292 


.085 


1050 


.512 


.263 


.418 


.175 


.361 


.131 


.324 


.105 


.295 


.088 


1075 


.518 


.269 


.423 


.179 


.365 


.134 


.328 


.108 


.299 


.090 


1100 


.524 


.275 


.428 


.183 


.37 


.138 


.332 


.11 


.303 


.092 



678 



Valve Gears. 



Valve Gears. 

The admission of the steam at the proper time to the cylinder may be 
effected by various forms of valve gear. 

The plain slide valve, operated by a single eccentric, is shown diagram- ' 
matically in the accompanying illustration. 





When the valve is so made that it just covers both ports when in the 
mid position, and the eccentric travel is just equal to twice the width of 
the port, there will be no expansion, the eccentric being placed exactly at 
right angles with the crank. It was soon found, however, that by making 
the valve with increased lap and by giving the eccentric more throw a 
certain degree of expansion could be obtained, together with an earlier 
release and compression, this resulting in better steam economy and 
smoother running. 

In order to accomplish this result without impeding the exhaust of the 
steam, the eccentric, r u must be given the so-called angle of advance, 
2° 1 . 2', beyond the mid position. The direction of rotation of the crank 
is then governed by this angle, the arrangement above giving rotation to 
the left, and the position, 1 . 2" for ri, giving right-hand rotation. 

The action of the slide valve may readily be represented graphically by 
use of Reuleaux's diagram. The angle of advance and lap being given, 
the point of cut-off can be determined by the following method : 




The circle, 1C°, represents the circle of the eccentric, and may also be 
taken as the crank circle on a reduced scale. C" and C" are two sym- 
metrically-placed positions of the piston at which it is desired that the 
cut-off shall take place. Through these points, with a radius 1 . 3 = I, 



Valve Diagrams. 



679 



i 



Co- 



describe arcs from, centres, 3" and 3'". Their intersections, E 2 and E s , 
with the circle give the angles at which the expansion, C°C" and C C", 
occurs,— in this instance T 7 o of the stroke. We now select the point, v 2 , of 
the crank circle at which the admission shall begin, join F2JE0, and draw 
the equator, 2.1.2', parallel to it, and the angle, 2 . 1 . C', will be the 
angle of advance, S, and the distance of 2 . 1 from E 2 V 2 , the outside lap, 
e 2 , for the port II. The width of port, a, must also be chosen, and must be 
sb taken that it is less than 7\ — e 2l and is represented by the parallel, A 2 . 
When the crank reaches I 2 ,—m this instance at T % 8 o of the stroke, — the 
exhaust begins, and the distance, i 2 i 2 , of the parallel, I 2 I 2 , from the equator 
is the inside lap. 

The construction is similar for the other half of the stroke. The angle, 
5, is already known, and hence the parallel, E% F 3 from E 3 , can be at once 
drawn and the admission point, F 3 , determined. The outside lap, e 3 , is 
somewhat less than e 2 , thus giving a correspondingly wider port opening. 
The inside lap, i s , is made equal to i 2 , and the bridges, b s and b 2 , are made 
equal, thus giving a symmetrical valve seat. A certain amount of dis- 
cretion is permissible in the selection of b 2 = 63, care being taken that there 
is sufficient bearing at the extreme valve stroke to insure tightness. The 
points, I 2 ' and J 3 ', are also of importance, as they determine the closing of 
the exhaust. The corresponding piston positions, C* v and C^ 7 , are not 
symmetrical, because i 3 = i 2 ; but the inequality in the compression is not 
serious. 

The above method of consider- 
ing the influence of the ratio — 
r 
is very simple. It is easy to sub- 
stitute any desired ratio — , but 

the variation is slight. It must be 
noted that the distance, 1 . 3, must 
be laid out to the actual scale of 
construction. 

The application of Zeuner's 
diagram to the same case is made 
in the following manner: The 
circle, lCb, represents, as before, 
the eccentric circle and the crank- 
pin path. The angle C . 1 . 2 = 
C . 1 . 2 = 90 — 8. With 1 as a 
centre describe circles with radii 
e and i, here made alike for both 
ends of the valve; also, one of 

radius e + a. Upon 1 . 2 and 1 . 2 as diameters describe circles, called the 
valve circles. 

The intersection of radii from 1 with these circles gives the distance of 
the valve from its middle position for various crank positions. For the 
position 1 V 2 , for instance, the admission for the left stroke begins, at 1E 2 
the expansion, at 1/the exhaust, etc. 

The Zeuner diagram gives the valve position by means of polar coordi- 
nates, while Reuleaux's diagram is based on parallel coordinates. To be 
strictly correct, the valve circles, 1 . 2 and 1 . 2', of the Zeuner diagram 
should fall upon each other. The arrangement shown has been adopted 
by Zeuner as more convenient in practice. 

It will be seen from the preceding that the rate of expansion can be 
varied by altering the eccentricity and the angle of advance. This may 
be carried so far that the direction of rotation is changed, giving what is 
termed a reversing motion. A variety of reversing motions have been 
devised, which accomplish the desired relation of parts by shifting a re- 
versing lever. Of these the most practical are the so-called link motions, of 
which a number will here be briefly shown. 

No. 1 is an outline diagram of Stephenson's link motion. The link, 3'3 ff , 
of convex curvature towards the valve, is given an oscillating motion by 
means of the two equal eccentrics, 1 . 2 f and 1 . 2", and is suspended from 
its middle point, 7, from the bell crank lever, S7'. The motion of the link 
is transmitted to the valve by means of the sliding block, 5, and rod, 6. 
No. 2 is Gooch's link motion. The link, 4, is driven by two eccentrics, as 




680 



Link Motions. 



before, but is curved in the opposite direction with a radius, 5 . 6, and is 
suspended from its middle point, 8, to a fixed pivot, 8', while the rod, 5 . 6, 
is shifted by means of the lever connection, S10 . 10'. 




No. 1. 



No. 2. 



No. 3 is the link motion of Pius Fink. In this form the link is operated 
by a single eccentric instead of two, as in the previous forms. This simple 
mechanism is not as widely used as its merits deserve. 




No. 3. 



No. 4 is the link motion of Allen. In this design the link, 4, is straight, 
and both the link and the radius rod are suspended and shifted by the 
lever connections, 8' . 8 and 9' . 9. 




No. 5. 



No. 6. 



No. 5 is Walschaert's link motion. The link, 4, vibrates upon a fixed 
centre, 9, and is operated by an eccentric, 1 . 2. The valve rod is moved 
from the main cross-head by the connections, 10 . 11 . 6 . 7, and also by the 
radius rod, 5 . 6, which latter is suspended from the bell crank, S . 12'. 

No. 6 is Marshall's valve gear. The curved link, 4, is rigidly secured and 
does not move. The eccentric, 1 . 2, moves the valve connection, 6 . 7, by 
means of the lever, 2.3.6, which vibrates about the point, 3, on the end 
of the radius rod, the other end of the rod being held by the link block, 5. 
Instead of the link, 4, a radius arm, 4 . 5, is often used, the centre, 4o, corre- 
sponding to the centre of curvature of the link, the action being the same 
in both cases. 

No. 7 is Brown's valve gear, which differs from the preceding by the 
substitution of a straight link of adjustable angle for the curved guide 
link. 

No. 8 is Angstrom's valve gear. The point, 3, of the preceding gear is 
guided by a parallel motion, and the point, 6, is between 2 and 3, instead 
of beyond. 



Slide Valves. 



681 



The eight preceding valve gears operate the valve approximately in the 
same manner as if a single eccentric of variable eccentricity and angular 




No. 7. 



No. ! 



advance were used, the eccentric rod being assumed of infinite length as 
compared with r. The path of the successive positions of the middle point 
of this imaginary eccentric is called the central curve of the valve gear. 








^--|-- 2" 




The general forms of the central curve are shown above. Form a is that 
for cases 1, 4, and 5 ; form b, for case 1, when the eccentric rods are crossed ; 
and form c, in which the curve becomes a straight line, is for cases 2, 3, 6, 
7, and 8. In the latter instance the lead is constant. 

The use of the central curve is involved in the mechanism of the valve 
gear of the single-valve automatic cut-off engines, in which the eccentric 
is shifted across the shaft by the action of a centrifugal or inertia governor. 



Slide Valves. 

The two principal forms of slide valves in use are the plan D valve and 
the Allen valve, the latter being designed to give a more rapid and full 
port opening. 

The action of the slide valve has already been discussed, and the 
amount of inside and outside lap may be determined for the desired steam 



W M 




f.&j.cuufc-* — \a d -4-b^a^bj] 



n^ IV 



distribution by the use of Reuleaux's or Zeuner's diagrams. The other 
dimensions are determined as follows : 

The width, a, of the steam ports is kept as small as is practicable, while 
the length at right angles to the plane of the drawing is made quite large. 
When a is given, the dimensions to be determined are the outside and 



682 



Slide Valves. 



inside lap, e and i ; the bridges, b ; the width of face, 6 , beyond the ports ; 
the width, a , of the exhaust port, IV; the travel, r ; the length of the 
valve, I ; and of the valve seat, l . The laps, e and i, are determined from 
the valve diagrams. 

In the same manner, also, is found the greatest distance, s, in which the 
edge of the valves passes the edge of the port. This gives the width of 
bearing, t, of the valve upon the bridge, since b = s + t. The value of t 
varies greatly ; the least permissible value is t = T V', and it is more fre- 
quently made %" to y 2 ". Approximately, we have, after assuming t as 
just given, a® -f t — (e -f- a + i) = a. We then have 



whence 
and 



a = 2a -f I -f i — t, 
r = a + e -f- s, 

I = 4a + SI + i + 2s + t 



The valve face must have an inner width of bearing, to (Fig. b), at least 
equal to t y whence for the total width of the valve face we have the value 

a + 26 + 2a + 2b , or 

l = 4a + 3e — i + 4s + t + 2^. 

The thickness of metal in the valve itself, when made of cast-iron, 

should be about = — — - + 0.4". 




(Jtr*<— bo-A 



The Allen Valve. 

This is a double valve, and consists of one D valve over another, with 
a steam passage between. As before, we have r = a + *e + s, and also make 
5 = 2e — t, — i.e., the inner edge of the outer valve, when the valve is in 
mid position, is at a distance = e from the edge of the valve seat. The 
consequence is that when the valve is moved a distance equal to e, say to 
the right, the passage through the valve opens to admit steam at the same 
instant as does the edge of the valve on the left. This gives a steam admis- 
sion twice as quickly, and an opening twice as great, as would otherwise 
be the case. 

The following positions, from a to /, will show the successive actions, the 
exhaust ports being omitted for simplicity. 

a. The admission is just about to take place both from the edge of the 
valve on the left and through the passage in the valve. If we apply 
Zeuner's diagram, we must, from the point A, which indicates the port 
opening, double the width given by the Zeuner circle until the entrance 
to the passage in the valve is wide open, as at b. By thus doubling the 
opening in the diagram we obtain the curve, AB\. 

b. From this position on, the opening at the left continues to grow 
wider, but that through the valve on the right does not; hence, on the 
Zeuner diagram, from this point we return to the opening which the 
regular valve circle gives, to which is added the constant opening, c = 
BB X = CC\, indicated by the curve, B\C\. This continues until the inner 
edge of the opening of the valve passage on the left reaches the edge of 
the bridge, as at c. 



Slide Valves. 



683 



L 



c. As the valve continues to move, the passage through it is gradually 
closed, but the steam port is opened to the same amount, and hence the 
actual port opening remains constant. This continues until the position, d, 
is reached, when the passage through the valve is entirely shut off. This 
is indicated in the diagram by the arc, CiD, struck from the centre at 1. 




d. The valve continues to move to the right until it is entirely upon the 
bridge, the corresponding portion of the diagram being the arc, DE, of 
the valve circle. 

e. The valve from this position moves on the bridge beyond the port 
until it has travelled a distance equal to s, as shown at/, during which time 
the port opening remains constant, as indicated in the diagram by the arc, 
EE f , struck from the centre, 1. From this point the same actions take 
place successively in the reversed order. 

It will be seen that Allen's valve gives a much quicker opening and 
also a much longer duration of the full opening than does the plain slide 
valve. It remains to be seen how these features can be used to the best 




advantage. This is best done by making the value of s negative, and 
also > t. This makes the port opening from C\ to Ci' in the diagram con- 
stant, as shown in the diagram. 

In order that the apparent contraction of the ports by the change in the 
sign of s shall not occur, the value of a is made greater than would other- 



684 



Condensers. 



wise be the case. Under these conditions we have for the exhaust port, Oq, 
the equation : 

Oq -\- 1 — e\ — a — i = a — s, 

in which s is given the magnitude equal to the distance which the edge of , 
the valve is moved beyond the edge of the bridges, as in Fig. /. We then 
have 

For the exhaust port, a = 2a + e\ + i — s — t ; 

For the bridge, b — e — e x + s — t ; 

For the passage through valve, c = e — t — e x ; 

For the total valve, I = 4a + 4e — e\-\-i — 3s + t. 

For a complete discussion of valves and valve gears see Zeuner's 
" Treatise on Valve Gears" and Auchincloss's " Link and Valve Motions ;" 
also, compare Reuleaux's "Constructor" and Unwin's "Machine Design." 



CONDENSERS. 

The gain in power by use of a condenser may be estimated upon the 
basis of an increase of 12 pounds per square inch to the mean effective 
pressure in the cylinder. Upon this basis the following table shows the 
gain in horse-power for cylinders of various diameters for every 100 feet 
piston speed. For any other speed, multiply by the speed, in feet, per 
minute and divide by 100 to obtain the gain in horse-power. 



Diameter of 
piston. 


Horse-power gained for 

every 100 feet of piston 

speed per minute. 


Diameter of 
piston. 


Horse-power gained for 

every 100 feet of piston 

speed per minute. 


5 


.71 


32 


29.24 


6 


1.03 


34 


33.01 


7 


1.40 


36 


37.01 


8 


1.83 


38 


41.24 


9 


2.31 


40 


45.70 


10 


2.86 


42 


50.38 


12 


4.11 


44 


55.29 


14 


5.60 


46 


60.43 


16 


7.31 


48 


65.80 


18 


9.25 


50 


71.40 


20 


11.42 


52 


77.23 


22 


13.82 


54 


83.28 


24 


16.45 


56 


89.56 


26 


19.30 


58 


96.08 


28 


22.39 


60 


102.81 


30 


25.70 







The size of a let condenser varies somewhat according to the speed of 
the engine, but is usually made from % to y 2 the volume of the steam cyl- 
Inder. The quantity of injection water required is from 25 to 30 times the 
weight ot steam to be condensed, according to the pressure of the exhaust 
and the temperature of the water. Too much water is poor economy, 
since the Increased burden on the air pump neutralizes the gain of the 
better vacuum, [tisbesl to provide for an ample supply, in case of emer- 
gency.and cut the injection down in actual use until the minimum amount 
to maintain a fair vacuum is ascertained. 

The temperature of the hot well Is best kept at about 100° F., although 
it sometimes may rise to 120° F. without materially impairing the vacuum. 



CONDENSEKS. 



685 



For surface condensers a cooling surface of 2 to 3 square feet per indi- 
cated horse-power is found satisfactory in practice, according to climate. 
The quantity of circulating water may be taken at about 30 times the 
weight of steam to be condensed. 

The size of the air pump may be determined by the following formula : 



Volume of air pump = 



I. H. P. 

revolutions 



Xc, 



where c = 700 for single-acting and jet condenser ; 
= 300 for single-acting surface condenser ; 
= 470 for double-acting horizontal pump ; 



or volume of single-acting air pump = 



volume of low-pressure cylinder 
23 ' 



Independent condensers are now extensively used. The following gen- 
eral dimensions are for standard designs, the Worthington being a jet 
condenser and the Wheeler a surface condenser. 



Sizes of Worthington Jet Condenser. 



a 


^ 




M 




d he 










o 
t-t 
oq 


o3 

^a 


o 
© 


"Sb.2 

o 3 

© s 

«M Pi 


©' 
Pi 

«w"p\ 


©' 

Pi 

^ Pi 


O a; 


O OQ 




° Si 


O ft 


o o 


° rT 


o © 


®.2 


®ii 


© 


© <M 


S-. +g 


fe/g 


^ d 
§■2 


© Jj 


-M T3 


-u t3 


,d 


IS O 


+; S q, 






■+j o3 


5 d 


2 d 




© 


5 e8 d 


° ce 


"© "o 


2>.d 


Bx 


a^ 


"So 


a §, 


3~ S 


a~ 


a. 2, 


a g 


eS >> 


eS >i 


d 


o3-£ 


e8 X S 


oS X 


os d 


<^^ 


•rt O 


■■H © 


© 


•-" Pi 


•rH O Pi 


— o> 




— <t3 


ft 


ft 


ft 


ft 


ft 


A 


ft" 


ft 


Inch. 




Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


5MX 4% > 


: 5 


% 


1M 


4 


2K. 


2 


6 X 5% > 


: 6 


1 


IK 


5 


3 


3 


?K X 7^ > 


: 6 


IK 


• 2 


8 


4 


4 


7% X 7 > 


: io 


IK 


2 


10 


4 


4 


7K x sy 2 > 


: io 


IK 


2 


12 


5 


5 


?K x i<h > 


: io 


IK 


2 


14 


7 


6 


9 X 12 > 


: io 


2 


2K 


14 


7 


8 


12 X 14 > 


: io 


2K 


3 


16 


8 


10 


12 X 15 > 


c io 


2K 


3 


16 


8 


10 


12 X 15 > 


< 15 


2K 


3 


18 


10 


10 


12 X 17 > 


C 15 


2K 


3 


18 


10 


12 


14 X 17 > 


a 15 


2K 


3 


18 


10 


12 


12 X 19 > 


C 15 


2K 


3 


18 


10 


12 


14 X 19 > 


C 15 


2K 


3 


18 


10 


12 


17 X 19 > 


C 15 


3 


4 


18 


10 


12 


14 X 22 > 


C 15 


3 


4 


20 


10 


14 


17 X 22 > 


( 15 


3 


4 


20 


10 


14 


18 X 22 > 


< 18 


3 


4 


20 


10 


14 


18 X 24 > 


( 18 


3 


4 


20 


10 


14 


18 X 26 > 


( 18 


3 


4 


24 


12 


16 


18 X 29 > 


C 18 


3 


4 


24 


14 


18 



686 



Condensers. 



Wheeler Surface Condenser, Mounted on Blake= 
Knowles Air and Circulating Pump. 









■.< 


Steam per hour. 


Cooling surface. 


Size of cylinders : steam, 
air, water. Stroke. 


Weight of outfit. 


Lb. 


Sq. ft. 


Inch. 


Lb. 


500 


80 


4X5X5X5 


1200 


800 


110 


4X5X5X5 


1350 


1000 


150 


4KX 5KX &y 2 X 6 


1700 


1500 


180 


4%X 5^X 5^X 6 


2000 


1800 


200 


5^X 6 X 6 X 7 


2600 


2000 


210 


5^X 6 X 6 X 7 


2700 


2250 


230 


5^X 6 X 6 X 7 


2800 


2500 


270 


6X8X8X7 


3300 


3000 


310 


6X8X8X7 


3500 


3500 


360 


6X8X8X7 


3600 


4000 


430 


7K X 8 X 8 X 10 


4600 


4500 


480 


iy 2 X 8 X 8 X 10 


4800 


5000 


530 


7^ X 8 X 8 X 10 


5600 


6000 


610 


8 X 9 X 9 X 10 


6000 


7000 


740 


8 X 9 X 9 X 10 


6300 


7500 


770 


8 X 10 X 10 X 12 


6900 


8000 


850 


8 X 10 X 10 X 12 


7200 


9000 


900 


8 X 10 X 10 X 12 


9100 


10500 


1000 


10 X 12 X 12 X 12 


9600 


11000 


1050 


10 X 12 X 12 X 12 


10700 


12000 


1200 


10 X 12 X 12 X 12 


13100 


14000 


1400 


12 X 14 X 14 X 12 


17000 


16000 


1600 


12 X 14 X 14 X 16 


19000 


18000 


1800 


14 X 16 X 16 X 16 


19800 


20000 


2000 


14 X 16 X 16 X 16 


20500 


22500 


2100 


14 X 16 X 16 X 16 


24000 


25000 


2360 


16 X 16 X 18 X 24 









Separate condensing plants have the advantage that they can be started 
before the main engines, and thus permit a vacuum to be secured at once, 
without blowing through. The speed of air and circulating pumps can be 
regulated according to the vacuum, which is not the case when they are 
operated by direct connection to the main engine. 

I" modem power plants, where there are many engines, pumps, and 
Other steam-driven auxiliaries, it is found advantageous to provide one 
large central condensing plant, with independent air and water pumps, 
into which all the engines discharge their exhaust. When compound 
engines are used the exhaust steam from pumps and similar machines in 
which the steam [s not used expansively mav well be discharged into the 
receiver ol the engine, being thus enabled to exert its expansive force upon 
the low-pressure piston, and then pass into the condenser. In this way 
much of the wastefulness of such auxiliaries may be prevented. « 



Internal-combustion Motors. 



687 



1 



INTERNAL=COMBUSTION MOTORS. 

Practically all of the internal-combustion motors now in active use are 
operated on the Beau de Rochas cycle, with a power impulse every fourth 
stroke. The sequence of operations is shown in the cuts, the correspond- 
ing portion of the indicator diagram being given in each case. 



/ / 1 \ \ 




m In the first outward stroke the mixed charge of air and gas is drawn 
in, and on the return stroke this is compressed. It is then ignited by an 
electric spark, hot tube, or similar device, and the expansion due to the 
explosion and combustion makes the second outward stroke,— this being 
the power stroke. The fourth phase in the cycle, the second inward stroke, 
is the exhaust. 

It is advantageous to use as high a compression pressure as possible, 
but the limit to this is found in the heat generated by compression. If 
the compression is too high the charge will be ignited by this heat and an 



688 



Internal-combustion Motors. 



injurious premature explosion occur. Various attempts have been made 
to obviate this difficulty. In the Banki engine a fine spray of water is : 
injected into the inlet pipe with the charge, and this absorbs much of the / 
heat of compression. The vapor of water thus produced expands with : 
the explosion, and there is thus a combined gas and steam action. In the J 
Diesel motor the charge drawn in is pure air, and this is compressed to 
about 500 pounds per square inch. A high temperature is thus produced, 
but there is no fuel in the cylinder to be ignited. At the end of the stroke 
the liquid fuel is injected and is ignited by the heat of the compressed air. 
In ordinary gas engines the compression is carried from 80 to 90 pounds 
per square inch. The maximum pressure in such engines is about 3.5 times 
the compression pressure. For compressions of 100 pounds per square inch 
or less the mean effective pressure may be obtained from the following 
formula : 

M.E.P. =2C-0.01C2, 

in which Cis the compression pressure. Thus, for 50 pounds compression, 
this would give 

M. E. P. = 100 — 25 = 75 pounds. 

Piston speed should not exceed 700 feet per minute,— more generally 500 
feet per minute is used. Maximum pressure should not be reached later 
than at one-tenth the stroke. The time of rise in pressure in a gas engine 
is : first, time taken for flame to strike back into the mixture ; second, time 
during which pressure rises after ignition. Cylinders of large dimensions 
have much larger ratio of volume to surface than small ones, and are 
therefore more economical. The size of valves should be such that the 
velocity of gases calculated upon mean piston speed does not exceed 100 
feet per minute. 

Internal-combustion motors have a much higher thermal efficiency 
than steam engines, on account of the greater temperature range and, 
also, because of the absence of losses from cylinder condensation, owing 
to the fact that the working fluid is a perfect gas. 

Gas engines frequently show on test thermal efficiencies of 22 to 25 per 
cent., while the Diesel motor has given a thermal efficiency of 38 per cent. 

The general proportions of gas-engine parts may be determined accord- 
ing to the general principles of machine design. There are, however, 



!c 



336.8 351.5' 




Gas-engine Diagram. 



certain parts winch may be given special consideration. Since gas engines 
may be used with tm 'Is of various calorific values, it is necessary to assume 
some standard upon which proportions may be based, and in the United 
States ii to often assumed that natural gas is the standard fuel, its calorific 
value being aboul L000B.T.U. per cubic foot. For gas of any other calorific 
value, a genera] rule is to make the compression ratio inversely as the 
caloriflc value oi the gas. Thus Eor a lean gas a higher degree of compres- 
sion will be required, and, although less power will be developed than 
w ith a richer gas, the thermal efficiency may be as high or even higher. 

For natural gas the compression space is made about 30 percent, of the 
piston displacement, so that the total volume of cylinder and clearance is 






Internal-combustion Motors. 689 

1.30 of the piston displacement, and the ratio of the clearance to the total 

volume is ' = 0.2308. 

Upon this assumption a typical gas-engine indicator diagram may be 
onstructed, from which the action in the cylinder may be seen. 

The following discussion is condensed from Roberts's "Gas-engine 
Hand-book." 

The compression curve has been found experimentally to be represented 
by the relation 

PV 1S *= K, 

in which P is the absolute pressure at any point ; V, the corresponding 
volume ; and K, a constant. If the volume of the cylinder is taken as 
unity, K is the absolute pressure of the atmosphere, or 14.7 pounds per 
square inch. 

With natural gas the pressure of explosion is about 4 times the com- 
pression pressure, both compression and explosion pressures being con- 
sidered above atmospheric. 

For the expansion curve the relation of pressure to volume is 

p F 1.35 = C, 

in which C is a constant depending upon the maximum pressure of 
explosion. 

To find the compression pressure with a clearance ratio of 0.2308, as 
determined above, we have 

pjn.3 = jt=14.7, 

14.7 14.7 

P = -^ = (02308)1 , 3 = 98.88 pounds. 

This is absolute pressure, and the pressure above atmospheric will be 

98.88 — 14.7 = 84.2 pounds per square inch. 

The explosion pressure will then be 84.2 X 4 = 336.8 pounds above atmos- 
phere, or 351.5 pounds absolute. Other points in the compression curve 
may then be computed by the formula. 

To apply the formula for the expansion curve, 

P 171.35 = Q 

the value of Cmust be found. This is the pressure at the end of the stroke 
when the volume is equal to 1 ; hence, we have 

p 171.35 = 351.5 x (0.2308) 1 - 35 = C, 
= 48.56 pounds absolute, 

as the terminal pressure. 

Intermediate points in the expansion curve may then be found, as 
shown in the diagram, from 

P 1/1.35 = 48.56. 

The mean effective pressure may then be measured from the diagram,—- 
preferably by the use of the planimeter. 

The power of the gas engine is generally determined by means of the 
brake, and the dimensions of parts are based on brake horse-power 
(B. H.P.). 
The brake horse-power may be expressed in general by the formula 

B . H ,P. = gX|Xj , 

in which 

D = diameter of cylinder, in inches; 
* L = stroke, in inches ; 

p = revolutions per minute ; 
G= constant, depending upon the fuel. 

For a four-cycle engine C may be taken as 19.000 for natural gas and 
18,000 for gasoline. The value of C may be determined from any engine in 

44 



690 Internal-combustion Motors. 

which the brake horse-power has been found, and then this value can be , 
used for subsequent computations with the same fuel. 1 

The stroke is usually made equal to 1.5D, and the piston speed about £ 
600 feet per minute. 

For the inlet and the exhaust passages we have 



S = piston speed, in feet, per minute ; 
A = piston area, in square inches ; 
a = inlet area ; 
a' = exhaust area. 

^ AS 

a- 6000 ; 

, _ AS 

a ~5100* 









The flow of water through the cylinder jacket is made 4 to 5 gallons 
per horse-power per hour. 

The 1902 Code of the American Society of Mechanical Engineers in- 
cludes the following : 

Rules for Conducting Tests of Gas and Oil Engines. 

Code of 1901. 

I. Objects of the Tests.— At the outset the specific object of the test 
should be ascertained, whether it be to determine the fulfilment of a con- 
tract guarantee, to ascertain the highest economy obtainable, to find the 
working economy and the defects as they exist, to ascertain the perform- 
ance under special conditions, or to determine the effect of changes in the 
conditions ; and the test should be arranged accordingly. 

II. General Condition of the Engine.— Examine the engine, and 
make notes of its general condition and any points of design, construction, 
or operation which bear on the objects in view. Make a special examina- 
tion of all the valves by inspecting the seats and bearing surfaces, and 
note their condition, and see if the piston rings are gas-tight. 

If the trial is made to determine the highest efficiency, and the exami- 
nation shows evidence of leakage, the valves and piston rings, etc., should 
be made tight and all parts of the engine put in the best possible working 
condition before starting on the test. 

III. Dimensions, etc. — Take the dimensions of the cylinder, or cylin- 
ders, whether already known or not. This should be done when they are 
hot, and in working order. If they are slightly worn, the average diam- 
eter should be determined. Measure, also, the compression space or 
clearance volume, which should be done, if practicable, by filling the 
spaces with water previously measured, the proper correction being made 
for the temperature. (See Section III., Steam-engine Code.) 

IV. Fuel.— Decide upon the gas or oil to be used, and, if the trial is to 
be made tor maximum etliciency, the fuel should be the best of its class 
that can readily he obtained, or one that shows the highest calorific power. 

(See Section IV., Stcimi-eiigine Code.) 

V. Calibration of Instruments Used in the Tests.— All instruments 
and apparatus should be calibrated and their reliability and accuracy veri- 
fied by comparison with recognized standards. Apparatus liable to change 
or to become broken during the tests, such as gauges, indicator springs, 
and thermometers, should be calibrated both before and after the experi- 
ments Thf accuracy of all scales should be verified by standard weights. 
In the case of gas- or water-meters, special attention should be given to 
their calibration, both before and after the trial, and at the same rate of ^ 
How and pressure a- exists during the trial. 

VI. Duration of Test.— The duration of a test should depend largely 
upon its Character and the objects in view, and in any case the test should 
be continued until the successive readings of the rates at which oil or gas 



Gas-engine Testing. 691 



is consumed, taken at, say, half-hourly intervals, become uniform and 
. thus verify each other. If the object is to determine the working economy, 
i and the period of time during which the engine is usually in motion is 
I some part of twenty-four hours, the duration of the test should be fixed 
jkfor this number of hours. If the engine is one using coal for generating 
gas, the test should cover a long enough period to determine with accuracy 
the coal used in the gas producer ; such a test should be of at least twenty- 
four hours' duration, and in most cases it should extend over several days. 

VII. Starting and Stopping a Test.— In a test for determining the 
maximum economy of an engine, it should first be run a sufficient time to 
bring all the conditions to a normal and constant state. Then the regular 
observations of the test should begin, and continue for the allotted time. 

If a test is made to determine the performance under working condi- 
tions, the test should begin as soon as the regular preparations have been 
made for starting the engine in practical work, and the measurements 
should then commence and be continued until the close of the period 
covered by the day's work. 

VIII. Measurement of Fuel.— If the fuel used is coal furnished to a 
gas producer, the same methods apply for determining the consumption 
as are used in steam-boiler tests. (See Code of Rules for Conducting 
Boiler Tests, "Transactions of the American Society of Mechanical Engi- 

l neers," Volume XXL, page 34.) 

If the fuel used be gas, the only practical method of measurement is 
the use of a meter through which the gas is passed. Gas bags should be 
placed between the meter and the engine to diminish the variations of 
pressure, and these should be of a size proportionate to the quantity used. 
Where a meter is employed to measure the air used by an engine, a re- 
ceiver with a flexible diaphragm should be placed between the engine and 

i the meter. The temperature and pressure of the gas should be measured, 
as also the barometric pressure and temperature of the atmosphere, and 
the quantity of gas should be determined by reference to the calibration 

I of the meter, taking into account the temperature and pressure of the gas. 

If the fuel is oil, this can be drawn from a tank which is filled to the 

original level at the end of the test, the amount of oil required for so 

1 doing being weighed ; or, for a small engine, the oil may be drawn from a 
calibrated vertical pipe. 

In an engine using an igniting flame the gas or oil required for it should 
be included in that of the main supply, but the amount so used should be 

! stated separately, if possible. 

IX. Measurement of Heat Units Consumed by the Engine.— The 

| number of heat units used is found by multiplying the number of pounds 

of coal or oil or the cubic feet of gas consumed by the total heat of com- 

1 bustion of the fuel, as determined by a calorimeter test. In determining 

I the total heat of combustion no deduction is made for the latent heat of 

j the water vapor in the products of combustion. There is a difference of 

I opinion on the propriety of using this higher heating value, and for pur- 

1 poses of comparison care must be taken to note whether this or the lower 

value has been used. The calorimeter recommended for determining the 

heat of combustion is the Mahler, for solid fuels or oil. or the Junker, for 

J gases, or some form of calorimeter known to be equally reliable. (See 

Poole on " The Calorific Power of Fuels.") 

It is sometimes desirable, also, to have a complete chemical analysis of 
J the oil or gas. The total heat of combustion may be computed, if desired, 
I from the results of the analysis, and should agree well with the calori- 
meter values. (See Section XVII., Boiler- test Code.) 

For the purpose of making the calorimeter test, if the fuel used is coal 
| for generating gas in a producer, or oil, samples should be taken at the 
time of the engine trial and carefully preserved for subsequent determina- 
tion. If gas is used, it is better to have a gas calorimeter on the spot, 
. samples taken, and the calorimeter test made while the trial is going on. 

X. Measurement of Jacket Water to Cylinder or Cylinders.— The 

jacket water may be measured by passing it through a water-meter or 
allowing it to flow from a measuring tank before entering the jacket, or by 
collecting it in tanks on its discharge. If measuring tanks are used, the 



692 Gas-engine Testing. 



same system of arrangement is recommended as that employed for feed- 
water measurements in boiler and steam-engine tests. (See Section XI., * 
Steam-engine Code. ) / 

XI. Indicated Horse=power.— The directions given for determining 
the indicated horse-power for steam engines apply in all respects to inter- 
nal-combustion engines. (See Section XIII., Steam-engine Code.) 

XII. Brake Horse=power.— The determination of the brake horse- 
power, which is very desirable, is the same for internal combustion as for 
steam engines. (See directions given in Section XV., Steam-engine 
Code.) 

XIII. Speed.— The same directions apply to internal-combustion en- 
gines as to steam engines for the determination of speed, and reference is 
made to Section XVII., Steam-engine Code, for suggestions on this sub- 
ject. 

In an engine which is governed by varying the number of explosions 
or working cycles, a record should be kept of the number of explosions 
per minute ; or if the engine is running at nearly maximum load, by 
counting the number of times the governor causes a miss in the explosions. 

XIV. Recording the Data.— The time of taking weights and every 
observation should be recorded, and note made of every event, however 
unimportant it may seem to be. The pressures, temperatures, meter read- 
ings, speeds, and other measurements should be observed every 20 or 30 
minutes when the conditions are practically uniform, and at more frequent 
intervals if they are variable. Observations of the gas or oil measurements 
should be taken with special care at the expiration of each hour, so as to 
divide the test into hourly periods and reveal the uniformity, or other- 
wise, of the conditions and results as the test goes forward. 

All data and observations should be kept on suitably-prepared blank 
sheets or in note-books. 

XV. Uniformity of Conditions.— When the object of the test is to 
determine the maximum economy, all the conditions relating to the opera- 
tion of the engine should be maintained as constant as possible during the 
trial. 






XVI. Indicator Diagrams and Their Analysis.— (a) Sample Dia- 
grams : Sample diagrams nearest to the mean should be selected from those 
taken during the trial and appended to the tables of the results. If there 
are separate compression or feed cylinders, the indicator diagrams from 
these should be taken and the power deducted from that of the main 
cylinder. 

XVII. Standards of Economy and Efficiency.— The hourly con- 
sumption of heat, determined as pointed out in Article IX., divided by the 
Indicated or the brake horse-power, is the standard expression of engine 
economy recommended. 

In making comparisons between the standard for internal-combustion 
engines and that for steam engines, it must be borne in mind that the 
former relates to energy concerned in the generation of the force employed, 
whereas In the steam engine it does not relate to the entire energy ex- 
pended during the process of combustion in the steam boiler. Thesteam 
engine standard does not. cover the losses due to combustion, while the 
Internal-combustion engine standard, in cases where a crude fuel such as 
oil is burned in the cylinder, docs cover these losses. To make a direct 
comparison between the two classes of engines considered as complete 
plants for the production of ]M>\ver, the losses in generating the working 
agent must he taken into account in both cases, and the comparison must 
he on the basis of the fuel used; and not only this, but on the basis of the 
Bame or equivalent fuel used in each case. In such a comparison, where ■ 
producer gas is used and the producer is included in the plant, the fuel 
consumption, which will be the weight of coal in both cases, may be 
directly compared. 

The thermal efficiency ratio per indicated horse-power or per brake 
hone-power for Internal-combustion engines is obtained in the same 



Gas-engine Testing. 693 



L 



manner as for steam engines referred to in Section XXI., Steam-engine 
Code, and is expressed by the fraction 



2545 



B. T. U. per horse-power per hour * 



XVIII. Heat Balance. — For purposes of scientific research, a heat 
balance should be drawn which shows the manner in which the total 
heat of combustion is expended in the various processes concerned in the 
working of the engine. It may be divided into three parts : first, the heat 
which is converted into the indicated or brake work; second, the heat 
rejected in the cooling water of the jackets ; and third, the heat rejected 
in the exhaust gases, together with that lost through incomplete com- 
bustion and radiation. 

To determine the first item, the number of foot-pounds of work per- 
formed by, say, 1 pound or 1 cubic foot of the fuel is determined ; and 
this quantity, divided by 778, which is the mechanical equivalent of 1 
B. T. U., gives the number of heat units desired. The second item is de- 
termined by measuring the amount of cooling water passed through the 
jackets, equivalent to 1 pound or 1 cubic foot of fuel consumed, and cal- 
culating the amount of heat rejected, by multiplying this quantity by the 
difference in the sensible heat of the water leaving the jacket and that 
entering. The third item is obtained by the method of differences,— that 
is, by subtracting the sum of the first two items from the total heat sup- 
plied. The third item can be subdivided by computing the heat rejected 
in the exhaust gases as a separate quantity. The data for this computation 
are found by analyzing the fuel and the exhaust gases, or by measuring 
the quantity of air admitted to the cylinder in addition to that of the gas 
or oil. 

XIX. Report of Test.— The data and results of a test should be re- 
ported in the manner outlined in one of the following tables, the first of 
which gives a complete summary when all the data are determined, and 
the second is a shorter form of report, in which some of the minor items 
are omitted. 

XX. Temperatures Computed at Various Points of the Indicator 
Diagram.— The computation of temperatures corresponding to various 
points in the indicator diagram is, at best, approximate. It is possible 
only where the temperature of one point is known or assumed, or where 
the amount of air entering the cylinder along with the charge of gas or 
oil and the temperature of the exhaust gases is determined. 



Data and Results of Test of Gas or Oil Engine. 

Arranged according to the Complete Form advised by the Engine Test 

Committee of the American Societv of Mechanical Engineers. 

Code of 1902. 



1. Made by of. 

on engine located at 

to determine 



2. Date of trial 

3. Type of engine (whether oil or gas) . 



4. Class of engine (mill, marine, motor for vehicle, pumping, or other) . . 

5. Number of revolutions for one cycle, and class of cycle . 

6. Method of ignition 



7. Name of builders 

8. Gas or oil used 

(a) Specific gravity deg. Fahr. 

(b) Burning-point deg. Fahr. 

(c) Flashing-point deg. Fahr. 



694 Gas-engine Testing. 



1st Cyl. 2d Cyl. 

9. Dimensions of engine : . 

(a) Class of cylinder (working or for compress- 

ing the charge) 

(b) Vertical or horizontal 

(c) Single- or double-acting 

(d) Cylinder dimensions 

Bore, in inches 

Stroke, in feet 

Diameter of piston-rod, in inches 

Diameter of tail-rod, in inches 

(e) Compression space or clearance, in per 

cent., of volume displaced by piston 
per stroke 

Head end 

Crank end 

Average 

(/) Surface, in square feet (average) 

Barrel of cylinders 

Cylinder heads 

Clearance and ports 

Ends of piston 

Piston-rod 

(g) Jacket surfaces or internal surfaces of cyl- 
inder heated by jackets, in square feet. 

Barrel of cylinder 

Cylinder heads 

Clearance and ports 

(h) Horse-power constant for 1 pound mean 
effective pressure and 1 revolution per 
minute 

10. Give description of main features of engine and plant, and illustrate 

with drawings of same given on an appended sheet. De- 
scribe method of governing. State whether the conditions 
were constant throughout the test. 

Total Quantities. 

11. Duration of test hours. 

12. Gas or oil consumed cu. ft. or lbs. 

13. Air supplied, in cubic feet cu. ft. 

14. Cooling water supplied to jackets cu. ft. 

15. Calorific value of gas or oil by calorimeter test, determined 

by calorimeter B. T. U. 

Hourly Quantities. 

1 »',. ( ias or oil consumed per hour lbs. 

17. Cooling water supplied per hour lbs. 

Pressures and Temperatures. 

18. Pressure at meter (for gas engine), in inches, of water ins. 

19. Barometric pressure of atmosphere: 

(a) Reading of height of barometer ins. 

(b) Reading of temperature of barometer , . deg. Fahr. 

(c) Reading of barometer corrected to 32° F ins. 

20. Temperature of cooling water : 

(<0 Inlet deg. Fahr. 

(i>) Outlet deg. Fahr. 

21. Temperature Oi pis at meter (for gas engine) deg. Fahr. 

22. Temperature of atmosphere: 

(a) Dry-bulb thermometer deg. Fahr. 

(/>) Wet-bulb thermometer deg. Fahr. 

(c) Degree of humidity per cent. 

23. Temperature of exhaust gases deg. Fahr. 

I low determined 



Gas-engine Testing. 695 

Data Relating to Heat Measurement. 

24. Heat units consumed per hour (pounds of oil or cubic feet 

of gas per hour multiplied by the total heat of 
combustion) B. T. U. 

25. Heat rejected in cooling water : 

(a) Total per hour B. T. U. 

(b) In per cent, of heat of combustion of the gas or oil 

consumed per cent. 

26. Sensible heat rejected in exhaust gases above temperature 

of inlet air : 

(a) Total per hour B. T. U. 

(6) In per cent, of heat of combustion of the gas or oil 

consumed per cent. 

27. Heat lost through incomplete combustion and radiation 

per hour : 

(a) Total per hour B. T. U. 

\b) In per cent, of heat of combustion of the gas or oil 

consumed '. per cent. 

Speed, Etc. 

28. Revolutions per minute rev. 

29. Average number of explosions per minute 

How determined , 

30. Variation of speed between no load and full load rev. 

31. Fluctuation of speed on changing from no load to full 

load, measured by the increase in the revolu- 
tions due to the change 

Indicator Diagrams. 

1st Cyl. 2d Cyl. 

32. Pressure, in pounds, per square inch above atmos- 

phere : 

(a) Maximum pressure 

(b) Pressure just before ignition 

(c) Pressure at end of expansion 

(d) Exhaust pressure 

33. Temperatures, in degrees Fahr., computed from 

diagrams : 

(a) Maximum temperature (not necessarily at 

maximum pressure) 

(b) Just before ignition 

(c) At end of expansion 

(d) During exhaust 

34. Mean effective pressure, in pounds, per square inch 

Power. 

35. Power, as rated by builders : 

(a) Indicated horse-power H. P. 

\b) Brake horse-power H. P. 

36. Indicated horse-power actually developed : 

First cylinder H. P. 

Second cylinder H. P. 

Total H. P. 

37. Brake horse-power, electric horse-power, or pump horse- 

power, according to the class of engine H. P. 

38. Friction indicated horse-power from diagrams, with no 

load on engine and computed for average speed H. P. 

39. Percentage of indicated horse-power lost in friction per cent. 

Standard Efficiency Results. 

40. Heat units consumed by the engine per hour : 

(a) Per indicated horse-power B. T. U. 

(b) Per brake horse-power B. T. U. 



696 Gas-engine Testing. 



41. Heat units consumed by the engine per minute : 

(a) Per indicated horse-power B. T. U. . 

(b) Per brake horse-power B. T. U. 

42. Thermal efficiency ratio : 

(a) Per indicated horse-power per cent. 

(b) Per brake horse-power per cent. 

Miscellaneous Efficiency Results. 

43. Cubic feet of gas or pounds of oil consumed per horse- 

power per hour : 

(a) Per indicated horse-power 

(6) Per brake horse-power 

Heat Balance. 

44. Quantities given, in per cents., of the total heat of com- 

bustion of the fuel : 

(a) Heat equivalent of indicated horse-power per cent. 

(b) Heat rejected in cooling water per cent. 

(c) Heat rejected in exhaust gases and lost through 

radiation and incomplete combustion per cent. 

Sum = 100 per cent. 
Subdivisions of Item (c) : 

(cl) Heat rejected in exhaust gases per cent. 

(c2) Lost through incomplete combustion per cent. 

(c3) Lost through radiation, and unaccounted for per cent. 

Sum = Item (c) 

Additional Data. 

Add any additional data bearing on the particular objects of the test or 
relating to the special class of service for which the engine is to be used. 
Also give copies of indicator diagrams nearest the mean and the corre- 
sponding scales. Where analyses are made of the gas or oil used as fuel, 
or of the exhaust gases, the results may be given in a separate table. 

Data and Results of Standard Heat Test of Gas or Oil 

Engine. 

Arranged according to the Short Form advised by the Engine Test Com- 
mittee of the American Society of Mechanical Engineers. Code of 1902. 

1. Made by of 

on engine located at 

to determine 






2. Date of trial 

3. Type and class of engine . 



4. Kind of fuel used 

(a) Specific gravity deg. Fahr. 

(b) Burning-point deg. Fahr. 

(c) Flashing-point deg. Fahr. 

5. Dimensions of engine : lst Cy1, 2d Cyl * 

(a) Class of cylinder (working or for compress- 

ing the charge) 

(b) Single- or double-acting 

(c) Cylinder dimensions : 

Bore, in inches 

Stroke, in feel 

Diameter of piston-rod, in inches 

(d) Average compression space or clearance, 

in per cent , 

(c) Horse-power constant for 1 pound mean 
effective pressure and 1 revolution per 
minute 






Gas-engine Testing. 697 

Total Quantities. 

6. Duration of test hours. 

7. Gas or oil consumed cu. f t. or lbs. 

8. Cooling water supplied to jackets cu. ft. or lbs. 

9. Calorific value of fuel by calorimeter test, determined by 

calorimeter B. T. U. 



Pressures and Temperatures. 

10. Pressure at meter (for gas engine), in inches, of water ins. 

11. Barometric pressure of atmosphere : 

(a) Reading of barometer ins. 

(b) Reading corrected to 32° F ins. 

12. Temperature of cooling water : 

(a) Inlet deg. Fahr. 

(6) Outlet deg. Fahr. 

(c) Degree of humidity deg. Fahr. 

13. Temperature of gas at meter (for gas engine) deg. Fahr. 

14. Temperature of atmosphere : 

(a) Dry-bulb thermometer deg. Fahr. 

(b) Wet-bulb thermometer deg. Fahr. 

15. Temperature of exhaust gases deg. Fahr. 

Data Relating to Heat Measurement. 

16. Heat units consumed per hour (pounds of oil or cubic feet 

of gas per hour multiplied by the total heat of 
combustion) B. T. U. 

17. Heat rejected in cooling water per hour B. T. U. 

Speed, Etc. 

18. Revolutions per minute rev. 

19. Average number of explosions per minute 

Indicator Diagrams. 

1st Cyl. 2d Cyl. 

20. Pressure, in pounds, per square inch above atmos- 

phere : 

(a) Maximum pressure 

(b) Pressure just before ignition 

(c) Pressure at end of expansion 

(d) Exhaust pressure 

(e) Mean effective pressure 

Power. 

21. Indicated horse-power : 

First cylinder . . . . H. P. 

Second cylinder H. P. 

Total H. P. 

22. Brake horse-power H. P. 

23. Friction horse-power by friction diagrams H. P. 

24. Percentage of indicated horse-power lost in friction per cent. 

Standard Efficiency and Other Results. 

■ 25. Heat units consumed by the engine per hour : 

(a) Per indicated horse-power B. T. U. 

(6) Per brake horse-power B. T. U. 

26. Pounds of oil or cubic feet of gas consumed per hour : 

(a) Per indicated horse-power lbs. or cu. ft. 

(b) Per brake horse-power , lbs. or cu. ft 



698 Electric Power. 



Additional Data. 

Add any additional data bearing on the particular objects of the test or 
relating to the special class of service for which the engine is to be used. 
Also give copies of indicator diagrams nearest the mean, and the corre- 
sponding scales. ' 

Note.— The volume of gas measured at any temperature should be 
reduced to the equivalent at a standard temperature and atmospheric 
pressure, corrected for the effect of moisture in the gas, which is ordinarily 
at the saturation- point or nearly so. It is recommended that a standard 
be adopted for gas-engine work, the same as that used in photometry,— 
namely, the equivalent volume of the gas when saturated with moisture 
at the normal atmospheric pressure at a temperature of 60° F. In order to 
reduce the reading of the volume containing moist gas at any other tem- 
perature to this standard, multiply by the factor 

459 .4 + 60 b — (29.92 — s) 
X 



459.4 + t ~ 29.4 

in which b is the height of the barometer, in inches, at 32° F. ; t, the tem- 
perature of the gas at the meter, in degrees Fahrenheit ; and s, the vacuum, 
in inches, of mercury corresponding to the temperature of t obtained from 
steam tables. 



ELECTRIC POWER TRANSMISSION. 

The principal applications of electricity in mechanical engineering are 
in the transmission of power and the independent driving of machines bv 
electric motors. It is yet a question for debate as to whether the actual 
transmission losses are materially reduced by the substitution of electric 
driving for shafting, belting, and pulleys, but there is no doubt as to the 
great advantages of electricity so far as the convenient arrangement of 
machinery and the utilization of floor-space are concerned. 

Some general data concerning electricity will here be given, and for 
special and fuller treatment the reader is referred to Foster's " Electrical 
Engineer's Pocket-book," Bell's "Electric Power Transmission," and the 
standard works of reference on electrical engineering. 



Electkical Units. 



699 



3.B 



Symbol. 



3 § 



bi*. 



g- p o P g- 

^ o a V s g- o 

>i P o 
© 



p © 
o © 



© Co 



i o g. 

1 B.< 



a ®pj pi 
s 2-^2: 

1*2 2: 



CD rt- 



^g-s-^^^asg ©= °i?p^ 
^p^jgilU^l^sg- 

*!9&8 i is* p& *~* !i 

O ffl S b £ c ^ 
M. « ft d * 2 B B 



! p ■" © CD 

i-S © h- o S-rt-2 ® 

p- o ^ ■? 



> 3 ~ m & 3 S ^ 
I d g «* p a ' p. 

> no m P 



S *d 2. e B* 
so ~> p 2. S 1 * 

g 2-i°Cfq o m. 22. 
2 a i-fi j p p oa 

P ft 



4 S 3 p- 



rt E Ms S S3 [ 



& a a 

° 2 3 



oS-fp B £ & 



r+ 3 3 CD 

p^n © 
® 3 ® © 

►p p*S2. o 

W 1-1." 

9^ ^ 1 g 



« 2 ® p 
cr 5G P _ 

© ert- 



OjOOO tB 



icg.' 



; ® d5 55 p s* 
! S ® 2 «* p 

> B 5" £ * 5- 



o 2- 



CD ^ 

P M 



: 3 CD 
P * 
*"* d h- (' "^ CD ^ O 

£SlS> 8,1* 

ft P ►— ■►£ ^ Mj 
P P- o © p" p- p 

g id m ® °2 B 2 



00 ©", ^ 

^crpo- 
£3p 



S-S 



P CD' 






P^d CD P 
CD e* 
gg C P 



B 50 © 

B £-0 
© p ©_ 

•d 5*3. 
2 P ° 



o B a 

p S-'g 

© ^ 

Oo P^ 



^ P P 



1 






H 


H 


to|w 


*4 


fel 


H 


kO|M 




4 

1 


1 



ft_0gO^ 

^BpB^I 
© ; ' °° 



fesl 

X 



fe9 <o 



II II 
X X 



2 Q 



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•d -< Lj^d^M 
© cd rn o osi 
* ^ 00^ p- 
o^B^2 
£"« • • 3 
2 g o o> 

P- S-' w # 



© i-j CO 
i - ^ C5 

ps- 



o ° ° 



M 


* 


© O 


w 










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ft J4 


p 






P- © 




P 
P 


^ 


P? 


^ - 


DD 


P 

P- 


QP 
r/^B 


S" 
hi 



700 



Electrical Analogies. 



Analogies Between the Flow of Water and Electricity. 

Water. Electricity. 

Head, difference of level, in feet. ( Volts ; electro-motive force ; differ- 



Difference of pressure per square 
inch, in pounds. 

Resistance of pipes, apertures, etc., 
increases with length of pipe, 
with contractions, roughness, 
etc. ; decreases with increase of 
sectional area. The law of in- 
crease and decrease is expressed 
by complex formulae. 



Rate of flow, as cubic feet per 
second, gallons per minute, etc., 
or volume divided by the time. 
In the mining regions sometimes 
expressed in "miner's inches." 

Quantity, usually measured in 
cubic feet or gallons, but is also 
equivalent to rate of flow X time, 
as cubic feet per second for so 
many hours. 



Work, or energy, measured in foot- 
pounds: product of weight of 
falling water into height of fall ; 
in pumping, product of quantity, 
in cubic feet, into the pressure, 
in pounds, per square foot against 
which the water is pumped. 



Power, rate of work. Horse-power, 
foot-pounds of work done in 1 
minute ■*- 30,000. 

In falling water, pounds falling in 
1 second -r- 150. In water flowing 
in pipes, rate of flow, in cubic 
feet, per second X pressure resist- 
ing the flow, in pounds, per square 
foot -f- 550. 



ence of potential or of pressure ;r^ 
E. or E. M. F. 

Ohms, resistance, R. The resist- 
ance increases directly as the 
length or the conductor or wire, 
and inversely as its sectional area. 
It varies with the nature or qual- 
ity of the conductor. 

Conductivity is the reciprocal of 
specific resistance. 

Amperes ; current ; current strength ; 
intensity of current ; rate of flow ; 
1 ampere = 1 coulomb per second. 



Amperes = 



volts 
ohms 



; c- 



E 



; E= CR. 



Coulomb, unit of quantity, Q = rate 
of flow X time, as ampere seconds ; 
1 ampere hour = 3600 coulombs. 



Joule, volt-coulomb, W, the unit 
of work = product of quantity by 
the electro-motive force = volt- 
ampere. 

If C (amperes) = rate of flow, and 
E (volts) = difference of pressure 
between two points in a circuit, 
energy expended = CEt, = C 2 Rt t 
since E = CR. 






Watt, unit of power, P = volts X 
amperes, = current or rate of 
flow X difference of potential. 

1 watt = 0.7373 foot-pound per 
second = 7 £ B of a horse-power. 



In the mechanical applications of electricity it must always be remem- 
bered that the volt corresponds to pressure and the ampere to flow, and the 
product— the VOli -ampere— is the watt, the unit of power, 746 of which are 
equal to a horse-power. The kilowatt = 1000 watts is equal to 

1000 , nA , 

-=7g- = 1.34 horse-power. 

The British Board of Trade Unit is equal to 1 kilowatt hour 

Strands of Copper Wire. 

(Roeblings.) 
Copper wires are twisted into concentric strands or into ropes of 7 
strands. A rope of 7 strands, each composed of 7 wires, is called a seven- 
by-seven rope, and Is usually written 7 X 7. The number of wires that can 
be made into a strand is limited by the capacity of the stranding machin- 
ery. 200 wires is the usual limit of a concentric strand, and 133 wires of a 
rope. 



Electrical Cables. 



701 



In a strand of circular milage, CM, composed of n wires of diameter, d, 
with a weight per 1000 feet, w, we have 
CM=d?Xn, 
w CM 

a 2 



d- 



J CM 
\ n ' 
0.00305 X CM. 



The weights of strands are calculated about 1 per cent, heavier than a 
solid wire of the same circular milage, while the resistance is calculated 
for the solid wire. 

The diameter of a strand may be calculated by multiplying the diame- 
ter of one wire by the factors given in the table, according to the number 
of wires composing the strand. 

Number of Wires and Diameter in Strand Required to 
Equal a Given Circular Milage. 

Diameter of Wires in Decimal Parts of an Inch. 



9) O 

1% 






Area, 


in circular mils. 




50 000 


100 000 


150 000 


200 000 


250 000 


300 000 


350 000 


7 


.0845 


.1203 


.1463 


.1690 


.1890 


.2070 


.2236 


19 


.0513 


.0725 


.0889 


.1025 


.1147 


.1256 


.1357 


37 


.0367 


.0519 


.0636 


.0735 


.0821 


.0900 


.0972 


61 


.0286 


.0405 


.0496 


.0572 


.0640 


.0701 


.0757 


127 


.0199 


.0280 


.0343 


.0396 


.0443 


.0486 


.0526 


169 


.0172 


.0243 


.0297 


.0344 


.0384 


.0421 


.0455 


217 


.0151 


.0214 


.0262 


.0304 


.0339 


.0371 


.0401 




400 000 


450 000 


500 000 


550 000 


600 000 


650 000 


700 000 


7 


.2390 


.2535 


.2672 


.2803 


.2927 


.3047 


.3163 


19 


.1450 


.1538 


.1622 


.1701 


.1776 


.1849 


.1919 


37 


.1039 


.1103 


.1162 


.1219 


.1273 


.1325 


.1375 


61 


.0809 


.0858 


.0905 


.0949 


.0991 


.1032 


.1071 


127 


.0561 


.0595 


.0627 


.0658 


.0687 


.0715 


.0742 


169 


.0486 


.0516 


.0543 


.0571 


.0595 


.0620 


.0643 


217 


.0429 


.0455 


.0480 


.0503 


.0525 


.0547 


.0567 




750 000 


800 000 


850 000 


900 000 


950 000 


1 000 000 


7 


.3273 


.3380 


.3484 


.3585 


.3684 


.3770 


19 


.1986 


.2050 


.2115 


.2176 


.2236 


.2294 


37 


.1423 


.1470 


.1515 


.1559 


.1602 


.1644 


61 


.1108 


.1145 


.1180 


.1214 


.1247 


.1280 


127 


.0768 


.0793 


.0818 


.0841 


.0864 


.0887 


169 


.0666 


.0687 


.0709 


.0729 


.0749 


.0769 


217 


.0588 


.0607 


.0625 


.0644 


.0661 


.0678 



702 



Copper Wire. 



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704 



National Electric Code. 



Table for the Conversion of Mils. ( T oVo inch) into 
Centimetres. 



Mils. 


Centi- 
metres. 


Mils. 


Centi- 
metres. 


Mils. 


Centi- 
metres. 


Mils. 


Centi- 
metres. 


1 


.00254 


26 


.06602 


51 


.1295 


76 


.1931 


2 


.00508 


27 


.06856 


52 


.1321 


77 


.1956 


3 


.00762 


28 


.07110 


53 


.1346 


78 


.1981 


4 


.01016 


29 


.07364 


54 


.1372 


79 


.2006 


5 


.01270 


30 


.07618 


55 


.1397 


80 


.2032 


6 


.01524 


31 


.07872 


56 


.1422 


81 


.2057 


7 


.01778 


32 


.08126 


57 


.1448 


82 


.2083 


8 


.02032 


33 


.08380 


58 


.1473 


83 


.2108 


9 


.02286 


34 


.08634 


59 


.1499 


84 


.2133 


10 


.02540 


35 


.08888 


60 


.1524 


85 


.2159 


11 


.02793 


36 


.09142 


61 


.1549 


86 


.2184 


12 


.03047 


37 


.09396 


62 


.1575 


87 


.2209 


13 


.03301 


38 


.09650 


63 


.1600 


88 


.2235 


14 


.03555 


39 


.09904 


64 


.1626 • 


89 


.2260 


15 


.03809 


40 


.1016 


65 


.1651 


90 


.2286 


16 


.04063 


41 


.1041 


66 


.1676 


91 


.2311 


17 


.04317 


42 


.1067 


67 


.1702 


92 


.2336 


18 


.04571 


43 


.1092 


68 


.1727 


93 


.2362 


, 19 


.04825 


44 


.1118 


69 


.1752 


94 


.2387 


20 


.05079 


45 


.1143 


70 


.1778 


95 


.2413 


21 


.05333 


46 


.1168 


71 


.1803 


96 


.2438 


22 


.05587 


47 


.1194 


72 


.1829 


97 


.2465 


23 


.05841 


48 


.1219 


73 


.1854 


98 


.2489 


24 


.06095 


49 


.1245 


74 


.1879 


99 


.2514 


25 


.06348 


50 


.1270 


75 


.1905 


100 


.2540 






44 National Electrical Code." 

Rules and Requirements of the National Board of Fire Underwriters for 
the Installation of Wiring and Apparatus for Electric Light, Heat, 
and Power as Recommended by the Underwriters' 
National Electric Association. 
Edition of 1901. 
The National Electrical Code is the result of the united efforts of the 
various electrical, insurance, architectural, and allied interests which 
have, through the National Conference on Standard Electrical Rules, com- 
posed of delegates from various national associations, unanimously voted 
to recommend it to their respective associations for approval or adoption. 

The following is a Lisl of the associations represented in the conference, 
all of which have approved of tin- Code: 

American Institute of Architects. 

American institute ot' Electrical Engineers, 

American Society of Mechanical Engineers, 
American Street Railway Association, 
Factory Mutual 1m re Insurance Companies, 
National Association of Fire Engineers, 
National Board of Fire Underwriters, 
National Electric Lighl Association, 
Underwriters' Nationul Electric Association. 



National Electeic Code. 705 

GENERAL PLAN GOVERNING THE ARRANGEMENT OF RULES. 

Class A. — Central Stations, Dynamo, Motor, and Storage=battery 
Rooms, Transformer Sub=stations, etc. Rules 1 to 11. 
*c, Class B.— Outside Work, all systems and voltages. Rules 12 and 13. 
Class C— Inside Work. Rules 14 to 39. Subdivided as follows : 

General Rules, applying to all systems and voltages. Rules 14 to 17. 
Constant=current Systems. Rules 18 to 20. 
Constant=potential Systems. 
All voltages. Rules 21 to 23. 
Voltage not over 550. Rules 24 to 31. 
Voltage between 550 and 3500. Rules 32 to 37. 
Voltage over 3500. Rules 38 and 39. 
Class D.— Specifications for Wires and Fittings. Rules 40 to 63. 
Class E.— Miscellaneous. Rules 64 to 67. 
Class F.— Marine Wiring. Rules 68 to 80. 

CLASS A.— STATIONS AND DYNAMO ROOMS. 

Includes Central Stations, Dynamo, Motor, and Storage-battery Rooms, 
Transformer Sub-stations, etc. 

1. Generators. 

(a) Must be located in a dry place. 

(6) Must never be placed in a room where any hazardous process is car- 
ried on, nor in places where they would be exposed to inflammable gases 
or flyings of combustible materials. 

(c) Must be insulated on floors or base frames, which must be kept filled 
I to prevent absorption of moisture, and also kept clean and dry. Where 

frame insulation is impracticable, the Inspection Department having juris- 
diction may, in writing, permit its omission, in which case the frame must 
be permanently and effectively grounded. 

A high-potential machine which, on account of great weight or for other 
reasons, cannot have its frame insulated from the ground, should be sur- 
rounded with an insulated platform. This maybe made of wood mounted 
on insulating supports, and so arranged that a man must always stand upon 
1 it in order to touch any part of the machine. 

In case of a machine having an insulated frame, if there is trouble from 
static electricity due to belt friction, it should be overcome by placing near 
the belt a metallic comb connected with the earth, or by grounding the 
frame through a very high resistance of not less than 200 ohms per volt 
generated by the machine. 

( d) Every constant-potential gen erator must be protected from excessive 
current by a safety fuse ; or equivalent device, of approved design in each 
lead wire. 

These devices should be placed on the machine or as near it as possible. 

Where the needs of the service make these devices impracticable, the 
Inspection Department having jurisdiction may, in writing, modify the 
requirements. 

(e) Must each be provided with a water-proof cover. 

(/) Must each be provided with a name-plate, giving the maker's name, 
the capacity in volts and amperes, and the normal speed, in revolutions, 
per minute. 

2. Conductors. 

From generators to switchboards, rheostats, or other instruments, and 
thence to outside lines. 

(a) Must be in plain sight or readily accessible. 

(b) Must have an approved insulating covering, as called for by rules in 
Class C for similar work, except that in central stations, on exposed cir- 
cuits, the wire which is used must have a heavy-braided, non-combustible 
outer covering. 

Bus bars may be made of bare metal. 

(c) Must be kept so rigidly in place that they cannot come in contact. 

45 



706 National Electric Code. 

(d) Must in all other respects be installed under the same precautions 
as required by rules in Class C for wires carrying a current of the same 
volume and potential. 

3. Switchboards. 

(a) Must be so placed as to reduce to a minimum the danger of com-* 
municating fire to adjacent combustible material. 

Special attention is called to the fact that switchboards should not be 
built down to the floor nor up to the ceiling, but a space of at least 10 or 
12 inches should be left between the floor and the board, and from 18 to 24 
inches between the ceiling and the board, in order to prevent fire from 
communicating from the switchboard to the floor or ceiling, and also to 
prevent the forming of a partially-concealed space very liable to be used 
for storage of rubbish and oily waste. 

(b) Must be made of non-combustible material or of hard wood, in skele- 
ton form, filled to prevent absorption of moisture. 

(c) Must be accessible from all sides when the connections are on the 
back, but may be placed against a brick or stone wall when the wiring is 
entirely on the face. 

(d) Must be kept free from moisture. 

(e) Bus bars must be equipped in accordance with rules for placing 
conductors. 

4. Resistance Boxes and Equalizers. 

(For Construction Rules, see No. 60.) 

(a) Must be placed on a switchboard, or, if not thereon, at a distance of 
a foot from combustible material, or separated therefrom by a non-inflam- 
mable, non-absorptive insulating material. 






5. Lightning Arresters. 

(For Construction Rules, see No. 63.) 

(a) Must be attached to each side of every overhead circuit connected 
witn the station. 

It is recommended to all electric light and power companies that 
arresters be connected at intervals over systems in such numbers and s< 
located as to prevent ordinary discharges entering (over the wires) build 
ings connected to the lines. 

(b) Must be located in readily-accessible places away from combustibL 
materials, and as near as practicable to the point where the wires enter th< 
building. 

Station arresters should generally be placed in plain sight on the swito 
board. 

In all cases, kinks, coils, and sharp bends in the wires between tin 
arresters and the outdoor lines must be avoided, as far as possible. 

(c) Must be connected with a thoroughly good and permanent ground 
connection by metallic strips or wires having a conductivity not less than 
that of a No. 6 B. & S. copper wire, which must be run as nearly in a 
straight line as possible from the arresters to the earth connection. 

Ground wires for lightning arresters must not be attached to gas-pip* 
within the buildings. 

It is often desirable to introduce a choke coil in circuit between the 
arresters and the dynamo. In no case should the ground wire from a 
lightning arrester be put into iron pipes, as these would tend to impede 
the discharge. 

6. Care and Attendance. 

(a) A competent man must be kept on duty where generators 
operating. 

(b) Oily waste must be kept in approved metal cans and removed daily. 
Approved waste cans shall be made of metal, with legs raising can 3 

inches from the floor, and with self-closing covers. * 



pes 

the 
i a 
3de 

are 



7. Testing of Insulation Resistance. 

fa) All circuits, except Buch as are permanently grounded in accordance 
with Rulel3A, must be provided with reliable ground detectors. Detectors 



National Electkic Code. 707 

which indicate continuously, and give an instant and permanent indication 
of a ground, are preferable. Ground wires from detectors must not be at- 
tached to gas-pipes within the building. 

(b) Where continuously-indicating detectors are not feasible, the 
^ .circuits should be tested at least once per day, and preferably of tener. 

(c) Data obtained from all tests must be preserved for examination by 
the Inspection Department having jurisdiction. 

These rules on testing to be applied at such places as may be designated 
by the Inspection Department having jurisdiction. 

8. Motors. 

(a) Must be insulated on floors or base frames, which must be kept filled 
to prevent absorption of moisture ; and must be kept clean and dry. Where 
frame insulation is impracticable, the Inspection Department having juris- 
diction may, in writing, permit its omission, in which case the frame must 
be permanently and effectively grounded. 

A high-potential machine, which on account of great weight or for other 
reasons cannot have its frame insulated, should be surrounded with an 
insulated platform. This may be made of wood mounted on insulating 
supports, and so arranged that a man must stand upon it in order to touch 
any part of the machine. 

In case of a machine having an insulated frame, if there is trouble from 
static electricity due to belt friction it should be overcome by placing near 
the belt a metallic comb connected to the earth, or by grounding the frame 
through a very high resistance of not less than 200 ohms per volt generated 
by the machine. 

(b) Must be wired under the same precautions as required by rules in 
Class C for wires carrying a current of the same volume and potential. 

The leads or branch circuits should be designed to carry a current at 
least 50 per cent, greater than that required by the rated capacity of the 
motor, to provide for the inevitable overloading of the motor at times 
j without overfusing the wires. 

(c) The motor and resistance box must be protected by a cutout and 
controlled by a switch (see No. 17, a), said switch plainly indicating 
whether "on" or "off." Where one-fourth horse-power or less is used on 
low-tension circuits a single-pole switch will be accepted. The switch and 
rheostat must be located within sight of the motor, except in such cases 
where special permission to locate them elsewhere is given in writing by 

\ the Inspection Department having jurisdiction. 

(d) Must have their rheostats or starting-boxes located as to conform to 
: the requirements of No. 4. 

In connection with motors, the use of circuit-breakers, automatic start- 
ing-boxes, and automatic under-load switches is recommended, and they 
must be used when required. 

(e) Must not be run in series-multiple or multiple-series, except on 
constant-potential systems, and then only by special permission of the 
Inspection Department having jurisdiction. 

(/) Must be covered with a water-proof cover when not in use, and, if 
deemed necessary by the Inspection Department having jurisdiction, must 
be inclosed in an approved case. 

From the nature of the question, the decision as to what is an approved 
case must be left to the Inspection Department having jurisdiction to 
determine in each instance. 

(g) Must, when combined with ceiling fans, be hung from insulated 
hooks, or else there must be an insulator interposed between the motor 
and its support. 

(h) Must each be provided with a name-plate, giving the maker's name, 
the capacity in volts and amperes, and the normal speed, in revolutions, 
per minute. 

9. Railway Power Plants. 

(a) Must be equipped in each feed wire before it leaves the station with 
an approved automatic circuit- breaker (see No. 52), or other device, which 
will immediately cut off the current in case of an accidental ground. This 
device must be mounted on a fire-proof base, and in full view and reach of 
the attendant. 



708 National Electric Code. 

10. Storage or Primary Batteries. 

(a) When current for light and power is taken from primary or second- 
ary batteries, the same general regulations must be observed as applied to 
similar apparatus fed from dynamo generators developing the same differ- 
ence of potential. 

(b) Storage-battery rooms must be thoroughly ventilated. 

(c) Special attention is directed to the rules for rooms where acid fumes 
exist. ( See No. 24, j and k. ) 

(d) All secondary batteries must be mounted on non-absorptive, non- 
combustible insulators, such as glass or thoroughly vitrified and glazed 
porcelain. 

(e) The use of any metal liable to corrosion must be avoided in cell 
connections of secondary batteries. 

11. Transformers. 

(For Construction Rules, see No. 62.) 

(a) In central or sub-stations the transformers must be so placed that 
smoke from the burning out of the coils or the boiling over of tho oil 
(where oil-filled cases are used) could do no harm. 

CLASS B.— OUTSIDE WORK. 

All Systems and Voltages. 
12. Wires. 

(a) Service wires must have an approved rubber insulator covering. 
(See No. 41.) Line wires, other than services, must have an approved 
weather-proof or rubber insulating covering (Nos. 41 and 44). All th< 
wires must have an insulation equal to that of the conductors they confine 

{b) Must be so placed that moisture cannot form a cross connection be- 
tween them, not less than a foot apart, and not in contact with any sub- 
stance other than their insulating supports. Service blocks must be covered 
over their entire surface with at least two coats of water-proof paint. 

(c) Must be at least 7 feet above the highest point of flat roofs, and at 
least 1 foot above the ridge of pitched roofs over which they pass or to 
which they are attached. 

(d) Must be protected by dead insulated guard iron or wires from possi- 
bility of contact with other conducting wires or substances to which 
current may leak. Special precautions of this kind must be taken where 
sharp angles occur, or where any wires might possibly come in contact 
with electric light or power wires. 

(e) Must be provided with petticoat insulators of glass or porcelain. 
Porcelain knobs or cleats and rubber hooks will not be approved. 

(/) Must be so spliced or joined as to be both mechanically and elec- 
trically secure without solder. The joints must then be soldered, to insure 
§ reservation, and covered with an insulation equal to that on the con 
uctors. 

All joints must be soldered, even if made with some form of paten 1 
splicing device. This ruling applies to joints and splices in all classes 
wiring covered by these rules. 

((/) Must, where they enter buildings, have drip loops outside, and the 
holes through which the conductors pass must be bushed with non-combus- 
tible, non-absorptive insulating tubes slanting upward towards the inside. 

(h) Telegraph, telephone, and similar wires must not be placed on thi 
simie cross- rm with electric; light or power wires; and when j)laced on 
the same pole with such wires the distance between the two inside pins of 
each cr08B-ariD must not he less than 26 inches. 

(/) The metallic sheaths to cables must be permanently and effectively 
connected to "earth." 

Trolley Wires. 

(./) Must not be smaller than No. B. & S. copper, or No. 4 B. & S. silicon 
bronze, and must readily stand the strain put upon them when in use. 

(/;) Must have a double insulation from the ground. In wooden-pole 
construction the pole will he considered as one insulation. 



I 

n 



* 



National Electric Code. 709 

(I) Must be capable of being disconnected at the power plant, or of 
being divided into sections, so that, in case of fire on the railway route, the 
current may be shut off from the particular section and not interfere with 
the work of the firemen. This rule also applies to feeders. 
H (m) Must be safely protected against accidental contact where crossed 
by other conductors. 

Guard wires should be insulated from the ground, and should be elec- 
trically disconnected in sections of not more than 300 feet in length. 

Ground Return Wires. 

(n) For the diminution of electrolytic corrosion of underground metal 
work, ground return wires must be so arranged that the difference of 
potential between the grounded dynamo terminal and any point on the 
return circuit will not exceed 25 volts. 

It is suggested that the positive pole of the dynamo be connected to the 
trolley line, and that whenever pipes or other underground metal work 
are found to be electrically positive to the rails or surrounding earth that 
they be connected by conductors arranged so as to prevent, as far as possi- 
ble,, current flow from the pipes into the ground. 

13. Transformers. 

(For Construction Rules, see No. 62. ) 

(a) Must not be placed inside of any building, excepting central 
stations, unless by special permission of the Inspection Department having 
jurisdiction. 

(b) Must not be attached to the outside walls of buildings, unless 
separated therefrom by substantial supports. 

13 A. Grounding Low=potential Circuits. 

The grounding of low-potential circuits under the following regulations 
is only allowed when so arranged that under normal conditions there will 
be no flow of current through the ground wire. 

Direct=current 3=Wire Systems. 

(a) Neutral wire may be grounded, and when grounded the following 
rules must be complied with : 

1. Must be grounded at the central station on a metal plate buried in 
coke beneath permanent moisture level, and also through all available 
underground water- and gas-pipe systems. 

2. In underground systems the neutral wire must also be grounded at 
each distributing box through the box. 

3. In overhead systems the neutral wire must be grounded every 500 
feet, as provided in Sections c, e, and /. 

The Inspection Department having jurisdiction may require grounding, 
if they deem it necessary. 

2-wire direct-current systems having no accessible neutral point are not 
to be grounded. 

Alternating Current Secondary Systems. 

(6) The neutral point of transformers or the neutral wire of distribu- 
ting systems may be grounded, and when grounded the following rules 
must be complied with : 

1. Transformers feeding 2-wire systems must be grounded at the centre 
of the secondary coils. 

2. Transformers feeding systems with a neutral wire must have the 
neutral wire grounded at the transformer and at least every 250 feet 
beyond. 

Inspection Department having jurisdiction may require grounding, if 
they deem it necessary. 

Ground Connections. 

(c) The ground wire in D. C. 3- wire systems must not at central stations 
be smaller than the neutral wire, and not smaller than No. 6 B. & S. else- 
where. 



710 National Electric Code. 

(d) The ground wire in A. C. systems must never be less than No. 6 
B. & S., and must always have equal carrying capacity to the secondary 
lead of the transformer, or the combined leads where transformers are 
banked. 

(e) The ground wire must be kept outside of buildings, but may be 
directly attached to the building or pole. The wire must be carried in as 
nearly a straight line as possible, and kinks, coils, and sharp bends must 
be avoided. 

(/) The ground connections for central stations, transformer sub- 
stations, and banks of transformers must be made through metal plates 
buried in coke below permanent moisture level, and connections should 
also be made to all available underground piping systems. For individual 
transformers and building services the ground connection may be made as 
above, or may be made to water or other piping systems running into the 
buildings. This connection may be made by carrying the ground wire into 
the cellar and connecting on the street side of meters, main clocks, etc. 

In connecting ground wires to piping systems, where possible, the wires 
should be soldered into one or more brass plugs, and the plugs forcibly 
screwed into a pipe-fitting, or, where the plugs are cast-iron, into a hole 
tapped to the pipe itself. For large stations, where connecting to under- 
ground pipes with bell and spigot joints, it is well to connect to several 
lengths, as the pipe joints may be of rather high resistance. Where such 
plugs cannot be used the surface of the pipe may be filed or scraped 
bright, the wire wound around it, and a strong clamp put over the wire 
and firmly bolted together. 

Where ground plates are used a No. 16 copper plate, about 3X6 feet in 
size, with about 2 feet of crushed coke or charcoal about pea size both 
under and over it, would make a ground of sufficient capacity for a mod- 
erate-size station, and would probably answer for the ordinary sub-station 
or bank of transformers. For a large central station considerable more 
area might be necessary, depending upon the other unground connections 
available. The ground wire should be riveted to such a plate in a number 
of places, and soldered for its whole length. Perhaps even better than i 
copper plate is a cast-iron plate with projecting forks, the idea of the fori 
being to distribute the connection to the ground over a fairly broad area, 
and to give a large surface contact. The ground wire can probably best be 
connected to such a cast-iron plate by brass plugs screwed into the plate 
to which the wire is soldered. In all cases the joint between the plate 
and the ground wire should be thoroughly protected against corrosion by 
suitable painting with water-proof paint or some equivalent. 

CLASS C— INSIDE WORK. 

All Systems and Voltages. 

GENERAL RULES.— ALL SYSTEMS AND VOLTAGES. 

14. Wires. 

(For Special Rules, see Nos. 18, 24, 32, 38, and 39.) 

(a) Must not be of smaller size than No. 14 B. & S., except as allowed 
under Rules 24, t, and 45, b. 

(&) Tie wires must have an insulation equal to that of the conductors 
they confine. 

(c) Must l >e so spliced or joined as to both mechanically and electrically 
secure without solder ; they must be then soldered to insure preservation, 
and the joint covered with an insulation equal to that on the conductors. 

Standard wires must be soldered before being fastened under clamps or 
binding screws; and, when they have a conductivity greater than No. 10 
B. & S. copper wire, they will be soldered into lugs. 

All joints musl be soldered, even if made with some form of patent 
splicing device. This ruling applies to joints and splices in all classes of 
wiring covered by these rules. 

['I) Must be separated from contact with walls, floors, timbers, or par- 
titions through which they may pass by non-combustible, non-absorptive 
Insulating tubes, such as glass or porcelain. 

Bushings must be long enough to bush the entire length of the hole in 
one continuous piece, or else the hole must first be bushed by a continuous 



a.. 



National Electric Code. 



711 



water-proof tube, which may be a conductor, such as iron pipe ; the tube 
then is to have a non-conducting bushing pushed in at each end, so as to 
keep the wire absolutely out of contact with the conducting pipe. 

(e) Must be kept free from contact with gas-, water-, or other metallic 

tun , piping, or any other conductors or conducting material which they may 

cross, by some continuous and firmly-fixed non-conductor, creating a 

separation of at least 1 inch. Deviations from this rule may sometimes be 

allowed by special permission. 

(/) Must be so placed in wet places that an air space will be left be- 
tween conductors and pipes in crossing, and the former must be run in 
such a way that they cannot come in contact with the pipe accidentally. 
Wires should be run over, rather than under, pipes upon which moisture is 
likely to gather, or which, by leaking, might cause trouble on a circuit. 

15. Underground Conductors. 

(a) Must be protected, when brought into a building, against moisture 
and mechanical injury, and all combustible material must be kept removed 
from the immediate vicinity. 

(6) Must not be so arranged as to shunt the current through a building 
around any catch-box. 

16. Carrying Capacity of Wires. 

Below is a table which must be followed in placing interior conductors, 
showing the allowable carrying capacity of wires and cables of 98 per 
cent, conductivity, according to the standard adopted by the American 
Institute of Electrical Engineers. 



o 


Table A, 
rubber- 
covered 
wires. (See 
No. 41. ) 

Amperes. 


Table B, 
weather- 
proof 
wires. (See 
Nos. 42-44.) 
Amperes. 


Circular 
mils. 


Circular 
mils. 


Table A, 
rubber- 
covered 
wires. (See 
No. 41.) 

Amperes. 


Table B, 
weather- 
proof 
wires. (See 
Nos. 42-44.) 
Amperes. 


18 


3 


5 


1624 


200 000 


200 


300 


16 


6 


8 


2 583 


300 000 


270 


400 


14 


12 


16 


4107 


400 000 


330 


500 


12 


17 


23 


6 530 


500 000 


390 


590 


10 


24 


32 


10 380 


600 000 


450 


680 


8 


33 


46 


16 510 


700 000 


500 


760 


6 


46 


65 


26 250 


800 000 


550 


840 


5 


54 


77 


33100 


900 000 


600 


920 


4 


65 


92 


41740 


1 000 000 


650 


1000 


3 


76 


110 


52 630 


1 100 000 


690 


1080 


2 


90 


131 


66 370 


1 200 000 


730 


1150 


1 


107 


156 


83 690 


1 300 000 


770 


1220 





127 


185 


105 500 


1 400 000 


810 


1290 


00 


150 


220 


133 100 


1 500 000 


850 


1360 


000 


177 


262 


167 800 


1 600 000 


890 


1430 


0000 


210 


312 


211 600 


1 700 000 
1 800 000 

1 900 000 

2 000 000 


930 

970 

1010 

1050 


1490 
1550 
1610 

1670 



The lower limit is specified for rubber-covered wires to prevent gradual 
deterioration of the high insulations by the heat of the wires, but not from 



712 National Electric Code. 

fear of igniting the insulation. The question of drop is not taken into 
consideration in the tables on page 711. 

The carrying capacity of 16- and 18- wire is given, but no smaller than 
14 is to be used, except as allowed under Rules 24, t, and 45, b. 

17. Switches, Cutouts, Circuit=breakers, etc. 

(For Construction Rules, see Nos. 51, 52, and 53.) 

(a) Must, whenever called for, unless otherwise provided (for excep- 
tions, see No. 8, c, and No. 22, c), be so arranged that the cutouts will 
protect, and the opening of the switch or circuit-breaker will disconnect, 
all of the wires, — that is, in a 2-wire system the 2 wires, and in a 3- wire 
system the 3 wires, must be protected by the cutout, and disconnected by 
the operation of the switch or circuit-breaker. 

(b) Must not be placed in the immediate vicinity of easily-ignitible 
stuff or where exposed to inflammable gases or dust or to flyings of com- 
bustible material. 

(c) Must, when exposed to dampness, either be inclosed in a water-proof 
box or mounted on porcelain knobs. 

CONSTANT=CURRENT SYSTEMS. 

Principally Series Arc Lighting. 

18. Wires. 

(See, also, Nos. 14, 15, and 16.) 

(a) Must have an approved rubber insulating covering. (See No. 41.) 
(ft) Must be arranged to enter and leave the building through an 
approved double-contact service switch (see No. 51), mounted in a non- 
combustible case, kept free from moisture, and easy of access to police or 
firemen. So-called "snap switches" must not be used on high-potential 
circuits. 

(c) Must always be in plain sight, and never incased, except when 
required by the Inspection Department having jurisdiction. 

(d) Must be supported on glass or porcelain insulators, which separate 
the wire at least 1 inch from the surface wired over, and must be kept 
rigidly at least 8 inches from each other, except within the structure of 
lamps, on hanger-boards, in cutout boxes, or like places, where a less 
distance is necessary. 

(e) Must, on side walls, be protected from mechanical injury by a sub- 
stantial boxing, retaining an air space of 1 inch around the conductors, 
closed at the top (the wires passing through bushed holes), and extending 
not less than 7 feet from the floor. When crossing floor-timbers in cellars 
or in rooms where they might be exposed to injury, wires must be attached 
by their insulating supports to the under side of a wooden strip not less 
than % inch in thickness. 

19. Arc Lamps. 

(For Construction Rules, see No. 57.) 

(a) Must be carefully isolated from inflammable material. 

(b) Must be provided at all times with a glass globe surrounding the 
arc, securely fastened upon a closed base. No broken or cracked globes to 
be used. 

(c) Must be provided with a wire netting (having a mesh not exceeding 
134 inches) around the globe and an approved spark arrester (see No. 58), 
when readily-inflammable material is in the vicinity of the lamps, to 
prevent escape of sparks, melted copper, or carbon. It is recommended 
that plain carbons, not copper-plated, be used for lamps in such places. 

Arc lamps, when used in places where they are exposed to flyings of 
easily-inflammable material, should have the carbons inclosed completely 
in a globe in such manner as to avoid the necessity for spark arresters. 

For the present, globe and spark arresters will not be required on so- 
called "inverted arc" lamps, but this type of lam p must not be used where 
exposed to flyings of easily-inflammabie materials. 



National Electric Code. 713 

(d) Where hanger-boards (see No. 56) are not used lamps must be hung 
from insulating supports other than their conductors. 

20. Incandescent Lamps in Series Circuits. 

(a) Must have the conductors installed as provided in No. 18, and each 
lamp must be provided with an automatic cutout. 

(b) Must have each lamp suspended from a hanger-board by means of 
rigid tube. 

(c) No electro-magnetic device for switches and no system of multiple- 
series or series-multiple lighting will be approved. 

(d) Under no circumstances can they be attached to gas fixtures. 

CONSTANT=POTENTIAL SYSTEMS. 

General Rules, all Voltages. 

21. Automatic Cutouts (Fuses and Circuit-breakers). 

(See No. 17, and for Construction, Nos. 52 and 53.) 

(a) Must be placed on all service wires, either overhead or underground, 
as near as possible to the point where they enter the building and inside 
the walls, and arranged to cut off the entire current from the building. 

Where the switch required by Rule No. 22 is inside the building, the 
cutout required by this section must be placed so as to protect it. 

(b) Must be placed at every point where a change is made in the size of 
wire (unless the cutout in the larger wire will protect the smaller). (See 
No. 16.) 

(c) Must be in plain sight or inclosed in an approved box (see No. 54) 
and readily accessible. They must not be placed in the canopies or shells 
of fixtures. 

(d) Must be so placed that no set of incandescent lamps, whether 
grouped on one fixture or several fixtures or pendants, requiring more 
than 660 watts shall be dependent upon one cutout. Special permission 
may be given in writing by the Inspection Department having jurisdiction 
for departure from this rule in case of large chandeliers, stage borders, 
and illuminated signs. 

(e) Must be provided with fuses, the rated capacity of which does not 
exceed the allowable carrying capacity of the wire; and, when circuit- 
breakers are used, they must not be set more than about 30 per cent, above 
the allowable carrying capacity of the wire, unless a fusible cutout is also 
installed in the circuit. (See No. 16.) 

22. Switches. 

(See No. 17, and for Construction, No. 51.) 

(a) Must be placed on all service wires, either overhead or underground, 
in a readily-accessible place, as near as possible to the point where the 
wires enter the building, and arranged to cut off the entire current. 

(6) Must always be placed in dry, accessible places, and be grouped, as 
far as possible. Knife switches must be so placed that gravity will tend to 
open rather than close the switch. 

(c) Must not be single-pole, except when the circuits which they con- 
trol supply not more than six 16-candle-power lamps or their equivalent. 

(d) Where flush switches are used, whether with conduit systems or 
not, the switches must be inclosed in boxes constructed of or lined with 
fire-resisting material. No push-buttons for bells, gas-lighting circuits, or 
the like shall be placed in the same wall-plate with switches controlling 
electric light or power wiring. 

23. Electric Heaters. 

(a) Must, if stationary, be placed in a safe situation, isolated from in- 
flammable materials, and be treated as sources of heat. 

(b) Must each have a cutout and indicating switch. (See No. 17, a.) 

(c) Must have the attachments of feed wires to the heaters in plain 
sight, easily accessible, and protected from interference, accidental or 
otherwise. 



714 National Electric Code. 

(d) The flexible conductors for portable apparatus, such as irons, etc., 
must have an approved insulating covering. (See No. 45, h.) 

(e) Must each be provided with name-plate, giving the maker's name 
and the normal capacity in volts and amperes. 

* 

LOW=POTENTIAL SYSTEMS. 

550 Volts or Less. 

Any circuit attached to any machine, or combination of machines, 
which develops a difference of potential between any two wires of over 10 
volts and less than 550 volts, shall be considered as a low-potential circuit, 
and as coming under this class, unless an approved transforming device is 
used, which cuts the difference of potential down to 10 volts or less. The 
primary circuit not to exceed a potential of 3500 volts. 



24. Wires. 

General Rules. 
(See, also, Nos. 14, 15, and 16.) 
(a) Must not be laid in plaster, cement, or similar finish. 






(b) Must never be fastened with staples. 

(c) Must not be fished for any great distance, and only in places where 
the inspector can satisfy himself that the rules have been complied with. 

(d) Twin wires must never be used, except in conduits or where flexi- 
ble conductors are necessary. 

(e) Must be protected on side walls from mechanical injury. When 
crossing floor-timbers in cellars or in rooms where they might be exposed 
to injury, wires must be attached by their insulating supports to the under 
side of a wooden strip not less than % inch in thickness and not less than 
3 inches in width. 

Suitable protection on side walls may be secured by a substantial box- 
ing, retaining an air space of 1 inch around the conductor, closed at the 
top (the wires passing through bushed holes), and extending not less than 
5 feet from the floor ; or by an iron -armored or metal-sheathed insulating 
conduit sufficiently strong to withstand the strain it will be subjected to ; 
or plain metal pipe lined with insulating tubing, which must extend % 
inch beyond the end of the metal tube. 

The pipe must extend not less than 5 feet above the floor, and may 
extend through the floor in place of a floor bushing. 

If iron pipes are used with alternating currents, the two or more wires 
of a circuit must be placed in the same conduit. In this case the insula- 
tion of each wire must be reinforced by a tough conduit tubing projecting 
beyond the ends of the iron pipe at least 2 inches. 

(/) When run immediately under roofs, or in proximity to water tanks 
or pipes, will be considered as exposed to moisture. 

Special Rules. 
For Open Work : 

In dry places : 

(g) Must have an approved rubber or "slow-burning," water-proof insu- 
lation. (See Nos. 41 and 42.) 

(h) Must be rigidly supported on non-combustible, non-absorptive insu- 
lators, which separate the wires from each other and from the surface 
wired over in accordance with following table : 






Voltage. 


Difference from surface. 


Distance between wires 


to 225 
225 to 550 


% inch 
1 inch 


2% inches 
4 inches 



Rigid supporting requires, under ordinary conditions, where wiring 
along flat surfaces, supj>orts at least every 4>£ feet. If the wires are liable 
to he disturbed, the distance between supports should be shortened. In 
buildings of mill construction, mains of No. 8 B. & S. wire or over, where 
not liable to be disturbed, may be separated about 4 inches and run from 



National Electric Code. 715 

timber to timber,— not breaking around,— and may be supported at each 
timber only. 

This rule will not be interpreted to forbid the placing of the neutral of 
a 3-wire system in the centre of a 3-wire cleat, provided the outside wires 
B j are separated in accordance with table on page 714. 

In damp places, such as breweries, sugar-houses, packing-houses, stables, 
dye-houses, paper or pulp mills, or buildings specially liable to moisture 
or acid or other fumes liable to injure the wires or their insulation, 
except where used for pendants : 

(i) Must have an approved rubber insulating covering. (See No. 41.) 
(j) Must be rigidly supported on non-combustible, non-absorptive insu- 
lators, which separate the wire at least 1 inch from the surface wired over, 
and they must be kept apart at least 2% inches. 

Rigid supporting requires, under ordinary conditions, where wiring 
over flat surfaces, supports at least every 4% feet. If the wires are liable 
to be disturbed, the distance between supports should be shortened. In 
buildings of mill construction, mains of No. 8 B. & S. wire or over, where 
not liable to be disturbed, may be separated about 4 inches and run from 
timber to timber,— not breaking around, — and may be supported at each 
timber only. 

(k) Must have no joints or splices. 

For Molding Work : 

(I) Must have approved rubber insulating covering. (See No. 41.) 
(m) Must never be placed in molding in concealed or damp places. 

For Conduit Work : 

(n) Must have an approved rubber insulating covering. (See No. 41.) 

(o) Must not be drawn in until all mechanical work on the building 
I has been, as far as possible, completed. 

(p) Must, for alternating systems, have the two or more wires of a 
circuit drawn in the same conduit. 

It is advised that this be done for direct-current systems also, so that 
they may be changed to alternating systems at any time, induction 
troubles preventing such a change unless this construction is followed. 

For Concealed "Knob and Tube" Work: 

(q) Must have an approved rubber insulating covering. (See No. 41. ) 

(r) Must be rigidly supported on non-combustible, non-absorptive in- 
sulators, which separate the wire at least 1 inch from the surface wired 
over, and must be kept at least 10 inches apart, and, when possible, should 
be run singly on separate timbers or studding. 

Rigid supporting requires, under ordinary conditions, where wiring 
along flat surfaces, supports at least every 4% feet. If the wires are liable 
to be disturbed, the distance between supports should be shortened. 

(s) When, from the nature of the case, it is impossible to place con- 
cealed wiring on non-combustible insulating supports of glass or porce- 
lain, an approved armored cable with single or twin conductors (see No. 48) 
may be used where the difference of potential between wires is not over 
300 volts, provided it is installed without joints between outlets, and the 
cable armor properly enters all fittings and is rigidly secured in place ; or, 
if the difference of potential between wires is not over 300 volts, and if 
wires are not exposed to moisture, they may be fished on the loop system 
if separately incased throughout in approved flexible tubing or conduits. 

For Fixture Work: 

(t) Must have an approved rubber insulating covering (see No. 46), and 
shall not be less in size than No. 18 B. & S. 

(u) Supply conductors, and especially the splices to fixture wires, must 
be kept clear of the grounded part of gas-pipes ; and, where shells are 
used, the latter must be constructed in a manner affording sufficient area 
to allow this requirement. 

(v) Must, when fixtures are wired outside, be so secured as not to be cut 
or abraded by the pressure of the fastenings or motion of the fixture. 



716 National Electric Code. 

25. Interior Conduits. 

(See, also, Nos. 24, n to p, and 49.) 

The object of a tube or conduit is to facilitate the insertion or extraction 
of the conductors, to protect them from mechanical injury, and, as far as • 
possible, from moisture. Tubes or conduits are to be considered merely as 
raceways, and are not to be relied upon for insulation between wire and 
wire, or between the wire and the ground. 

la) No conduit tube having an internal diameter of less than % inch 
shall be used. (If conduit is lined, measurement to be taken inside of 
lining.) 

(b) Must be continuous from one junction box to another or to fixtures, 
and the conduit tube must properly enter all fittings. 

(c) Must be first installed as a complete conduit system, without the 
conductors. 

(d) Must be equipped at every outlet with an approved outlet box. 

(e) Metal conduits, where they enter junction boxes, and at all other 
outlets, etc., must be fitted with a capping of approved insulating material, 
fitted so as to protect wire from abrasion. 

(/) Must have the metal of the conduit permanently and effectively 
grounded. 

26. Fixtures. 

(See, also, No. 24, t to v.) 

(a) Must, when supported from the gas-piping of a building, be in- 
sulated from the gas-pipe system by means of approved insulating joints 
(see No. 59) placed as close as possible to the ceiling. 

It is recommended that the gas outlet pipe be protected above the insu- 
lating joint by a non-combustible, non-absorptive insulating tube, having 
a flange at the lower end where it comes in contact with the insulating 
joint; and that, where outlet tubes are used, they be of sufficient length 
to extend below the insulating joint, and that they be so secured that they 
will not be pushed back when the canopy is put in place. Where iron 
ceilings are used care must be taken to see that the canopy is thoroughly 
and permanently insulated from the ceiling. 

(b) Must have all burs, or fins, removed before the conductors are 
drawn into the fixture. 

(c) The tendency to condensation within the pipes should be guarded 
against by sealing the upper end of the fixture. 

(d) No combination fixture in which the conductors are concealed in a 
space less than % inch between the inside pipe and the outside casing will 
be approved. 

(e) Must be tested for "contacts" between conductors and fixture, for 
"short circuits," and for ground connections before it is connected to its 
supply conductors. 

(/) Ceiling blocks for fixtures should be made of insulating material ; 
if not, the wires in passing through the plate must be surrounded with 
non-combustible, non-absorptive insulating material, such as glass or 
porcelain. 

(g) Under no conditions shall there be a difference of potential of more 
than 300 volts between wires contained in or attached to the same fixture. 

27. Sockets. 

(For Construction Rules, see No. 55.) 

(a) In rooms where inflammable gases may exist the incandescent 
lamp and socket must be inclosed in a vapor-tight globe and supported on 
a pipe hanger, wired with approved rubber-covered wire (see No. 41) 
soldered directly to the circuit. 

(b) In damp or wet places, or over specially-inflammable stuff, water- 
proof sockets must be used. 

When water-proof sockets are used they should be hung by separate- 
stranded, rubber-cove icd wires, not smaller than No. 14 B. & S., which 
should preferably be twisted together when the drop is over 3 feet. These 
wires should be soldered direct to the circuit wires, but supported inde- 
pendently of them. 



National Electkic Code. 717 

28. Flexible Cord. 

(a) Must have an approved installation and covering. (See No. 45.) 

(b) Must not be used where the difference of potential between the two 
wires is over 300 volts. 

■ t (c) Must not be used as a support for clusters. 

(d) Must not be used except for pendants, wiring of fixtures, and porta- 
ble lamps or motors. 

(e) Must not be used in show windows. 

(/) Must be protected by insulating bushings where the cord enters the 
socket. 

(g) Must be so suspended that the entire weight of the socket and lamp 
will be borne by knots under the bushing in the socket, and above the 
point where the cord comes through the ceiling-block or rosette, in order 
that the strain may be taken from the joints and binding screws. 

29. Arc Lights on Low=potential Circuits. 

(a) Must have a cutout (see No. 17, a) for each lamp of each series of 
lamps. 

The branch conductors should have a carrying capacity about 50 per 
cent, in excess of the normal current required by the lamp, to provide for 
heavy current required when lamp is started or when carbons become 
stuck, without overfusing the wires. 

(b) Must only be furnished with such resistances or regulators as are 
inclosed in non-combustible material, such resistances being treated as 
sources of heat. Incandescent lamps must not be used for resistance 
devices. 

(c) Must be supplied with globes and protected by spark arresters and 
wire netting around globe, as in the case of arc lights on high-potential 
circuits. (See Nos. 19 and 58.) 

30. Economy Coils. 

(a) Economy and compensator coils for arc lamps must be mounted on 
non-combustible, non-absorptive insulating supports, such as glass or porce- 
lain, allowing an air space of at least 1 inch between frame and support, 
and in general to be treated like sources of heat. 

31. Decorative Series Lamps. 

(a) Incandescent lamps run in series shall not be used for decorative 
purposes inside of buildings, except by special permission in writing from 
the Inspection Department having jurisdiction. 

32. Car=wiring. 

(a) Must be always run out of reach of the passengers, and must have 
an approved rubber insulating covering. (See No. 41.) 

33. Car=houses. 

(a) Must have the trolley wires securely supported on insulating 
hangers. 

{b) Must have the trolley hangers placed at such distance apart that in 
case of a break in the trolley wire contact cannot be made with the floor. 

(c) Must have cutout switch located at a proper place outside of the 
building, so that all trolley circuits in the building can be cut out at one 
point; and line circuit-breakers must be installed, so that when this cutout 
switch is open the trolley wire will be dead at all points within 100 feet of 
the building. The current must be cut out of the building whenever the 
same is not in use or the road not in operation. 

(d) Must have all lamps and stationary motors installed in such a way 
that one main switch can control the whole of each installation, — lighting 
or power,— independently of main feeder-switch. No portable incandes- 
cent lamps or twin wire allowed, except that portable incandescent lamps 
may be used in the pits, connections to be made by two approved rubber- 



718 National Electric Code. 

covered, flexible wires (see No. 41), properly protected against mechanical 
injury ; the circuit to be controlled by a switch placed outside of the pit. 

(e) Must have all wiring and apparatus installed in accordance with 
rules under Class C for constant-potential systems. 

(/) Must not have any system of feeder distribution centring in the 
building. 

(a) Must have the rails bonded at each joint with no less than No. 2 
B. & S. annealed copper wire ; also, a supplementary wire to be run for 
each track. 

(h) Must not have cars left with trolley in electrical connection with 
the trolley wire. 

34. Lighting and Power from Railway Wires. 

(a) Must not be permitted, under any pretense, in the same circuit with 
trolley wires with a ground return, except in electric railway cars, electric 
car houses, and their power stations ; nor shall the same dynamo be used 
for both purposes. 



HIGH=POTENTIAL SYSTEMS. 

550 to 3500 Volts. 

Any circuit attached to any machine, or combination of machines, 
which develops a difference of potential between any two wires of over 
300 volts and less than 3500 volts, shall be considered as a high-potential 
circuit, and as coming under that class, unless an approved transforming 
device is used which cuts the difference of potential down to 300 volts or 
less. 

35. Wires. 

(See, also, Nos. 14, 15, and 16.) 

(a) Must have an approved rubber insulating covering. (See No. 41.) 

(b) Must be always in plain sight and never incased, except where 
required by the Inspection Department having jurisdiction. 

(c) Must be rigidly supported on glass or porcelain insulators, which 
raise the wire at least'l inch from the surface wired over, and must be kept 
apart at least 4 inches for voltages up to 750, and at least 8 inches for volt- 
ages over 750. 

Rigid supporting requires, under ordinary conditions, where wiring 
along flat surfaces, supports at least about every 4% feet. If the wires are 
unusually liable to be disturbed, the distance between supports should be 
shortened. 

In buildings of mill construction, mains of No. 8 B. & S. wire or over, 
where not liable to be disturbed, may be separated about 6 inches for 
voltages up to 750, and about 10 inches for voltages above 750, and run 
from timber to timber,— not breaking around,— and may be supported at 
each timber only. 

(d) Must be protected on side walls from mechanical injury by a sub- 
stantial boxing, retaining an air space of 1 inch around the conductors, 
closed at the top (the wires passing through bushed holes), and extending 
not less than 7 feet from the floor. When crossing floor-timbers in cellars 
or in rooms where they might be exposed to injury, wires must be attached 
by their insulating supports to the under side of a wooden strip not less 
than % inch in thickness. 

36. Transformers (when permitted inside buildings). (See No. 13.) 
(For Construction Rules, see No. 62.) 

(a) Must be located at a point as near as possible to that at which the 
primary wires enter the building. 

(b) Must be placed in an inclosurc constructed of or lined with fire- 
resisting material ; the inclosure to be used only for this purpose and to be 
kept securely locked, and access to the same allowed only to responsible 
persons. 



National Electkic Code. 719 

(c) Must be effectually insulated from the ground, and the inclosure in 

which they are placed must be practically air-tight, except that it shall be 

thoroughly ventilated to the out-door air, if possible, through a chimney 

or flue. There should be at least 6 inches air space on all sides of the 

c transformer. 

37. Series Lamps. 

(a) No system of multiple-series or series-multiple for light or power 
will be approved. 

(b) Under no circumstances can lamps be attached to gas fixtures. 



EXTRA HIGrUPOTENTIAL SYSTEMS. 

Over 3500 Volts. 

Any circuit attached to any machine, or combination of machines, 
which develops a difference of potential between any two wires of over 
3500 volts, shall be considered as an extra high-potential circuit, and as 
coming under that class, unless an approved transforming device is used, 
which cuts the difference of potential down to 3500 volts or less. 

38. Primary Wires. 

(a) Must not be brought into or over building, except power and sub- 
stations. 

39. Secondary Wires. 

(a) Must be installed under rules for high-potential systems when their 
immediate primary wires carry a current of over 3500 volts, unless the 
primary wires are entirely underground within city and village limits. 

The presence of wures carrying a current with a potential of over 3500 
volts in the streets of cities, towns, and villages is considered to increase 
the fire hazard. Extra high-potential circuits are also objectionable in any 
location where telephone, telegraph, and similar circuits run in proximity 
to them. As the underwriters have no jurisdiction over streets and roads, 
they can only take this indirect way of discouraging such systems ; but 
further, it is strongly urged that municipal authorities absolutely refuse to 
grant any f ranchise'for right of way for overhead wires carrying a current 
of extra high-potential through streets or roads which are used to any 
great extent for public travel or for trunk-line, telephone, or telegraph 
circuits. 



CLASS D.-FITTINGS, MATERIALS, AND DETAILS OF CON= 
STRUCTION. 

All Systems and Voltages. Insulated Wires, Rules 40 to 48. 

40. General Rules. 

(a) Copper for insulated conductors must never vary in diameter so as 
to be more than T5 2 5 o inch less than the specified size. 

(b) Wires and cables of all kinds designed to meet the following specifi- 
cations must be plainly tagged or marked as follows : 

1. The maximum voltage at which the wire is designed to be used. 

2. The words " National Electrical Code Standard." 

3. Name of the manufacturing company, and, if desired, trade-name of 
the wire. 

4. Month and year when manufactured. 

41. Rubber=covered. 

(a) Copper for conductors must be thoroughly tinned. 



From 


18 to 


From 


14 to 


From 


7 to 


From 


lto 


From 


0,000 to 



720 National Electric Code. 

Insulation for Voltages Between and 600. 

(b) Must be of rubber or other approved substance, and be of a thick- 
ness not less than that given in the following table for B. & S. gauge sizes : 

16, inclusive, ^ inch. 
8, inclusive, -fy inch. 
2, inclusive, j 1 ^ inch. 
0,000, inclusive, ^ inch. 
500,000 C. M., scinch. 
From 500,000 to 1,000,000 C. M., & inch. 
Larger than 1,000,000 CM., % inch. 

Measurements of insulating wall are to be made at the thinnest portion 
of the dielectric. 

(c) The completed coverings must show an insulation resistance of at 
least 100 megohms per mile during 30 days' immersion in water at 70° F. 

(d) Each foot of the completed covering must show a dielectric strength 
sufficient to resist throughout 5 minutes the application of an electro- 
motive force of 3000 volts per ^ inch thickness of insulation under the 
following conditions : 

The source of alternating electro-motive force shall be a transformer of 
at least 1 kilowatt capacity. The application of the electro-motive force 
shall first be made at 4000 volts for 5 minutes, and then the voltage in- 
creased by steps of not over 3000 volts, each held for 5 minutes, until the 
rupture of the insulation occurs. The tests for dielectric strength shall be 
made on a sample of wire which has been immersed for 72 hours in water, 
1 foot of which is submerged in a conducting liquid held in a metal trough, 
one of the transformer terminals being connected to the wire and the 
other to the metal of the trough. 

Insulation for Voltages Between 600 and 3500. 

(e) The thickness of the insulating walls must not be less than those 
given in the following table for B. & S. gauge sizes : 

From 14 to 1, inclusive, & inch. 

From to 500,000 C. M., -^ inch, covered by a tape or a braid. 

Larger than 500,000 C. M., ^ inch, covered by a tape or a braid. 

(/) The requirements as to insulation and break-down resistance for 
wires for low-potential systems shall apply, with the exception that an in- 
sulation resistance of not less than 300 megohms per mile shall be required. 

{g) Wire for arc-light circuits exceeding 3500 volts potential shall have 
an insulating wall not less than ^ inch in thickness, and shall withstand 
a break-down test of at least 30,000 volts, and have an insulation of at 
least 500 megohms per mile. 

The tests on this wire to be made under the same conditions as for low- 
potential wires. 

Specifications for insulations for alternating currents exceeding 3500 
volts have been considered, but on account of the somewhat complex 
conditions in such work it has so far been deemed inexpedient to specify 
general insulations for this use. 

(h) All of the above insulations must be protected by a substantial 
braided covering, properly saturated with a preservative compound, and 
sufficiently strong to withstand all the abrasion likely to be met with in 
practice, and sufficiently elastic to permit all wires smaller than No. 7 
B. & S. gauge to be bent around a cylinder with twice the diameter of the 
wire, without injury to the braid. 






42. Slow=burning, Weather=proof. 

(a) The insulation shall consist of two coatings, the inner one to be 
fire-proof in character, the outer to be weather-proof. The inner fire-proof 
coating must comprise at least ^ of the total thickness of the wall. The 



: 



National Electric Code. 721 

_ 

completed covering must be of a thickness not less than that given in the 
following table for B. & S. gauge sizes : 

From 14 to 8, inclusive, -^ inch. 

From 7 to 2, inclusive, T V inch. 

From 2 to 0,000, inclusive, B 5 ¥ inch. 

From 0,000 to 500,000 C. M., T&inch. 
From 500,000 to 1,000,000 C. M., & inch. 
Larger than 1,000,000 C. M., % inch. 

Measurements of insulating wall are to be made at the thinnest portion 
of the dielectric. 

(b) The inner fire-proof coating shall be layers of cotton or other thread, 
the outer one of which must be braided. All the interstices of these layers 
are to be filled with the fire-proofing compound. This is to be material 
whose solid constituent is not susceptible to moisture, and which will not 
burn even when ground in an oxidizable oil, making a compound which, 
while proof against fire and moisture, at the same time has considerable 
elasticity, and which, when dry, w T ill suffer no change at a temperature of 
250° F., and which will not burn at even higher temperature. 

(c) The weather-proof coating shall be a stout braid thoroughly satu- 
rated with a dense, moisture-proof compound thoroughly slicked down, 
applied in such manner as to drive any atmospheric moisture from the 
cotton braiding, thereby securing a covering to a greater degree water- 
proof and of high insulating power. This compound to retain its elasticity 
at zero Fahrenheit, and not to drip at 160° F. 

This wire is not as burnable as the old "weather-proof," nor as subject 
to softening under heat, but still is able to repel the ordinary amount of 
moisture found in-doors. It would not usually be used for outside work. 

43. Slow=burning. 

(a) The insulation shall be the same as the "slow-burning, weather- 
proof," except that the outer braiding shall be impregnated with a fire- 
proofing compound similar to that required for the interior layers, and 
with the outer surface finished smooth and hard. 

This "slow-burning" wire shall only be used with special permission of 
the Inspection Department having jurisdiction. 

This is practically the old " Underwriters' " insulation. It is specially 
useful in hot, dry places, where ordinary insulations would perish; also 
where wires are bunched, as on the back of a large switchboard or in a 
wire tower, so that the accumulation of rubber or weather-proof insula- 
tion would result in an objectionably large mass of highly-inflammable 
material. 

Its use is restricted, as its insulating qualities are not high and are 
damaged by moisture. 

44. Weather=proof. 

(a) The insulating covering shall consist of at least 3 braids thoroughly 
impregnated with a dense moisture repellent, which will not drip at a 
temperature lower than 180° F. The thickness of insulation shall be not 
less than that of "slow-burning, weather-proof." The outer surface shall 
be thoroughly slicked down. 

This wire is for out-door use, where moisture is certain and where fire- 
proof qualities are not necessary. 

45. Flexible Cord. 

(a) Must be made of stranded copper conductors, each strand to be not 
larger than No. 26 or smaller than No. 30 B. & S. gauge, and each stranded 
conductor must be covered by an approved insulation and protected from 
mechanical injury by a tough, braided outer covering. 

For Pendent Lamps : 

In this class is to be included all flexible cord which under usual condi- 
tions hangs freely in air, and which is not likely to be moved sufficiently 
to come in contact with surrounding objects. 

46 



722 National Electric Code. 

(6) Each stranded conductor must have a carrying capacity equivalent 
to not less than a No. 18 B. & S. gauge wire. 

(c) The covering of each stranded conductor must be made up as 
follows : 

1. A tight, close wind of fine cotton. i 

2. The insulation proper, which shall be either water-proof or slow- 
burning. 

3. An outer cover of silk or cotton. 

The wind of cotton tends to prevent a broken strand puncturing the 
insulation and causing a short circuit. It also keeps the rubber from 
corroding the copper. 

(d) Water-proof insulation must be solid, at least ^ inch thick, and 
must show an insulation resistance of 50 megohms per mile throughout 
2 weeks' immersion in water at 70° F., and stand the test prescribed for 
low-tension wires as far as they apply. 

(e) Slow-burning insulation must be at least 3 X 2 inch in thickness, and 
composed of substantial, elastic, slow-burning materials, which will suffer 
no damage at a temperature of 250° F. 

(/) The outer protecting braiding should be so put on and sealed in 
place that when cut it will not fray out, and where cotton is used it should 
be impregnated with a flame-proof paint, which will not have an injurious 
effect on the insulation. 

For Portables : 

In this class is included all cord used on portable lamps, small portable 
motors, etc. 

(g) Flexible cord for portable use must have water-proof insulation, as 
required in Section d for pendent cord, and in addition be provided with a 
reinforcing cover especially designed to withstand the abrasion it will be 
subject to in the uses to which it is to be put. 

For Portable Heating Apparatus : 

(h) Must be made up as follows : 

1. A tight, close wind of fine cotton. 

2. A thin layer of rubber about T £ n inch thick, or other cementing 
material. 

3. A layer of asbestos insulation at least 5 3 j inch thick. 

4. A stout braid of cotton. 

5. An outer reinforcing cover especially designed to withstand abrasion. 
This cord is in no sense water-proof, the thin layer of rubber being 

specified in order that it may serve merely as a seal to help hold in place 
the fine cotton and asbestos, and it should be so put on as to accomplish 
this. 

46. Fixture Wire. 

(a) Must have a solid insulation, with a slow-burning, tough, outer 
covering, the whole to be 5 V inch in thickness, and show an insulation re- 
sistance between conductors, and between either conductor and the 
ground, of at least 1 megohm per mile, after 1 week's submersion in water 
at 70° F., and after 3 minutes' electrification with 550 volts. 

47. Conduit Wire. 

Must comply with the following specifications : 

(a) For metal conduits, having a lining of insulating material, single 
wires must comply with No. 41, and all duplex, twin, and concentric con- 
ductors must comply with No. 41, and must also have each conductor 
separately braided or taped, and a substantial braid covering the whole. 

(b) Porunlined metal conduits, conductors must conform to the speci- 
fications given for lined conduits, and in addition have a second outer ^ 
fibrous covering at least J$ inch in thickness, and sufficiently tenacious to 
withstand the abrasion of being hauled through the metal conduit. 

The braid required around each, conductor in duplex, twin, and con- 
centric cables is to hold the rubber insulation in place and prevent 
jamming and flattening. 



National Electric Code. 723 

48. Armored Cable. 

(a) The armor of such cables must be at least equal in thickness and of 
equal strength to resist penetration by nails, etc., as the armor of metal 
covering of metal conduits. (See No. 49, b.) 

(b) The conductors in same — single wire or twin conductors — must have 
an insulating covering as required by No. 41, any tiller used to secure a 
round exterior must be impregnated with a moisture repellent, and the 
whole bunch of conductors and fillers must have a separate exterior 
covering of insulating material at least -^ inch in thickness, conforming 
to the insulation standard given in No. 41, and covered with a substantial 
braid. 

Very reliable insulation is specified, as such cables are liable to hard 
usage, and in part of their length may be subject to moisture, while they 
may not be easily removable, so that a breakdown of insulation is likely 
to be expensive. 

49. Interior Conduits. 

(For Wiring Rules, see Nos. 24 and 25.) 

(a) Each length of conduit, whether insulated or uninsulated, must 
have the maker's name or initials stamped in the metal or attached thereto 
in a satisfactory manner, so that the inspectors can readily see the same. 

Metal Conduits with Lining of Insulating Material. 

(b) The metal covering or pipe must be equal in strength to the ordinary 
commercial forms of gas-pipe of the same size, and its thickness must be 
not less than that of standard gas-pipe, as shown by the following table : 



Size, in 


Thickness of 


Size, in 


Thickness of 


inches. 


wall, in inches. 


inches. 


wall, in inches. 


Vz 


.109 


m 


.140 


% 


.111 


VA 


.145 


% 


.113 


2 


.154 


1 


.134 







An allowance of T | \o inch for variation in manufacturing and loss of 
thickness by cleaning will be permitted. 

(c) Must not be seriously affected externally by burning out a wire 
inside the tube when the iron pipe is connected to one side of the circuit. 

(d) Must have the insulating lining firmly secured to the pipe. 

(e) The insulating lining must not crack or break when a length of the 
conduit is uniformly bent at temperature of 212° F. to an angle of 90°, 
with a curve having a radius of 15 inches for pipes of 1 inch and less, and 
15 times the diameter of pipe for larger pipes. 

(/) The insulating lining must not soften injuriously at a temperature 
below 212° F., and must leave water in which it is boiled practically 
neutral. 

(g) The insulating lining must be at least ^ inch in thickness ; and the 
materials of which it is composed must be of such a nature as will not 
have a deteriorating effect on the insulation of the conductor, and be suffi- 
ciently tough and tenacious to withstand the abrasion test of drawing long 
I lengths of conductors in and out of same. 

(h) The insulating lining must not be mechanically weak after 3 days' 
j submersion in water, and when removed from the pipe entire must not 
; absorb more than 10 per cent, of its weight of water during 100 hours of 
, submersion. 

(i) All elbows or bends must be so made that the conduit or lining of 
same will not be injured. The radius of the curve of the inner edge of 
any elbow not to be less than 3% inches. Must have not more than the 
equivalent of 4 quarter-bends from outlet to outlet, the bends at the 
outlets not being counted. 

Unlined Metal Conduits. 

(j) Plain iron or steel pipes of equal thickness and strengths specified 
for lined conduits in No. 49, b, may be used as conduits, provided their in- 



724 National Electric Code. 

terior surfaces are smooth and free from burs ; pipe to be galvanized, or 
the interior surfaces coated or enamelled to prevent oxidation with some 
substance which will not soften, so as to become sticky and prevent wire 
from being withdrawn from the pipe. 

(k) All elbows or bends must be so made that the conduit will not be** 
injured. The radius of the curve of the inner edge of any elbow not to 
be less than 3% inches. Must have not more than the equivalent of 4 
quarter-bends from outlet to outlet, the bends at the outlet not being 
counted. 

50. Wooden Moldings. 

(For Wiring Rules, see No. 24.) 

(a) Must have, both outside and inside, at least two coats of water-proof 
paint, or be impregnated with a moisture repellent. 

(6) Must be made of two pieces, a backing and capping, so constructed 
as to thoroughly incase the wire, and provide a %-inch tongue between 
the conductors, and a solid backing, which, under grooves, shall not be 
less than % inch in thickness, and must afford suitable protection from 
abrasion. 

It is recommended that only hardwood molding be used. 



51. Switches. 

(SeeNos. 17 and 22.) 

(a) Must be mounted on non-combustible, non-absorptive insulating 
bases, such as slate or porcelain. 

(b) Must have carrying capacity sufficient to prevent undue heating. 

(c) Must, when used for service switches, indicate, on inspection, 
whether the current be "on" or "off." 

(d) Must be plainly marked, where it will always be visible, with the 
name of the maker and the current and voltage for which the switch is 
designed. 

(e) Must, for constant-potential systems, operate successfully at 50 per 
cent, overload in amperes, with 25 per cent, excess voltage under the most 
severe conditions they are liable to meet with in practice. 

(/) Must, for constant-potential systems, have a firm and secure con- 
tact; must make and break readily, and not stop when motion has once 
been imparted by the handle. 

(g) Must, for constant-current systems^ close the main circuit and dis- 
connect the branch wires when turned " off ;" must be so constructed that 
they shall be automatic in action, not stopping between points when 
started, and must prevent an arc between the points under all circum- 
stances. They must indicate, upon inspection, whether the current be 
"on" or "off." 

52. Cutouts and Circuit=breakers. 

(For Installation Rules, see Nos. 17 and 21.) 

(a) Must be supported on bases of non-combustible, non-absorptive 
insulating material. 

(b) Cutouts must be provided with covers, when not arranged in ap- 
proved cabinets, so as to obviate any danger of the melted fuse metal 
coming in contact with any substance which might be ignited thereby. 

(c) Cutouts must operate successfully, under the most severe conditions 
they are liable to meet with in practice, on short circuits with fuses rated 
at 50 per cent, above, and with a voltage 25 per cent, above the current 
and voltage for which they are designed. 

(d) Circuit-breakers must operate successfully, under the most severe g 
conditions they are liable to meet with in practice, on short circuits when J 
set ai 50 per cent, above the current, and with a voltage 25 per cent, above 
that for which they an; designed. 

(e) Must be plainly marked, where it will always be visible, with the 
name of the maker 'and current and voltage for which the device is 
designed. 



National Electric Code. 725 

53. Fuses. 

(For Installation Rules, see Nos. 17 and 21.) 

(a) Must have contact surfaces or tips of harder metal having perfect 
electrical connection with the fusible part of the strip. 

(b) Must be stamped with about 80 per cent, of the maximum current 
they can carry indefinitely, thus allowing about 25 per cent, overload 
before fuse melts. 

With naked, open fuses of ordinary shapes and not over 500 amperes 
capacity, the maximum current which will melt them in about 5 minutes 
may be safely taken as the melting-point, as the fuse practically reaches 
its maximum temperature in this time. With larger fuses a longer time is 
necessary. 

Inclosed fuses, where the fuse is often in contact with substances having 
good conductivity to heat and often of considerable volume, require a 
much longer time to reach a maximum temperature, on account of the 
surrounding material, which heats up slowly. 

These data are given to facilitate testing. 

(c) Fuse terminals must be stamped with the maker's name, initials, or 
some known trade-mark. 

54. Cutout Cabinets. 

(a) Must be so constructed, and cutouts so arranged, as to obviate any 
danger of the melted fuse metal coming in contact with any substance 
which might be ignited thereby. 

A suitable box can be made of marble, slate, or wood, strongly put 
together, the door to close against a rabbet, so as to be perfectly dust-tight ; 
and it should be hung on strong hinges and held closed by a strong hook 
or catch. If the box is wood, the inside should be lined with sheets of 
asbestos-board about T ^ inch in thickness, neatly put on and firmly secured 
in place by shellac and tacks. The wire should enter through holes 
bushed with porcelain bushings ; the bushings tightly fitting the holes in 
the box, and the wires tightly fitting the bushings (using tape to build up 
the wire, if necessary), so as to keep out the dust. 

55. Sockets. 

(See No. 27.) 

Sockets of all kinds, including wall receptacles, must be constructed in 
accordance with the following specifications : 

(a) Standard Sizes.— The standard lamp socket shall be suitable for 
use on any voltage not exceeding 250, and with any size lamp up to 50 
candle-power. For lamps larger than 50 candle-power a standard keyless 
socket may be used, or, if a key is required, a special socket designed for 
the current to be used must be made. Any special sockets must follow the 
general spirit of these specifications. 

(6) Marking.— The standard socket must be plainly marked 50 candle- 
power, 250 volts, and with either the manufacturer's name or registered 
trade-mark. Special large sockets must be marked with the current and 
voltage for which they are designed. 

(c) Shell.— Metal used for shells must be moderately hard, but not 
hard enough to be brittle or so soft as to be easily dented or knocked out 
of place. Brass shells must be at least 0.013 inch in thickness, and shells of 
any other material must be thick enough to give the same stiffness and 
strength of brass. 

(d) Lining.— The inside of the shells must be lined with insulating 
material, which shall absolutely prevent the shell from becoming a part 
of the circuit, even though the wires inside the socket should start from 
their position under binding screws. 

The material used for lining must be at least -^ inch in thickness, and 
must be tough and tenacious. It must not be injuriously affected by the 
heat from the largest lamp permitted in the socket, and must leave the 
water in which it is boiled practically neutral. It must be so firmly 
secured to the shell that it will not fall out with ordinary handling of the 
socket. It is preferable to have the lining in one piece. 



726 National Electric Code. 

(e) Cap.— Caps, when of sheet brass, must be at least 0.013 inch in thick- 
ness, and when cast or made of other metals must be of equivalent 
strength. The inlet piece, except for special sockets, must be tapped and 
threaded for ordinary %-inch pipe. It must contain sufficient metal for a 
full, strong thread, and, when not of the same piece as the cap, must be 
joined to it in a way to give the strength of a single piece. 

There must be sufficient room in the cap to enable the ordinary wire- 
man to easily and quickly make a knot in the cord, and push it into place 
in cap without crowding. All parts of the cap upon which the knot is 
likely to bear must be smooth and well insulated. 

(/) Frame and Screws.— The frame holding moving parts must be 
sufficiently heavy to give ample strength and stiffness. 

Brass pieces containing screw threads must be at least 0.06 of an inch 
in thickness. 

Binding-post screws must not be smaller than No. 5 wire, and about 40 
threads per inch. 

(g) Spacing.— Points of opposite polarity must everywhere be kept not 
less than B 3 ¥ inch apart unless separated by a reliable insulation. 

(fi) Connections. — The connecting points for the flexible cord must be 
made to very securely grip a No. 16 or 18 B. & S. conductor. A turned-up 
lug, arranged so that the cord may be gripped between the screw and the 
lug in such a way that it cannot possibly come out, is strongly advised. 

(i) Lamp=holder. — The socket must firmly hold the lamp in place, so 
that it cannot be easily jarred out, and must provide a contact good 
enough to prevent undue heating with maximum current allowed. The 
holding-pieces, springs, and the like, if a part of the circuit, must not be 
sufficiently exposed to allow them to be brought in contact with anything 
outside of lamp and socket. 

(j) Base. — The inside parts of the socket, which are of insulating 
material (except the lining), must be made of porcelain. 

(k) Key.— The socket key-handle must be of such a material that it 
will not soften from the heat of a 50 candle-power lamp hanging down- 
ward, in air at 70° F., from the socket, and must be securely, but not 
necessarily rigidly, attached to the metal spindle it is designed to turn. 

(I) Sealing.— All screws in porcelain pieces, which can be firmly sealed 
in place, must be so sealed by a water-proof compound which will not 
melt below 200° F. 

(m) Putting Together.— The socket must, as a whole, be so put 
together that it will not rattle to pieces. Bayonet joints or equivalent are 
recommended. 

(n) Test.— The socket, when slowly turned "on and off" at the rate 
of about 2 or 3 times per minute, must "make and break" the 
circuit 6000 times before failing when carrying a load of 1 ampere at 220 
volts. 

(o) Keyless Sockets.— Keyless sockets of all kinds must comply with 
requirements for key sockets as far as they apply. 

(p) Sockets of Insulating Materials. — Sockets made of porcelain or 
other insulating material must conform to the above requirements as far 
as they apply, and all parts must be strong enough to withstand a moderate 
amount of hard usage without breaking. 

(q) Inlet Bushing.— When the socket is not attached to fixtures the 
threaded inlet must be provided with a strong insulating bushing having 
a smooth hole of at least J| inch in diameter. The corners of the bushing 
must be rounded and all inside fins removed, so that in no place will the 
cord be subjected to the cutting or wearing action of a sharp edge. 



56. Hanger=boards. 

(a) Hanger-boards must be so constructed that all wires and current- 
carrying devices thereon shall be exposed to view and thoroughly insu- 
lated by being mounted on a non-combustible, non-absorptive insulating 
substance. All switches attached to the same must be so constructed that 
they shall be automatic in their action, cutting off both poles to the lamp, 



National Electric Code. 727 

not stopping between points when started, and preventing an arc between 
points under all circumstances. 

57. Arc Lamps. 

(For Installation Rules, see No. 19.) 

(a) Must be provided with reliable stops to prevent carbons from falling 
out in case the clamps become loose. 

(&) Must be carefully insulated from the circuit in all their exposed 
parts. 

(c) Must, for constant-current systems, be provided with an approved 
hand switch, also an automatic switch that will shunt the current around 
the carbons should they fail to feed properly. 

The hand switch to be approved, if placed anywhere except on the 
lamp itself, must comply with requirements for switches on hanger-boards, 
as laid down in No. 56. 

58. Spark Arresters. 

(See No. 19, c.) 

(a) Spark arresters must so close the upper orifice of the globe that it 
will be impossible for any sparks thrown off by the carbons to escape. 

59. Insulating Joints. 

(See No. 26, a.) 

(a) Must be entirely made of material that will resist the action of 
illuminating gases and will not give way or soften under the heat of an 
ordinary gas-flame or leak under a moderate pressure. They shall be so 
arranged that a deposit of moisture will not destroy the insulating effect, 
and shall have an insulating resistance of at least 250,000 ohms between 
the gas-pipe attachments, and be sufficiently strong to resist the strain they 
will be liable to be subjected to in being installed. 

(b) Insulating joints having soft rubber in their construction will not 
be approved. 

60. Resistance Boxes and Equalizers. 

(For Installation Rules, see No. 4.) 

(a) Must be equipped with metal or with other non-combustible 
frames. 

The word "frame" in this section relates to the entire case and sur- 
roundings of the rheostat, and not alone to the upholding supports. 

61. Reactive Coils and Condensers. 

(a) Reactive coils must be made of non-combustible material, mounted 
on non-combustible bases, and treated, in general, like sources of heat. 

(b) Condensers must be treated like apparatus operating with equiva- 
lent voltage and currents. They must have non-combustible cases and 
supports, and must be isolated from all combustible materials, and, in 
general, treated like sources of heat. 

62. Transformers. 

(For Installation Rules, see Nos. 11, 13, and 33.) 

(a) Must not be placed in any but metallic or other non-combustible 
cases. 

(b) Must be constructed to comply with the following tests : 

1. Shall be run for 8 consecutive* hours at a full load in watts under 
conditions of service, and at the end of that time the rise in temperature, 
as measured by the increase of resistance of the primary coil, shall not 
exceed 135° F. 

2. The insulation of transformers, when heated, shall withstand con- 
tinuously for 5 minutes a difference of potential of 10,000 volts (alter- 



728 National Electric Code. 

nating) between primary and secondary coils and core, and between the 
primary coils and core and a no-load "run" at double voltage for 30 
minutes* 

63. Lightning Arresters. 

(For Installation Rules, see No. 5.) 

(a) Must be mounted on non-combustible bases, and must be so con- 
structed as not to maintain an arc after the discharge has passed, and 
must have no moving parts. 



CLASS E.-MISCELLANEOUS. 

64. Signalling Systems (governing wiring for telephone, telegraph, 

district messenger, and call-bell circuits, fire and burglar 

alarms, and all similar systems). 

(a) Outside wires should be run in underground ducts or strung on 
poles, and, as far as possible, kept off of buildings, and must not be placed 
on the same cross-arm with electric light or power wires. 

(b) When outside wires are run on same pole with electric light or 
power wires the distance between the two inside pins of each cross-arm 
must not be less than 26 inches. 

(c) All aerial conductors and underground conductors which are 
directly connected to aerial wires must be provided with some approved 
protective device, which shall be located as near their point of entrance 
to the building as possible, and not less than 6 inches from curtains or 
other inflammable material. 

(d) If the protector is placed inside of building, wires— from outside 
supports to binding-posts of protector— shall comply with the following 
requirements : 

1. Must be of copper, and not smaller than No. 16 B. & S. gauge. 

2. Must have an approved rubber insulating covering. (See No. 41.) 

3. Must have drip loops in each wire immediately outside the building. 

4. Must enter buildings through separate holes sloping upward from the 
outside. When practicable, holes to be bushed with non-absorptive, non- 
combustible insulating tubes extending through their entire length. 
Where tubing is not practicable, the wires shall be wrapped with two 
layers of insulating tape. 

5. Must be supported on porcelain insulators, so that they will not come 
in contact with anything other than their designed supports. 

6. A separation between wires of at least 2% inches must be main- 
tained. 

In case of crosses, these wires may become a part of a high-voltage 
circuit, so that similar care to that given high-voltage circuits is needed in 
placing them. Reliable porcelain bushings at the entrance holes are de- 
sirable, and are only waived under adverse conditions, because the state of 
the art in this type of wiring makes an absolute requirement inadvisable. 

(e) The ground wire of the protective device shall be run in accord- 
ance with the following requirements : 

1. Shall be of copper, and not smaller than No. 16 B. & S. 

2. Must have an approved rubber insulating covering. (See No. 41. ) 

3. Shall run in as straight a line as possible to a good, permanent 
ground, to be made by connecting to water- or gas-pipe, preferably water- 
pipe. If gas-pipe is used, the connection, in all cases, must be made 
between the meter and service pipes. In the absence of other good 
ground, the ground shall be made by means of a metallic plate or bunch 
of wires buried in permanently moist earth. 

4. Shall be kept at least 3 inches from all other conductors, and sup- 
ported on porcelain insulators, so as not to come in contact with anything 
other than its designated supports. 

In attaching a ground wire to a pipe it is often difficult to make a 
thoroughly-reliable solder joint. It is better, therefore, where possible, to 
carefully solder the wire to a brass plug, which may then be firmly screwed 
into a pipe fitting. 

Where such joints are made underground they should be thoroughly 
painted and taped to prevent corrosion. 



National Electric Code. 729 

(/) The protector, to "be approved, must comply with the following 
requirements : 

1. Must be mounted on non-combustible, non-absorptive insulating 
bases, so designed that when the protector is in place all parts which may 
be alive will be thoroughly insulated from the wall holding the protector. 

2. Must have the following parts : 

A lightning arrester which will operate with a difference of potential 
between wires of not over 500 volts, and so arranged that the chance of 
accidental grounding is reduced to a minimum. 

A fuse designed to open the circuit in case the wires become crossed 
with light or power circuits. The fuse must be able to open the circuit 
without arcing or serious flashing when crossed with any ordinary com- 
mercial light or power circuit. 

A heat coil which will operate before a sneak current can damage the 
instrument the protector is guarding. 

The heat coil is designed to warm up and melt out with a current large 
enough to endanger the instruments, if continued for a long time, but so 
small that it would not blow the fuses ordinarily found necessary for such 
instruments. These smaller currents are often called "sneak" currents. 

3. The fuses must be so placed as to protect the arrester and heat coils, 
and the protector terminals must be plainly marked " line," "instrument," 
"ground." 

(g) Wires beyond the protector, except where bunched, must be neatly 
arranged and securely fastened in place in any convenient, workmanlike 
manner. They must not come nearer than 6 inches to any electric light 
or power wire in the building, unless incased in approved tubing so 
secured as to prevent its slipping out of place. 

The wires would ordinarily be insulated, but the kind of insulation is 
not specified, as the protector is relied upon to stop all dangerous currents. 
Porcelain tubing or circular loom conduit may be used for incasing wires 
where required as above. 

(h) Wires connected with outside circuits, where bunched together 
within any building, or inside wires, where laid in conduits or ducts with 
electric light or power wires, must have fire-resisting coverings, or else 
must be inclosed in an air-tight tube or duct. 

It is feared that if a burnable insulation were used a chance spark 
might ignite it and cause a serious fire, for many installations contain a 
large amount of very readily-burnable matter. 

65. Electric Gas=lighting. 

Where electric gas-lighting is to be used on the same fixture with the 
electric light : 

(a) No part of the gas-piping or fixture shall be in electric connection 
with the gas-lighting circuit. 

(6) The wires used with the fixtures must have a non-inflammable in- 
sulation, or, where concealed between the pipe and shell of the fixture, 
the insulation must be such as required for fixture wiring for the electric 
light. 

(c) The whole installation must test free from "grounds." 

(d) The two installations must test perfectly free from connection with 
each other. 

66, Insulation Resistance. 

The wiring in any building must test free from grounds,— i.e., the com- 
plete installation must have an insulation between conductors and between 
all conductors and the ground (not including attachments, sockets, recep- 
tacles, etc.) of not less than the following : 

Up to 5 amperes 4,000,000 ohms. 

Up to 10 amperes 2,000,000 ohms. 

Up to 25 amperes 800,000 ohms. 

Up to 50 amperes 400,000 ohms. 

Up to 100 amperes 200,000 ohms. 

Up to 200 amperes 100,000 ohms. 

Up to 400 amperes 25,000 ohms. 

Up to 800 amperes 25,000 ohms. 

Up to 1600 amperes 12,500 ohms. 



730 National Electric Code. 

All cutouts and safety devices in place in the preceding. 
Where lamp sockets, receptacles, and electroliers, etc., are connected, 
one-half of the preceding insulation will be required. 

67. Soldering Fluid. 

(a) The following formula for soldering fluid is suggested : 

Saturated solution of zinc chloride 5 parts. 

Alcohol 4 parts. 

Glycerine 1 part. 

CLASS F.— MARINE WORK. 
68. Generators. 

(a) Must be located in a dry place. 

(b) Must have their frames insulated from their bed-plates. 

(c) Must each be provided with a water-proof cover. 

(d) Must each be provided with a name-plate, giving the maker's name, 
the capacity in voltage and amperes, and normal speed in revolutions per 
minute. 

69. Wires. 

(a) Must have an approved insulating covering. 

The insulation for all conductors, except for portables, to be approved, 
must be at least % inch in thickness and be covered with a substantial 
water-proof and flame-proof braid. The physical characteristics shall not 
be affected by any change in temperature up to 200° F. After 2 weeks' 
submersion in salt water at 70° F. it must show an insulation resistance of 
1 megohm per mile after 3 minutes' electrification with 550 volts. 

(b) Must have no single wire larger than No. 12 B. & S. Wires to be 
stranded when greater carrying capacity is required. No single solid wire 
smaller than No. 14 B. & S., except in fixture wiring, to be used. 

Stranded wires must be soldered before being fastened under clamps or 
binding screws, and when they have a conductivity greater than No. 10 
B. & S. copper wire they must be soldered into lugs. 

(c) Must be supported in approved molding, except at switchboards 
and portables, 

Special permission may be given for deviation from this rule in dynamo- 
rooms. 

(d) Must be bushed with hard-rubber tubing % inch in thickness when 
passing through beams and non-water-tight bulkheads. 

(e) Must have, when passing through water-tight bulkheads and 
through all decks, a metallic stuffing- tube lined with hard rubber. In 
case of deck tubes they shall be boxed near deck to prevent mechanical 
injury. 

(/) Splices or taps in conductors must be avoided as far as possible. 
Where it is necessary to make them they must be so spliced or joined as to 
be both mechanically and electrically secure without solder. They must 
then be soldered to insure preservation, covered with an insulating com- 
pound equal to the insulation of the wire, and further protected by a 
water-proof tape. The joint must then be coated or painted with a water- 
proof compound. 

70. Portable Conductors, 

(a) Must be made of two stranded conductors, each having a carding 
capacity equivalent to not less than No. 14 B. & S. wire, and each covered 
with an approved insulation and covering. 

Where not exposed to moisture or severe mechanical injury, each 
stranded conductor must have a solid insulation at least ^ inch in thick- 
ness, and must show an insulation resistance between conductors, and 
between either conductor and the ground, of at least 1 megohm per mile 
after 1 week's submersion in water at 70° F. and after 3 minutes' electrifi- 
cation with 590 volts, and be protected by a slow-burning, tough-braided, 
outer covering. 



National Electric Code. 



731 



Where exposed to moisture and mechanical injury — as for use on decks, 
holds, and fire-rooms— each stranded conductor shall have a solid insula- 
tion, to be approved, of at least ^ inch in thickness and protected by a 
tough braid. The two conductors shall then be stranded together, using 
, a jute filling. The whole shall then be covered with a layer of flax, either 
woven or braided, at least ■& inch in thickness, and treated with a non- 
inflammable, water-proof compound. After 1 week's submersion in water 
at 70° F., at 550 volts and a 3 minutes' electrification, must show an insula- 
tion between the two conductors, or between either conductor and the 
ground, of 1 megohm per mile. 

71. Bel! or Other Wires. 

(a) Shall never run in same duct with lighting or power wires 





72. Table of Capacity 


of Wires. 




B. & S. G. 


Area, actual 
C. M. 


Number of 
strands. 


Size of strands, 
B. &S. G. 


Amperes. 


19 


1288 
1624 
2 048 

2 583 

3 257 
4107 
6 530 
9 016 

11368 

14 336 

18 081 

22 799 

30 856 

38 912 

49 077 

60 088 

75 776 

99 064 

124 928 

157 563 

198 677 

250 527 

296 387 

373 737 

413 639 








18 






3 


17 








16 






6 


15 








14 






12 


12 






17 




7 
7 
7 
7 
7 

19 
19 
19 
37 
37 
61 
61 
61 
61 
61 
91 
91 
127 


19 
18 
17 
16 
15 
18 
17 
16 
18 
17 
18 
17 
16 
15 
14 
15 
14 
15 


21 




25 




30 




35 




40 




50 

60 






70 




85 




100 
1^0 






145 




170 




200 




235 




270 




320 




340 







When greater conducting area than that of a single wire is required, 
the conductor shall be stranded in a series of 7, 19, 37, 61, 91, or 127 wires, 
as may be required, the strand consisting of 1 central wire, the remainder 
laid around it concentrically, each layer to be twisted in the opposite 
direction from the preceding. 



73. Switchboards. 

(a) Must be made of non-combustible, non-absorptive insulating ma- 
terial, such as marble or slate. 

(6) Must be kept free from moisture, and must be located so as to be 
accessible from all sides. 



732 National Electkic Code. 

(c) Must have a main switch, main cutout, and ammeter for each 
generator. 

Must also have a voltmeter and ground detector. 

(d) Must have a cutout and switch for each side of each circuit leading 
from board. , 

74. Resistance Boxes. 

(a) Must be made of non-combustible material. 

(b) Must be located on switchboard or away from combustible material. 
When not placed on switchboard they must be mounted on non-inflamma- 
ble, non-absorptive insulating material. 

(c) Must be so constructed as to allow sufficient ventilation for the uses 
to which they are put. 



75. Switches. 






(a) Must have non-combustible, non-absorptive insulating bases. 

(b) Must operate successfully, at 50 per cent, overload in amperes with 
25 per cent, excess voltage, under the most severe conditions they are liable 
to meet with in practice, and must be plainly marked, where they will 
always be visible, with the name of the maker and the current and voltage 
for which the switch is designed. 

(c) Must be double pole when circuits which they control supply mon 
than six 16-candle-power lamps or their equivalent. 

(d) When exposed to dampness they must be inclosed in a water-tight 
case. 

76. Cutouts. 

(a) Must have non-combustible, non-absorptive insulating bases. 

(b) Must operate successfully, under the most severe conditions the; 
are liable to meet with in practice, on short circuit, with fuse rated at " 
per cent, above and with a voltage 25 per cent, above the current and 
voltage they are designed for, and must be plainly marked, where the: 
will always be visible, with the name of the maker and current and vol' 
age for which the device is designed. 

(c) Must be placed at every point where a change is made in the size of 
the wire (unless the cutout in the larger wire will protect the smaller). 

(d) In places such as upper decks, holds, cargo spaces, and fire-rooms a 
water-tight and fire-proof cutout may be used, connecting directly to 
mains when such cutout supplies circuits requiring not more than 660 
watts energy. 

(e) When placed anywhere except on switchboards and certain places, 
as cargo spaces, holds, fire-rooms, etc., where it is impossible to run from 
centre of distribution, they shall be in a cabinet lined with fire-resisting 
material. 

(/) Except for motors, searchlights, and diving-lamps, shall be so placed 
that no group of lamps requiring a current of more than 6 amperes shall 
ultimately be dependent upon 1 cutout. 

A single-pole, covered cutout may be placed in the molding when same 
contains conductor supplying circuits requiring not more than 220 watts 
energy. 

77. Fixtures. 

(a) Shall be mounted on blocks made from well-seasoned lumbe: 
treated with two coats of white lead or shellac. 

{b) Where exposed to dampness the lamp must be surrounded by a 
vapor-proof globe. 

(c) Where exposed to mechanical injury the lamp must be surrounded 
by a globe protected by a stout wire guard. 

(d) Shall be wired with same grade of insulation as portable conductors 
which are not exposed to moisture or mechanical injury. 

78. Sockets. 

(a) No portion of the lamp socket or lamp base exposed to contact with 
outside objects shall be allowed to come into electrical contact with either 
of the conductors. 






National Electric Code. 733 

79. Wooden Moldings. 

(a) Must be made of well-seasoned lumber and be treated inside and 
out with at least two coats of white lead or shellac. 

(6) Must be made of two pieces, a backing and a capping, so con- 
' structed as to thoroughly incase the wire and provide a %-inch tongue 
between the conductors, and a solid backing which, under grooves, shall 
not be less than % inch in thickness. 

(c) Where molding is run over rivets, beams, etc., a backing strip must 
first be put up and the molding secured to this. 

(d) Capping must be secured by brass screws. 

80. Motors. 

(a) Must be wired under the same precautions as with a current of same 
volume and potential for lighting. The motor and resistance box must be 
protected by a double-pole cutout and controlled by a double-pole switch, 
except in cases where % horse- power or less is used. 

The leads or branch circuits should be designed to carry a current at 
least 50 per cent, greater than that required by the rated capacity of the 
motor, to provide for the inevitable overloading of the motor at times. 

(b) Must be thoroughly insulated. Where possible, should be set on 
base frames made from rilled, hard, dry wood, and raised above surround- 
ing deck. On hoists and winches they shall be insulated from bed-plates 
by hard rubber, fibre, or similar insulating material. 

(c) Shall be covered with a water-proof cover when not in use. 

(d) Must each be provided with a name-plate giving maker's name, the 
capacity in volts and amperes, and the normal speed, in revolutions, per 
minute. 

GENERAL SUGGESTIONS. 

In all electric work, conductors, however well insulated, should always 
be treated as bare, to the end that under no conditions, existing or likely 
to exist, can a grounding or short circuit occur, and so that all leakage 
from conductor to conductor, or between conductor and ground, may be 
reduced to the minimum. 

In all wiring special attention must be paid to the mechanical execu- 
tion of the work. Careful and neat running, connecting, soldering, taping 
of conductors, and securing and attaching of fittings are specially con- 
ducive to security and efficiency, and will be strongly insisted on. 

In laying out an installation, except for constant-current systems, the 
work should, if possible, be started from a centre of distribution, and the 
switches and cutouts controlling and connected with the several branches 
be grouped together in a safe and easily-accessible place, where they can 
be readily got at for attention or repairs. The load should be divided as 
evenly as possible among the branches, and all complicated and unneces- 
sary wiring avoided. 

The use of wire-ways for rendering concealed wiring permanently 
accessible is most heartily endorsed and recommended ; and this method 
of accessible, concealed construction is advised for general use. 

Architects are urged, when drawing plans and specifications, to make 
provision for the channelling and pocketing of buildings for electric light 
or power wires, and in specifications for electric gas-lighting to require a 
2- wire circuit, whether the building is to be wired for electric lighting or 
not, so that no part of the gas fixtures or gas-piping be allowed to be used 
for the gas-lighting circuit. 



General Wiring Formulas. 

(General Electric Company.) 

The following general formulas may be used to determine the size of 
copper conductors, volts loss in lines, current per conductor, and the weight 
of copper per circuit for any system of electrical distribution : 



734 



Wiring Formulas. 



Area of conductor, circular mils. = 
Current in main conductors 
Volts loss in line 
Pounds of copper 



Where 



JX WXC . 
P X E* ' 
WX T m 
E ' 
PX EX B m 
100 
J> 2 X WXCXA 
" P X E*X 1,000,000 ' 



W — total watts delivered ; 
D = distance of transmission (one way), in feet ; 
p = loss in line, in per cent., of power delivered, — that is, of TV, 
E = voltage between main conductors at receiving or consumer's 
end of circuit. 

For continuous current C= 2160, T= 1, B = 1, and A = 6.04. 

Values of A, C, and T. 





o 

CO 

CD 


Values of C. 


Values of T. 


System. 


Percentage of power 
factor. 


Percentage of power 
factor. 




100 


95 


90 


85 


80 


100 95 1 90 85 1 80 


Single phase 


6.04 
12.08 
9.06 


2160 
1080 

1080 


2400 
1200 
1200 


2660 
1330 
1330 


3000 
1500 
1500 


3380 
1690 
1690 


1.00 1.05 1.11 1.17 1.25 


Two-phase (4- wire) . . . 
Three-phase (3- wire) . . 


.50 .53 .55 .59 .62 
.58 .61 .64 .68 .72 



The following formula will be found a convenient one for calculating 
the copper required for long-distance, three-phase transmission circuits. 



Pounds of copper : 



M* >/ Kw. X 300,000,000 
PXE* 



M ■ 

Kw. ■ 



■■ distance of transmission, in miles ; 
= the power delivered, in kilowatts. 



Power factor is assumed to be approximately 95 per cent. 



Application of Formulas. 

The value of C for any particular power factor is obtained by dividing 
2160, the value for continuous current, by the square of that power factor 
for single-phase, and by twice the square of that power factor for 3-wire 
three-phase, or 4-wire two-phase. 

The value of B depends upon the size of wire, frequency, and power 
factor. It is equal to 1 for continuous current and for alternating current 
with 100 per cent, power factor, and sizes of wire given in the preceding 
table of wiring constants. 

The figures given are for wires 18 inches apart, and are sufficiently 
accurate for all practical purposes, provided the displacement in phase 
between current and electro-motive force at the receiving end is not very 
much greater than that at the generator; in other words, provided the 
reactance of the line is not excessive or the line loss unusually high. For 
example, the constants should not be applied at 125 cycles if the largest 
conductors are used and the loss is 20 per cent, or more of the power 
delivered. 



Wiring Formulas. 



735 



; 























M 


to 


CO 


hU 


Number of wire, 


o 


SO 


QO 


<I 


OS 


C7» 


4* 


CO 


bO 


M 


c5 


© 


© 


© 


B. & S. gauge. 






















,_, 


M 


4-1 


to 












M 




CO 


4^ 


rn 


OS 


00 


cs 


CO 


© 




Area of wire, 




o 


CO 


© 


© 


03 


CO 




to 


OS 




On 


CO 


~sl 






CO 


o 


S 


OO 


ro 


l_l 


<I 


OS 


CO 


OS 


On 


o 


oo 


© 


circular mils 




QO 


SO 






o 








CD 


03 




© 


© 






bO 


o 


CO 


© 


O 


to 


to 


CO 


CO 


^ 


OO 


CO 


Oi 


© 






CO 


co 


^ 


OS 


^7 


o 


(O 


On 


to 
o 


ro 

Oi 


CO 


4- 

O 


Oi 

o 


© 


Weight of bare 




SO 


SO 




SO 


o 


OS 


so 


o 


CO 


CO 


to 


GO 


wire per 1000 


*- 


© 


SO 


O 


4- 


to 


4- 


CO 


SO 


MS* 


o 


so 


h-i 


-i . 


feet. 








SO 


CO 


so 


CO 


o 


„c 


CO 


CO 


o 


-J 


to 


CO 




SO 


<I 


bs 


£>. 


CO 


CO 


ro 


J_J 


M 


M 


b 


b 


b 


CO f° 


Resistance of 


wire 


SO 


00 


to 

s 


SO 

at 


CD 

CO 


ro 


4s» 


SO 
OS 


On 
On 


to 

CO 


SO 
-7 


-4 

-7 


os 

4-1 


per 1000 feet 


at 




~. 


■JO 


4- 


o 


CO 


to 


os 


hi*. 


On 


On 
OO 


!K 


20° C. 
































*o 






o 


O 


o 


o 


o 


o 


o 


o 


o 


o 








to 


Cfl 


CD 
CD 
















to 


oo 


CJ1 


-vj 


o 


^4- 


^J0 


CO 






M 


M 


M 


M 


M 


M 


h-» 


H, 


M 


M 


h-» 


4- 1 


M 


4- 1 


*o 


O 


o 


o 


O 


o 


o 


o 


o 


o 


o 




M 


ro 


bO 


o 




Cn 
















bO 


rf*. 


-J 


H ' 


OS 


to 


SO 




O 




































































M 


I- 1 


I- 1 


M 


M 


M 


M 


I— 1 


4^ 


M 


4- 1 


H 


4^ 


M 


00 


r* 2, 


CD 


O 


o 


o 


O 


O 


O 


O 


o 


O 

to 


o 

On 


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736 Electrical Standardization. 

At lower frequencies, however, the constants are reasonably correct, 
even under such extreme conditions. They represent about the true 
values at 10 per cent, line loss, are close enough at all losses less than 10 
per cent., and often, at least for frequencies up to 40 cycles, close enough 
for even much larger losses. Where the conductors of a circuit are much f 
nearer each other than 18 inches the volts loss will be less than that given 
by the formula, and if close together, as with a multiple-conductor cable, 
the loss will be only that due to the resistance. 

The value of T depends on the system and power factor. It is equal to 
1 for continuous current and for single-phase current of 100 per cent, 
power factor. 

The value of A and the weights of wires in the tables are based on 
0.00000302 of a pound as the weight of a foot of copper wire of 1 circular 
mil area. 

In using the formulas and constants on page 734, it should particularly 
be observed that P stands for the per cent, loss in the line of the delivered 
power, not for the per cent, loss in the line of the power at the generator, 
and that E is the power at the end of the line and not at the generator. 

When the power factor cannot be more accurately determined, it may 
be assumed to be as follows for any alternating system operating under 
average conditions: Incandescent lighting and synchronous motors, 95 
per cent. ; lighting and induction motors together, 85 per cent. ; induction 
motors alone, 80 per cent. 

In continuous-current 3-wire systems the neutral wire for feeders 
should be made of one-third the section obtained by formula for either of 
the outside wires. In both continuous- and alternating-current systems 
the neutral conductor for secondary mains and house-wiring should be 
taken as large as the other conductors. 

The 3 wires of a three-phase circuit and the 4 wires of a two-phase 
circuit should all be of the same size, and each conductor should be of the 
cross-section given by the first formula. 

Report of the Committee on Standardization. 

American Institute of Electrical Engineers. 
1898. 

GENERAL PLAN. 

Efficiency. Sections 1 to 24. 

I. Commutating Machines. Sections 6 to 11. 
II. Synchronous Machines. Sections 10 to 11. 

III. Synchronous Commutating Machines. Sections 12 to 15. 

IV. Rectifying Machines. Sections 16 to 17. 

V. Stationary Induction Apparatus. Sections 18 to 19. 
VI. Rotary Induction Apparatus. Sections 20 to 23. 
VII. Transmission Lines. Section 24. 
Rise of Temperature. Sections 25 to 31. 
Insulation. Sections 32 to 41. 
Regulation. Sections 42 to 61. 
Variation and Pulsation. Sections 62 to 65. 
Rating. Sections 66 to 73. 

Classification of Voltages and Frequencies. Sections 74 to 78. 
Overload Capacities. Sections 79 to 82. 
Appendices. 

I. Efficiency. 
II. Apparent Efficiency. 

III. Power Factor and inductance Factor. 

IV. Notation. 

V. Table of Sparking Distances. ' 

Electrical apparatus will be treated under the following heads : 
I. Commutating Machines, which comprise a constant magnetic 
field, a closed-coil armature, and a multi-segmental commutator con- 
nected thereto. 



Electrical Standardization. 737 

Under this head may be classed the following : Direct-current gener- 
ators, direct-current motors, direct-current boosters, motor-generators, 
dynamotors, converters, and closed-coil arc machines. 

A booster is a machine inserted in series in a circuit to change its 
voltage, and may be driven either by an electric motor or otherwise. In 
the former case it is a motor-booster. 

A motor-generator is a transforming device consisting of two machines, 
a motor and a generator, mechanically connected together. 

A dynamotor is a transforming device combining both motor and gen- 
erator action in one magnetic field, with two armatures, or with an 
armature having two separate windings. 

For converters, see III. 

II. Synchronous Machines, which comprise a constant magnetic 
field and an armature receiving or delivering alternating currents in 
synchronism with the motion of the machine, — i.e., having a frequency 
equal to the product of the number of pairs of poles and the speed of the 
machine, in revolutions, per second. 

III. Synchronous Commutating Machines.— These include: 1, syn- 
chronous converters, — i.e., converters from alternating to direct, or from 
direct to alternating current; and 2, double-current generators, — i.e., 
generators producing both direct and alternating currents. 

A converter is a rotary device transforming electric energy from one 
form into another without passing it through the intermediary form of 
mechanical energy. 

A converter may be either : 

(a) A direct-current converter, converting from a direct current to a 
direct current, or 

(b) A synchronous converter, formerly called a rotary converter, con- 
verting from an alternating to a direct current, or vice versa. Phase 
converters are converters from an alternating-current system to an alter- 
nating-current system of the same frequency but different phase. 

Frequency converters are converters from an alternating-current system 
of one frequency to an alternating-current system of another frequency, 
with or without changes of phase. 

IV. Rectifying Machines, or Pulsating-current Generators, which 
produce a unidirectional current of periodically varying strength. 

V. Stationary Induction Apparatus, — i.e., stationary apparatus 
changing electric energy from one form into another without passing it 
through an intermediary form of energy. These comprise : 

(a) Transformers, or stationary induction apparatus, in which the 
primary and secondary windings are electrically insulated from each 
other. 

(b) Auto-transformers, formerly called compensators,— i.e., stationary 
induction apparatus, in which part of the primary winding is used as a 
secondary winding, or conversely. 

(c) Potential regulators, or stationary induction apparatus having a 
coil in shunt and a coil in series with the circuit, so arranged that the 
ratio of transformation between them is variable at will. 

These may be divided into : 

1. Compensator potential regulators, in which the number of turns of 
one of the coils is changed. 

2. Induction potential regulators, in which the relative positions of 
primary and secondary coils is changed. 

3. Magneto-potential regulators, in which the direction of the magnetic 
flux with respect to the coils is changed. 

(d) Reactive coils, or reactance coils, formerly called choking coils,— 
i.e., stationary induction apparatus, used to produce impedance or phase 
displacement. 

VI. Rotary Induction Apparatus, which consists of primary and 
secondary windings, rotating with respect to each other. They comprise : 

(a) Induction motors. 
(6) Induction generators. 
I c) Frequency changers. 
(d) Rotary phase converters. 

47 



738 Electrical Standardization. 

EFFICIENCY. 

1 . The "efficiency" of an apparatus is the ratio of its net power output 
to its gross power input.* 

2. Electric power should be measured at the terminals of the apparatus. 

3. In determining the efficiency of alternating-current apparatus the 
electric power should be measured when the current is in phase with the 
E. M. F., unless otherwise specified^ except when a definite phase difference 
is inherent in the apparatus, as in induction motors, etc. 

4. Mechanical power in machines should be measured at the pulley, 
gearing, coupling, etc., thus excluding the loss of power in said pulley, 
gearing, or coupling, but including the bearing friction and windage. The 
magnitude of bearing friction and windage may be considered as inde- 
pendent of the load. The loss of power in the belt and the increase of 
bearing friction due to belt tension should be excluded. Where, however, 
a machine is mounted upon the shaft of a prime mover in such a manner 
that it cannot be separated therefrom, the frictional losses in bearings and 
in windage, which ought by definition to be included in determining the 
efficiency, should be excluded, owing to the practical impossibility of 
determining them satisfactorily. The brush friction, however, should be 
included. 

(a) Where a machine has auxiliary apparatus, such as an exciter, the 
power lost in the auxiliary apparatus should not be charged to the machine, 
but to the plant, consisting of machine and auxiliary apparatus taken to- 
gether. The plant efficiency in such cases should be distinguished from 
the machine efficiency. 

5. The efficiency may be determined by measuring all the losses indi- 
vidually and adding their sum to the output to derive the input, or sub- 
tracting their sum from the input to derive the output. All losses should be 
measured at, or reduced to, the temperature assumed in continuous opera- 
tion, or in operation under conditions specified. (See Sections 25 to 31.) 

In order to consider the application of the foregoing rules to various 
machines in general use, the latter may be conveniently divided into 
classes, as follows : 

I. Commutating Machines. 

6. In commutating machines the losses are : 

(a) Bearing friction and windage. (See Section 4.) 

(b) Molecular magnetic friction and eddy currents in iron and copper. 
These losses should be determined with the machine on open circuit, and 
at a voltage equal to the rated voltage + It in a generator and — Ir in a 
motor, where i" denotes the current strength and r denotes the internal 
resistance of the machine. They should be measured at the correct speed 
and voltage, since they do not usually vary in proportion to the speed or 
to any definite power of the voltage. 

(c) Armature resistance losses, JV, where I is the current strength in 
the armature and r' is the resistance between armature brushes, excluding 
the resistance of brushes and brush contacts. 

(d) Commutator brush friction. 

(e) Commutator brush-contact resistance. It is desirable to point out 
that with carbon brushes the losses, {d) and (e), are usually considerable in 
low-voltage machines. 

(/) Field excitation. With separately-excited fields the loss of power 
in the resistance of the field coils alone should be considered. With shunt 
fields or series fields, however, the loss of power in the accompanying 
rheostat should also be included, the said rheostat being considered as an 
essential part of the machine, and not as separate auxiliary apparatus. 

(b) and (c) are losses in the armature, or " armature losses ;" (d) and (e), 
"commutator losses;" (/), "field losses." 

7. The difference between the total losses under load and the sum of 
the losses above specified should be considered as "load losses," and are 
usually trivial in commutating machines of small field distortion. When 

* An exception should be noted in the case of storage batteries or apparatus 
for storing energy, in which the efficiency, unless otherwise qualified, should be 
understood as the ratio of the energy output to the energy intake in a normal 
cycle. 



Electrical Standardization. 739 

the field distortion is large, as is shown by the necessity for shifting the 
brushes between no load and full load, or with variations of load, these 
load losses may be considerable, and should be taken into account. In this 
case the efficiency may be determined either by input and output measure- 
ments or the load losses may be estimated by the method of Section II. 

8. Boosters should be considered and treated like other direct-current 
machines in regard to losses. 

9. In motor-generators, dynamotors, or converters the efficiency is the 
electric output 

electric input * __ _ . __ __. 

II. Synchronous Machines. 

10. In synchronous machines the output or input should be measured 
with the current in phase with the terminal E. M. F., except when other- 
wise expressly specified. 

Owing to the uncertainty necessarily involved in the approximation of 
load losses, it is preferable, whenever possible, to determine the efficiency 
of synchronous machines by input and output tests. 

1 1 . The losses in synchronous machines are : 

(a) Bearing friction and windage. (See Section 4.) 

(b) Molecular magnetic friction and eddy currents in iron, copper, and 
other metallic parts. These losses should be determined at open circuit of 
the machine at the rated speed and at the rated voltage, + Ir in a synchro- 
nous generator, — Ir in a synchronous motor, where I = current in arma- 
ture, r = armature resistance. It is undesirable to compute these losses 
from observations made at other speeds or voltages. 

These losses may be determined either by driving the machine by a 
motor, or by running it as a synchronous motor and adjusting its fields so 
as to get minimum current input, and measuring the input by wattmeter. 
The former is the preferable method ; and in polyphase machines the latter 
method is liable to give erroneous results in consequence of unequal dis- 
tribution of currents in the different circuits, caused by inequalities of the 
impedance of connecting leads, etc. 

(c) Armature-resistance loss, which may be expressed by pl 2 r, where 
r = resistance of one armature circuit or branch, I = the current in such 
armature circuit or branch, and p = the number of armature circuits or 
branches. 

(d) Load losses as defined in Section 7. While these losses cannot well 
be determined individually, they may be considerable, and, therefore, their 
joint influence should be determined by observation. This can be done by 
operating the machine on short circuit and at full-load current,— that is, 
by determining what maybe called the " short-circuit core loss." With 
the low field intensity and great lag of current existing in this case the 
load losses are usually greatly exaggerated. 

One-third of the short-circuit core loss may, as an approximation, and 
in the absence of more accurate information, be assumed as the load loss. 

(e) Collector-ring friction and contact resistance. These are generally 
negligible, except in machines of extremely low voltage. 

(/) Field excitation. In separately-excited machines the I-r of the 
field-coils proper should be used. In self-exciting machines, however, the 
loss in the field rheostat should be included. (See Section 6,/.) 

III. Synchronous Commutating Machines. 

12. In synchronous converters the power on the alternating-current 
side is to be measured with the current in phase with the terminal E. M. F., 
unless otherwise specified. 

13. In double-current generators the efficiency of the machine should 
be determined as a direct-current generator, in accordance with Section 6, 
and as an alternating-current generator, in accordance with Section 11. 
The two values of efficiency may be different, and should be clearly dis- 
tinguished. 

14. In synchronous converters the losses should be determined when 
driving the machine by a motor. These losses are : 

(a) Bearing friction and windage. (See Section 4.) 

(6) Molecular magnetic friction and eddy currents in iron, copper, and 
metallic parts. These losses should be determined at open circuit and at 
the rated terminal voltage, no allowance being made for the armature 



740 Electkical. Standakdization. 

resistance, since the alternating and the direct currents flow in opposite 
directions. 

(c) Armature resistance. The loss in the armature is ql 2 r, where 1 = 
direct current in armature, r = armature resistance, and q, a factor which 
is equal to 1.37 in single-phasers, 0.56 in three-phasers, 0.37 in quarter- 
phasers, and 0.26 in six-phasers. 

(d) Load losses. The load losses should be determined in the same 
manner as described in Section 11, d, with reference to the direct-current 
side. 

(e) and (/) Losses in commutator and collector-friction and brush- 
contact resistance. (See Sections 6 and 11.) 

(g) Field excitation. In separately-excited fields the I 2 r loss in the 
field-coils proper should be taken, while in shunt and series fields the 
rheostat loss should be included, except where fields and rheostats are 
intentionally modified to produce effects outside of the conversion of 
electric power, as for producing phase displacement for voltage control. 
In this case 25 per cent, of the I 2 r loss in the field proper at non-inductive 
alternating circuit should be added as proper estimated allowance for 
normal rheostat losses. ( See Section 6, /. ) 

15. Where two similar synchronous machines are available their 
efficiency can be determined by operating one machine as a converter 
from direct to alternating, and the other as a converter from alternating 
to direct, connecting the alternating sides together and measuring the 
difference between the direct-current input and the direct-current output. 
This process may be modified by returning the output of the second 
machine through two boosters into the first machine and measuring the 
losses. Another modification might be to supply the losses by an alternator 
between the two machines, using potential regulators. 

IV. Rectifying Machines, or Pulsating=current Generators. 

j6. These include open-coil arc machines, constant-current rectifiers, 
constant-potential rectifiers. 

The losses in open-coil arc machines are essentially the same as in Sec- 
tions 6 to 9 (closed-coil commutating machines). In alternating-current 
rectifiers, however, the output must be measured by wattmeter and not 
by voltmeter and ammeter, since owing to the pulsation of current and 
E. M. F. a considerable discrepancy may exist between watts and volt- 
amperes, amounting to as much as 10 or 15 per cent. 

1.7. In constant-current rectifiers, transforming from constant-potential 
alternating to constant direct current by means of constant-current trans- 
formers and rectifying commutators, the losses in the transformers are to 
be included in the efficiency, and have to be measured when operating the 
rectifier, since in this case the losses are generally greater than when feed- 
ing an alternating secondary circuit. In constant-current transformers 
the load losses are usually larger than in constant-potential transformers, 
and thus should not be neglected. 

The most satisfactory method of determining the efficiency in rectifiers 
is to measure electric input and electric output by wattmeter. The input 
is usually not non-inductive, owing to a considerable phase displacement 
and to wave distortion. For this reason the apparent efficiency should 
also be considered, since it is usually much lower than the true efficiency. 
The power consumed by the synchronous motor or other source driving 
the rectifier should be included' in the electric input. 

V. Stationary Induction Apparatus. 

18. Since the efficiency of induction apparatus depends upon the wave 
shape of E. M. F., it should be referred to a sine wave of E. M. F., except 
where expressly specified otherwise. The efficiency should be measured 
with non-inductive load and at rated frequency, except where expressly 
specified otherwise. The losses are : 

(a) Molecular magnetic friction and eddy currents measured at open 
circuit and at rated voltage — Ir, where / = rated current, r = resistance 
of primary circuit. 

(b) Resistance losses. The sum of the V-r of primary and of secondary 
in a transformer, or of the two sections of the coil in the compensator 
or auto-transformer, where I = current in the coil or section of coil, r = 
resistance. 



Electrical Standardization. 741 

(c) Load losses, — i.e., eddy currents in the iron, and especially in the 
copper conductors, caused by the current. They should be measured by 
short-circuiting the secondary of the transformer and impressing upon the 
primary an E. M. F. sufficient to send full-load current through the trans- 
former. The loss in the transformer under these conditions, measured by 
wattmeter, gives the load losses = I-r losses in both primary and secondary 
coils. 

{d) Losses due to the methods of cooling, as power consumed by the 
blower in air-blast transformers and power consumed by the motor driving- 
pumps in oil- or water-cooled transformers. Where the same cooling 
apparatus supplies a number of transformers, or is installed to supply 
future additions, allowance should be made therefor. 

19. In potential regulators the efficiency should be taken at the maxi- 
mum voltage for which the apparatus is designed, and with non-inductive 
load, unless otherwise specified. 

VI. Rotary Induction Apparatus. 

20. Owing to the existence of load losses, and since the magnetic 
density in the induction motor under load changes in a complex manner, 
the efficiency should be determined by measuring the electric input by 
wattmeter and the mechanical output at the pulley, gear, coupling, etc. 

21. The efficiency should be determined at the rated frequency, and 
the input measured with sine waves of impressed E. M. F. 

22. The efficiency may be calculated from the apparent input, the 
power factor, and the power output. The same applies to induction gen- 
erators. Since phase displacement is inherent in induction machines, their 
apparent efficiency is also important. 

23. In frequency changers, — i.e., apparatus transforming from a poly- 
phase system to an alternating system of different frequency, with or with- 
out a change in the number of phases and phase converters,— i.e., apparatus 
converting from an alternating system, usually single-phase, to another 
alternating system, usually polyphase, of the same frequency, the efficiency 
should also be determined by measuring both output and input. 

VII. Transmission Lines. 

24. The efficiency of transmission lines should be measured with non- 
inductive load at the receiving end, w T ith the rated receiving pressure and 
frequency, also with sinusoidal impressed E. M. F.'s, except where ex- 
pressly specified otherwise, and with the exclusion of transformers or 
other apparatus at the ends of the line. 

RISE OF TEMPERATURE. 
General Principles. 

25. Under regular service conditions the temperature of electrical 
machinery should never be allowed to remain at a point at which per- 
manent deterioration of its insulating material takes place. 

26. The rise of temperature should be referred to the standard condi- 
tions of a room-temperature of 25° C, a barometric pressure of 760 milli- 
metres, and normal conditions of ventilation,— that is, the apparatus under 
test should neither be exposed to draught nor inclosed, except where ex- 
pressly specified. 

27. If the room -temperature during the test differs from 25° C, the 
observed rise of temperature should be corrected by % per cent, for each 
degree C* Thus, with a room-temperature of 35° C. the observed rise of 
temperature has to be decreased by 5 per cent. , and with a room-tempera- 
ture of 15° C. the observed rise of temperature has to be increased by 5 
per cent. The thermometer indicating the room-temperature should be 
screened from thermal radiation emitted by heated bodies or from draughts 
of air. When it is impracticable to secure normal conditions of ventilation 

* This correction is also intended to compensate, as nearly as is at present 
practicable, for the error involved in the assumption of a constant temperature 
coefficient of resistivity, — i.e., 0.4 per cent, per degree C, taken with varjing 
initial temperatures. 



742 Electrical Standardization. 

on account of an adjacent engine or other sources of heat, the thermometer 
for measuring the air-temperature should be placed so as fairly to indicate 
the temperature which the machine would have if it were idle, in order 
that the rise of temperature determined shall be that caused by the opera- 
tion of the machine. 

28. The temperature should be measured after a run of sufficient dura- 
tion to reach practical constancy. This is usually from 6 to 18 hours, 
according to the size and construction of the apparatus. It is permissible, 
however, to shorten the time of the test by running a lesser time on an 
overload in current and voltage, then reducing the load to normal, and 
maintaining it thus until the temperature has become constant. 

In apparatus intended for intermittent service, as railway motors, start- 
ing rheostats, etc., the rise of temperature should be measured after a 
shorter time, depending upon the nature of the service, and should be 
specified. 

In apparatus which, by the nature of their service, may be exposed to 
overload, as railway converters, and in very high voltage circuits a smaller 
rise of temperature should be specified than in apparatus not liable to 
overloads or in low-voltage apparatus. In apparatus built for conditions 
of limited space, as railway motors, a higher rise of temperature must be 
allowed. 

29. In electrical conductors the rise of temperature should be deter- 
mined by their increase of resistance. For this purpose the resistance may 
be measured either by galvanometer test or by drop-of-potential method. 
A temperature coefficient of 0.4 per cent, per degree C. may be assumed 
for copper.* Temperature elevations measured in this way are usually in 
excess of temperature elevations measured by thermometers. 

30. It is recommended that the following maximum values of tempera- 
ture elevation should not be exceeded : 

Commutating machines, rectifying machines, and synchronous ma- 
chines : 

Field and armature, by resistance, 50° C. 

Commutator and collector rings and brushes, by thermometer, 55° C. 
Bearings and other parts of machine, by thermometer, 40° C. 
Rotary induction apparatus : 

Electric circuits, 50° C, by resistance. 

Bearings and other parts of the machine, 40° C, by thermometer. 
In squirrel-cage or short-circuited armatures, 55° C, by thermometer, 
may be allowed. 

Transformers for continuous service,— electric circuits, by resistance, 50° 
C. ; other parts, by thermometer, 40° C, under conditions of normal venti- 
lation. 

Reactive coils, induction and magneto regulators and transformers of 
15 kilowatts or less,— electric circuits, by resistance, 55° C. ; other parts, by 
thermometer, 45° C. 

Where a thermometer, applied to a coil or winding, indicates a higher 
temperature elevation than that shown by resistance measurement, the 
thermometer indication should be accepted. In using the thermometer 
care should be taken so to protect its bulb as to prevent radiation from it, 
and, at the same time, not to interfere seriously with the normal radiation 
from the part to which it is applied. 

3 1 . In the case of apparatus intended for intermittent service, the tem- 
perature elevation which is attained at the end of the period corresponding 
to the term of full load should not exceed 50° C, by resistance, in electric 
circuits. In the case of transformers intended for intermittent service or 
not operating continuously at full load, but continuously in circuit, as in 
the ordinary case of lighting transformers, the temperature elevation above 
the surrounding air-temperature should not exceed 50° C, by resistance, 
in electric circuits, and 40° C, by thermometer, in other parts, after the 
period corresponding to the term of full load. In this instance the best 
load should not be applied until the transformer has been in circuit for a 
sufficient time to attain the temperature elevation due to core loss. With 
transformers for commercial lighting the duration of the full-load test 



*By the formula /»' T = Iif (1 + 0.0040). Where /fy is the resistance at room- 
temperature, /i' T the resistance when heated, and the temperature elevation 
(T-t) in degrees centigrade. 



Electrical Standardization. 



743 



may be taken as 3 hours, unless otherwise specified. In the case of railway, 
crane, and elevator motors the conditions of service are necessarily so 
varied that no specific period corresponding to the full load term can be 
stated. 

INSULATION. 

32. The ohmic resistance of the insulation is of secondary importance 
only, as compared with the dielectric strength or resistance to rupture by 
high voltage. 

Since the ohmic resistance of the insulation can be very greatly increased 
by baking, — but the dielectric strength is liable to be weakened thereby,— 
it is preferable to specify a high dielectric strength rather than a high insula- 
tion resistance. The high-voltage test for dielectric strength should always 
be applied. 

Insulation Resistance. 

33. Insulation resistance tests should, if possible, be made at the press- 
ure for which the apparatus is designed. 

The insulation resistance of the complete apparatus must be such that 

the rated voltage of the apparatus will not send more than yrjr^w^r of the 

full load current, at the rated terminal voltage, through the insulation. 
Where the value found in this way exceeds 1 megohm, 1 megohm is 
sufficient. 

Dielectric Strength. 

34. The dielectric strength or resistance to rupture should be deter- 
mined by a continued application of an alternating E. M. F. for one 
minute. The source of alternating E. M. F. should be a transformer of 
such size that the charging current of the apparatus as a condenser does 
not exceed 25 per cent, of the rated capacity of the transformer. 

35. The high-voltage tests should not be applied when the insulation is 
low owing to dirt and moisture, and should be applied before the machine 
is put into commercial service. 

36. It should be pointed out that tests at high voltages considerably in 
excess of the normal voltages are admissible on new machines, to deter- 
mine whether they fulfil their specifications, but should not be made subse- 
quently at a voltage much exceeding the normal, as the actual insulation 
of the machine may be weakened by such tests. 

37. The test for dielectric strength should be made with the completely- 
assembled apparatus, and not with its individual parts, and the voltage 
should be applied as follows : 

1. Between electric circuits and surrounding conducting material, and, 

2. Between adjacent electric circuits, where such exist, as in trans- 
formers. 

The tests should be made with a sine wave of E. M. F., or, where this 
is not available, at a voltage giving the same striking distance between 
needle-points in air as a sine wave of the specified E. M. F., except where 
expressly specified otherwise. As needles, new sewing-needles should be 
used. It is recommended to shunt the apparatus during the test by a spark 
gap of needle-points set for a voltage exceeding the required voltage by 10 
per cent. 

38. The following voltages are recommended for apparatus, not in- 
cluding transmission lines or switchboards : 



Rated terminal voltage. 


Capacity. 


Testing voltage. 


Not exceeding 400 volts. 


Under 10 kilowatts. 


1000 volts. 


Not exceeding 400 volts. 


10 kilowatts and over. 


1500 volts. 


400 and over, but less than 800 volts. 


Under 10 kilowatts. 


1500 volts. 


400 and over, but less than 800 volts. 


10 kilowatts and over. 


2000 volts. 


800 and over, but less than 1200 volts. 


Any. 


3500 volts. 


1200 and over, but less than 2500 volts. 


Any. 


5000 volts. 


2500 and over. 


Any. 


Double the 
normal rated 
voltages. 



744 Electrical Standardization. 

Synchronous motor fields and fields of converters started from the alter- 
nating current side should be tested at 5000 volts. 

Synchronous motors and synchronous converter field-coils should be 
tested at 5000 volts, since in the starting of such machines a high voltage 
is induced in their field-coils. 

Alternator field circuits should be tested under a breakdown test voltage 
corresponding to the rated voltage of the exciter referred to an output 
equal to the output of the alternator,— i.e., the exciter should be rated for 
this test as having an output equal to that of the machine it excites. 

Condensers should be tested at twice their rated voltage and at their 
rated frequency. 

The above values are effective values, or square roots of mean square 
reduced to a sine wave of E. M. F. 

39. In testing insulation between different electric circuits, as between 
primary and secondary of transformers, the testing voltage must be 
chosen corresponding to the high- voltage circuit. 

40. In transformers of from 10,000 volts to 20,000 volts it should be con- 
sidered as sufficient to operate the transformer at twice its rated voltage 
by connecting first the one and then the other terminal of the high-voltage 
winding to the core and to the low- voltage winding. The test of dielectric 
resistance between the low-voltage winding and the core should be in 
accordance with the recommendation in Section 39 for similar voltages 
and capacities. 

4 1 . When machines or apparatus are to be operated in series, so as to 
employ the sum of their separate E. M. F.'s, the voltage should be referred 
to this sum, except where the frames of the machine are separately insu- 
lated, both from ground and from each other. 

REGULATION. 

42. The term "regulation" should have the same meaning as the term 
"inherent regulation," at present frequently used. 

43. The regulation of an apparatus intended for the generation of con- 
stant potential, constant current, constant speed, etc., is to be measured by 
the maximum variation of potential current, speed, etc., occurring within 
the range from full load to no load under such constant conditions of 
operation as give the required full-load values, the conditions of full load 
being considered in all cases as the normal condition of operation. 

44. The regulation of an apparatus intended for the generation of a 
potential, current, speed, etc., varying in a definite manner between full 
load and no load, is to be measured by the maximum variation of poten- 
tial, current, speed, etc., from the satisfied condition, under such constant 
conditions of operation as give the required full-load values. 

If the manner in which the variation in potential, current, speed, etc., 
between full load and no load is not specified, it should be assumed to be a 
simple linear relation. 

The regulation of an apparatus may, therefore, differ according to its 
qualification for use. Thus, the regulation of a compound-wound generator 
specified as a constant-potential generator will be different from that it 
possesses when specified as an over-compounded generator. 

45. The regulation is given in percentage of the full-load value of po- 
tential, current, speed, etc., and the apparatus should be steadily operated 
during the test under the same conditions as at full load. 

46. The regulation of generators is to be determined at constant speed, 
of alternating apparatus at constant impressed frequency. 

47. The regulation of a generator unit, consisting of a generator united 
with a prime mover, should be determined at constant conditions of the 
prime mover, — i.e., constant steam pressure, head, etc. It would include 
the inherent speed variations of the prime mover. For this reason the 
regulation of a generator unit is to be distinguished from the regulation 
of either the prime mover or of the generator contained in it and taken 
separately. 

48. In apparatus generating, transforming, or transmitting alternating 
currents, regulation should be understood to refer to non-inductive load,— 
that is, to a load in which the current is in phase with the E. M. F. at the 
output side of the apparatus, except where expressly specified otherwise. 

49. In alternating apparatus receiving electric power, regulation should 
refer to a sine wave of E. M. F., except where expressly specified otherwise. 



Electrical Standardization. 745 

50. In commutating machines, rectifying machines, and synchronous 
machines, as direct-current generators and motors, alternating-current and 
polyphase generators, the regulation is to be determined under the follow- 
ing conditions : 

(a) At constant excitation in separately-excited fields, 
(6) With constant resistance in shunt-field circuits, and 
(c) With constant resistance shunting series fields,— i.e., the'field adjust- 
ment should remain constant, and should be so chosen as to give the 
required full-load voltage at full-load current. 

51. In constant-potential machines the regulation is the ratio of the 
maximum difference of terminal voltage from the rated full-load value 
(occurring within the range from full load to open circuit) to the full -load 
terminal voltage. 

52. In constant-current machines the regulation is the ratio of the 
maximum difference of current from the rated full-load value (occurring 
within the range from full load to short circuit) to the full-load current. 

53. In constant-power machines the regulation is the ratio of maximum 
difference of power from the rated full-load value (occurring within the 
range of operation specified) to the rated power. 

54. In over-compounded machines the regulation is the ratio of the 
maximum difference in voltage from a straight line connecting the no-load 
and full-load values of terminal voltage as function of the current to the 
full-load terminal voltage. 

55. In constant-speed, continuous-current motors the regulation is 
the ratio of the maximum variation of speed from its full-load value 
(occurring within the range from full load to no load) to the full-load 
speed. 

56. In transformers the regulation is the ratio of the rise of secondary 
terminal voltage from full load to no load (at constant primary impressed 
terminal voltage) to the secondary terminal voltage. 

57. In induction motors the regulation is the ratio of the rise of speed 
from full load to no load (at constant impressed voltage) to the full-load 
speed. 

The regulation of an induction motor is, therefore, not identical with 
the slip of the motor, which is the ratio of the drop in speed from syn- 
chronism to synchronous speed. 

58. In converters, dynamotors, motor-generators, and frequency- 
changers the regulation is the ratio of the maximum difference of terminal 
voltage at the output side from the rated full-load voltage (at constant 
impressed voltage and at constant frequency) to the full-load voltage on 
the output side. 

59. In transmission lines, feeders, etc., the regulation is the ratio of 
maximum voltage difference at the receiving end between no load and 
full non-inductive load to the full-load voltage at the receiving end, with 
constant voltage impressed upon the sending end. 

60. In steam engines the regulation is the ratio of the maximum varia- 
1 tion of speed in passing from full load to no load (at constant steam 
| pressure at the throttle) to the full-load speed. 

61 . In a turbine or other water motor the regulation is the ratio of the 
maximum variation of speed from full load to no load (at constant head 
of water, — i.e., at constant difference of level between tail-race and head- 

! race) to the full-load speed. 

VARIATION AND PULSATION. 

62. In prime movers which do not give an absolutely uniform rate of 
. rotation or speed, as in steam engines, the "variation" is the maximum 

angular displacement in position of the revolving member from the posi- 
tion it would occupy at uniform rotation, expressed in degrees, — that is, 
with one revolution at 300° ; and the pulsation is the ratio of the maxi- 
mum change of speed in an engine cycle to the average speed. 

63. In alternators, or alternating-current circuits in general, the varia- 
, tion is the maximum difference in phase of the generated wave of E. M. F. 

from a wave of absolutely constant frequency, expressed in degrees, and 
is due to the variation of the prime mover. The pulsation is the ratio of 
the maximum change of frequency during an engine cycle to the average 
frequency. 



746 Electrical Standardization. 



64. If n = number of poles, the variation of an alternator is —times 



the variation of its prime mover if direct connected, and — p times the 

variation of the prime mover if rigidly connected thereto in the velocity^ 
ratio, p. 

65. The pulsation of an alternating-current circuit is the same as the 
pulsation of the prime mover of its alternator. 



T 4 

* 1 



RATING. 

66. Both electrical and mechanical power should be expressed in kilo- 
watts, except when otherwise specified. Alternating-current apparatus 
should be rated in kilowatts on the basis of non-inductive condition,— i.e., 
with the current in phase with the terminal voltage. 

67. Thus, the electric power generated by an alternating-current 
apparatus equals its rating only at non-inductive load, — that is, when the 
current is in phase with the terminal voltage. 

68. Apparent power should be expressed in kilovolt-amperes, as dis- 
tinguished from real power in kilowatts. 

69. If a power factor other than 100 per cent, is specified, the rating 
should be expressed in kilovolt-amperes and power factor at full load. 

70. The full-load current of an electric generator is that current which, 
with the rated full-load terminal voltage, gives the rated kilowatts ; but in 
alternating-current apparatus, only at non-inductive load. 

7 1 . Thus, in machines in which the full-load voltage differs from the 
no-load voltage, the full-load current should refer to the former. 

If P = rating of an electric generator and E = full-load terminal volt- 
age, the full-load current is : 

p 

I = -=■ in a continuous-current machine or single-phase alternator ; 

p 

I — ^ / — in a three-phase alternator : 

p 
I = -^= in a quarter-phase alternator. 

72. Constant-current machines, such as series arc-light generators, 
should be rated in kilowatts based on terminal volts and amperes at full 
load. 

73. The rating of a fuse or circuit-breaker should be the current strength 
at which it will open the circuit, and not the working-current strength. 

CLASSIFICATION OF VOLTAGES AND FREQUENCIES. 

74. Tn direct-current, low-tension generators the following average 
terminal voltages are in general use, and are recommended : 

125 volts. 250 volts. 550 volts. 

75. In direct-current and alternating-current, low-pressure circuits the 
following average terminal voltages are in general use, and are recom- 
tnended : 

110 volts. 220 volts. 

In direct-current power circuits, for railway and other service, 500 volts 
may be considered as standard. 

76. In alternating-current, high -pressure circuits at the receiving end 
Che following pressures are in general use, and are recommended : 

LOOOvolte. SOOOTOltS. 10,000 volts. 20,000 volts. 

1 



77. In alternating-current, high-pressure generators or generating sys- 
tems the following terminal voltages are in general use, and are recom- 
mended : 

1160 volts. 2300 volts. 3450 volts. 



Electrical Standardization. 747 

These pressures allow of a maximum drop in transmission of 15 per 
cent, of the pressure at the receiving end. If the drop required is greater 
than 15 per cent., the generator should be considered as special. 

78. In alternating-current circuits the following approximate fre- 
- .^uencies are recommended as desirable : 

25- or 30-, 40-. 60-. 120-.* 

These frequencies are already in extensive use, and it is deemed advisa- 
ble to adhere to them as closely as possible. 



OVERLOAD CAPACITIES. 

79. All guaranties on heating, regulation, sparking, etc., should apply 
to the rated load, except where expressly specified otherwise, and in alter- 
nating-current apparatus to the current in phase with the terminal E. M. F., 
except where a phase displacement is inherent in the apparatus. 

80. All apparatus should be able to carry a reasonable overload without 
self-destruction by heating, sparking, mechanical weakness, etc., and with 
an increase of temperature elevation not exceeding 15° C. above those 
specified for full loads. (See Sections 25 to 31.) 

81. Overload guaranties should refer to normal conditions of operation 
regarding speed, frequency, voltage, etc., and to non-inductive conditions 
in alternating apparatus, except where a phase displacement is inherent 
in the apparatus. 

82. The following overload capacities are recommended : 

1. In direct-current generators and alternating-current generators, 25 
per cent, for % hour. 

2. In direct-current motors and synchronous motors, 25 per cent, for % 
hour, 50 per cent, for 1 minute, except in railway motors and other appara- 
tus intended for intermittent service. 

3. Induction motors, 25 per cent, for % hour, 50 per cent, for 1 minute. 

4. Synchronous converters, 50 per cent, for % hour. 

5. Transformers, 25 per cent, for % hour, except in transformers con- 
nected to apparatus for which a different overload is guaranteed, in which 
case the same guaranties shall apply for the transformers as for the appara- 
tus connected thereto. 

6. Exciters of alternators and other synchronous machines, 10 per cent, 
more overload than is required for the excitation of the synchronous 
machine at its guaranteed overload and for the same period of time. 



APPENDIX I. 

EFFICIENCY. 

Efficiency of Phase=displacing Apparatus. 

In apparatus producing phase displacement, as, for example, syn- 
chronous compensators, exciters of induction generators, reactive coils, 
condensers, polarization cells, etc., the efficiency should be understood to 
be the ratio of the volt-ampere activity to the volt-ampere activity plus 
power loss. 

The efficiency may be calculated by determining the losses individually, 
adding to them the volt-ampere activity, and then dividing the volt- 
ampere activity by the sum. 

1. In synchronous compensators and exciters of induction generators 
the determination of losses is the same as in other synchronous machines 
under Sections 10 and 11. 

2. In reactive coils the losses are molecular friction, eddy losses, and 
I 2 r loss. They should be measured by wattmeter. The efficiency of re- 



* The frequency of 120 — may be considered as covering the already-existing 
commercial frequencies between 120 — aud 140 , and the frequency of 60 — as 
covering the already-existing commercial frequencies between 60 — and 70 — . 



748 Electrical Standardization. 

active coils should be determined with a sine wave of impressed E. M. F., 
except where expressly specified otherwise. 

3. In condensers the losses are due to dielectric hysteresis and leakage, 
and should be determined by wattmeter with a sine wave of E. M. F. 

4. In polarization cells the losses are those due to electric resistivity 
and a loss in the electrolyte of the nature of chemical hysteresis, and are| 
usually very considerable. They depend upon the frequency, voltage, and 
temperature, and should be determined with a sine wave of impressed 
E. M. F., except where expressly specified otherwise. 



APPENDIX II. 
APPARENT EFFICIENCY. 

In apparatus in which a phase displacement is inherent to their opera- 
tion, apparent efficiency should be understood as the ratio of net power 
output to volt-ampere input. 

Such apparatus comprise induction motors, reactive synchronous con- 
verters, synchronous converters controlling the voltage of an alternating- 
current system, self-exciting synchronous motors, potential regulators, and 
open magnetic circuit transformers, etc. 

Since the apparent efficiency of apparatus generating electric power de- 
pends upon the power factor of the load, the apparent efficiency, unless 
otherwise specified, should be referred to a load power factor of unity. 



APPENDIX III. 
POWER FACTOR AND INDUCTANCE FACTOR. 

The power factor in alternating circuits or apparatus may be defined as 
the ratio of the electric power in watts to volt-amperes. 

The inductance factor is to be considered as the ratio of wattless volt- 
amperes to total volt-amperes. 

Thus, if p = power factor, q = inductance factor ; then 

p 2 + q 2 = 1. 
The power factor is the 

(energy component of current or E. M. F.) 
total current or E. M. F. 

and the inductance factor is the 

(wattless component of current or E. M. F.) _ true power 



(total current of E. M. F. ) volt-amperes 

Since the power factor of apparatus supplying electric power depends 
upon the power factor of the load, the power factor of the load should be 
considered as Unity, unless otherwise specified. 



APPENDIX IV. 

The following notation is recommended : 

E, e = voltage, E. M. F., potential difference ; R, r = resistance ; 

I, i = current ; X, x = reactance ; 

p = power ; Z, z = impedance ; 

(j> = magnetic flux ; L,l = inductance ; 

/3 = magnetic density ; C, c = capacity. 

Vector quantities, when used, should be denoted by capital italics. 



Electric Driving. 



749 



APPENDIX V. 

Table of sparking distances in air between opposed sharp needle-points, 
for various effective sinusoidal voltages, in inches and in centimetres. 



► V I 












[ Kilovolts. 
Square root of 
mean square. 


Distance. 


Kilovolts. 
Square root of 
mean square. 


Distance. 


Inches. 


Centimetres. 


Inches. 


Centimetres. 


5 


.225 


.57 


60 


4.65 


11.8 


10 


.470 


1.19 


70 


5.85 


14.9 


15 


.725 


1.84 


80 


7.10 


18.0 


20 


1.000 


2.54 


90 


8.35 


21.2 


25 


1.300 


3.3 


100 


9.60 


24.4 


30 


1.625 


4.1 


110 


10.75 


27.3 


35 


2.00 


5.1 


120 


11.85 


30.1 


40 


2.45 


6.2 


130 


12.95 


32.9 • 


45 


2.95 


7.5 


140 


13.95 


35.4 


50 


3.55 


9.0 


150 


15.0 


38.1 



Cary T. Hutchinson, Charles P. Steinmetz, 
A. E. Kennelly, Lewis B. Stilwell, 

John Lieb, Jr., Elihtj Thomson, 

F. B. Crocker, Chairman. 



ELECTRIC DRIVING. 

The general opinion is in favor of independent driving, each tool hav- 
ing its own motor attached. In some cases a group of small machines may 
be operated to advantage from a short line-shaft driven by an electric 
motor, but in the great majority of cases the independent driving is to be 
preferred. 

The advantages of independent driving are well set forth in a paper by 
F. B. Duncan before the Engineers' Society of Western Pennsylvania. 

1. Greater output per machine due to positive nature of drive ; in many 
cases this is at least 50 per cent. 

2. Ability to accurately determine— by means of recording instruments 
centrally located, with a multi-point switch— whether tools are being kept 
at work 'in proper manner, thereby affording a graphic record of the time 
each machine is in operation and its consumption of power. This will 
also enable the detection of tools that are in bad condition due to abnor- 
mal friction of bearings or moving parts. 

3. The flexibility of placement of machine tools to suit the passage of 
the work through the shop. 

4. Better light and absence of dirt due to belts, shafting, pulley hangers, 
etc., and less first cost of building owing to the lighter overhead construc- 
tion permissible when no shafting, pulleys, hangers, or belt tension have to 
be taken care of. 

5. Free head room for crane service. 

6. Ability to shut down or start up any one machine independently of 
all others. 

Mr. Duncan also gives the following data sheet of power required by a 
number of different machine tools. These represent average practice, 
,using ordinary tool steels, but for the modern high-speed tool steels the 
cutting speeds may be increased to 80 to 100 feet per minute for cast- or 
' wrought-iron, in which case the power required will be about three times 
that given on pages 750-753. 

For planers the maximum power is that required for reversing the 
platen, as will be seen. 



750 Electkic Driving. 



Data Sheet of Motor Power on Standard Machine Tools. 
No. 1. 



Description of machine, Planer. 
Make of machine, Niles Tool Company. 
Size of machine, 10' X 10' X 20'. 
Number of cutting tools, 3. 
Size of cut, %" X Vs", each tool. 
Cutting speed, 18 feet j 



I 



t per minute. 

Material machined, cast-iron. 

Weight on platen, 40 tons. 

Power for cut, 26.54 H. P. 

Power for reverse, 42.93 H. P. 

Power for return, 23.56 H. P. 

Ratio of return, 3 to 1. % 

Method of drive, motor belted to counter-shaft. 

Kind of motor, Direct-current Compound-wound. 

Remarks.— -Not enough fly-wheel effect on counter-shaft to equalize load 

at moment of reversal. A 30 H. P. motor was used for above drive with 

good results. 



No. 2. 

Description of machine, Planer. 

Make of machine, Pond Machine Company. 

Size of machine, 8' X 8' X 20'. 

Number of cutting tools, 3. 

Size of cut, yj' XW', each tool. 

Cutting speed, 18 feet per minute. 

Material machined, cast-iron. 

Weight on platen, 32 tons. 

Power for cut, 16 H. P. 

Power for reverse, 28.15 H. P. 

Power for return, 14.80 H. P. 

Ratio of return, 3 to 1. 

Method of drive, motor belted to counter-shaft. 

Kind of motor, Direct-current Compound-wound. 

Remarks. — Not enough fly-wheel effect on counter-shaft to equalize load 

at moment of reversal. A 25 H. P. motor was used on this machine with 

good results. 



No. 3. 

Description of machine. Planer. 

Make of machine, Pond Machine Company. 

Size of machine, 66' X 60' X 12'. 

Number of cutting tools, 2. 

Size of cut, W' x T y. 

Cutting speed, 21 feet per minute. 

Material machined, open-hearth steel castings. 

Weight on platen, 4 tons. 

Power for cut, 10 H. P. 

Power for reverse, 16 H. P. 

Power for return, 14 H. P. 

Ratio of return, 3%to 1. 

Method of drive, Direct-current Compound-wound Motor, mounted 

on housing of planer with 42-inch, 1500-pound fly-wheel, running 

at 400 revolutions per minute, mounted on motor-shaft. Fly-wheel 

used as driving pulley for return of platen. 

Remarks.— A series of recording ammeter cards taken on this planer 

showed it was idle an average of 2% hours per day, showing a saving of 

power by use of individual motor drive. The above 2% hours was generally 

made up of short periods for setting work, taking measurements, etc. 



Electric Driving. 751 



No. 4. 

Description of machine, Planer. 
Make of machine, Gray. 
t Size of machine, 28" X 32" X 6'. 

Number of cutting tools, 1. 
Size of cut, %''X y B ". 
Cutting speed, 22 feet per minute. 
Material machined, cast-iron. 
Weight on platen, 3 tons. 
Power for cut, 3.1 H. P. 
Power for reverse, 4.4 H. P. 
Power for return, 3.8 H. P. 
Ratio of return, 4 to 1. 

Method of drive, Direct-current Compound-wound Motor, mounted 
on platen housings, with fly-wheel 30 inches in diameter, 496 
pounds, 800 revolutions per minute, mounted on motor-shaft and 
used as pulley for return of platen. 
Remarks.— Average load on motor, 2.48. A 3 H. P. motor at 800 revolu- 
tions per minute gave first-class service. Rheostat used in series with 
shunt field to raise cutting speed on light work to 30 feet per minute. 



No. 5. 

Description of machine, Turret Lathe. 

Make of machine, Gisholt Machine Company. 

Size of machine, 28 inches swing. 

Number of cutting tools, 5. 

Size of cut, %'' X T V, 1 tool ; %" X &", 4 tool. 

Cutting speed, 25 feet. 

Material machined, Tropenas cast-steel. 

Power for cut, 3.9 H. P. 

Weight of casting, 400 pounds. 

Method of drive, Direct-current Compound-wound Motor, 600 revolu- 
tions per minute, geared to headstock gear in place of cone pulley. 
Speed variations on motor 100 per cent, in all,— 25 per cent, "by 
armature control below normal, and 75 per cent, increase above 
normal by resistance in shunt field. Eleven points in controller, 
giving, with the three gear speeds, 33 changes of speed in all. An 
increase in output of 100 per cent, was obtained on this machine 
by changing from belt to geared motor drive. 



No. 6. 

Description of machine, Drill-press. 

Make of machine, W. F. & John Barnes. 

Size of machine, 21 inches. 

Motor power required, 1 H. P. 

Method of drive, Direct-current Compound Motor, mounted on frame 
of press and belted down to driving pulley. Starter and reversing 
switch mounted on frame of press within reach of operator seated 
at table. 



No. 7. 

Description of machine, Radial Drill-press. 

Make of machine, Niles Tool Works. 

Size of machine, No. 1, 5-foot arm from centre of column. 

Motor power required (maximum), 2.03 H. P. 

Size of motor used, 2 H. P., 600 revolutions per minute. 

Method of drive, Vertical Direct-current Compound-wound Motor, 

mounted on top of column and geared to driving-shaft. Raw-hide 

pinion used on motor-shaft. 



752 Electkic Driving. 



No. 8. 

Description of machine, Double-end Emery-wheel Stand. 
Size of wheel, 18" X 2". 

Speed of wheels, 950 revolutions per minute. 
Kind of work, 2 laborers grinding castings. 
Maximum horse-power, 6 H. P. momentarily. 
Average horse-power, 3.5 H. P. 

Horse-power motor required, 5 H. P. open, with dust-proof covers. 
Method of drive, Direct-current Compound-wound Motor, mounted 
on grinder-shaft between the wheels. 

No. 9. 

Description of machine, Vertical Boring Mill. 

Make of machine, Pond Machine Company. 

Size of machine, 10-foot table. 

Number of cutting tools, 2. 

Size of cut, %» X tV 

Cutting speed, 20 feet per minute. 

Material machined, cast-iron. 

Weight on table, 3.5 tons. 

Motor power required, 8.58 H. P. 

Method of drive, Direct-current Compound- wound Motor, belted to 

counter-shaft. 12 H. P. motor gave good results on heaviest cuts 

and weights of castings. 

No. 10. 

Description of machine, Slotter. 
Make of machine, Bement & Miles. 
Number of cutting tools, 1. 
Size of cut, %» X &". 
Speed of tool, 20 feet per minute. 
Material machined, open-hearth steel castings. 
Motor power required, 6.98 H. P. 

Method of drive, Direct-current Compound-wound Motor, belted to 
counter-shaft. 

No. 11. 

Description of machine, Flat Turret Lathe. 

Make of machine, Jones & Lamson. 

Size of machine, 2" X 24", their standard. 

Motor power required, 1% H. P. for satisfactory service. 

No. 12. 

Description of machine, Tool Grinder. 

Make of machine, Gisholt Machine Company. 

Size of wheel, their standard cup wheel. 

Speed of wheel, 16 to 18 revolutions per minute. 

Maximum horse-power required, 7 for short periods. 

Average horse-power required, 4. 

Method of drive, Direct-current Compound-wound Inclosed Motor, 
mounted on grinder-shaft, with field rheostat in series with shunt 
coils to increase speed from 1600 to 1800. A 5 H. P. open motor 
with inclosing covers gave good satisfaction on this grinder. 

No. 13. 

Description of machine, Engine Lathe. 

Make of machine, Hendey Norton. 

Size of machine, 16 inches. 

Motor power required, approximate, 2 H. P. at maximum. 

Method of drive, Direct-current Compound-wound Motor, mounted 

on support, bolted to bed of lathe, and equipped with clutch and 

cone pulley, with belt to headstock cone. 



Electkig Cranes. 753 






No. 14. 

Description of machine, Engine Lathe. 
Make of machine, Putnam. 
Size of machine, 18" X 6' between centres. 
Motor power required, 2.1 H. P. 

Method of drive, Direct-current Compound- wound Motor, geared to 
counter-shaft. 

No. 15. 

Description of machine, Engine Lathe. 
Make of machine, Pond Machine Company. 
Size of tool, 36" X 10' between centres. 
Motor power required, 10H.P. 

Method of drive, Direct-current Compound-wound Motor, direct- 
geared to counter-shaft. 
On all the preceding machines, where motors are geared, raw-hide 
pinions were used on motor-shaft. 

Electric Cranes, 

In discussing electric driving before the Engineer's Society of Western 
Pennsylvania, Mr. S. S. Wales gives data as to the power required for 
electric cranes. 

As in a general crane specification the actual weights of material and 
gear reduction, etc., are not known, some arbitrary assumptions will have 
to be made and some empirical formulse will be used, but as both are 
founded on facts and experience some reliance may be placed in them. 

An electric crane is divided into three general parts,— bridge, trolley, 
and hoist,— each of which has its own motor and controlling system, and 
each subj ected to different conditions of work. 

For the bridge, where the ratio of axle-bearings to diameter of wheel is 
between 1 to 5 and 1 to 6, the following table will answer our purpose for 
weights and traction for different spans. 
Let 

L = working load of crane, in tons ; 
W= weight of bridge alone, in tons ; 
w = weight of trolley alone, in tons ; 
S = speed, in feet, per minute ; 
P = pounds per ton required. 

Span. W. P. 

25 feet. .3 L. 30 pounds. 

50 feet. .6 L. 35 pounds. 

75 feet. 1.0 L. 40 pounds. 

100 feet. 1.5 L. 45 pounds. 

For the trolley we would assume the weight and traction as shown in 
the following table : 

L. W. P. 

1 to 25 tons. .3 L. 30 pounds. 

25 to 75 tons. .4 L. 35 pounds. 

75 to 150 tons. .5 L. 40 pounds. 



Now the power required for bridge will be 

(Z + TT+ w) X -PX 



33000 



horse-power, 



which result will be used in connection with the motor characteristic to 
determine the gear reduction from motor to track wheel. As the nominal 
horse-power rating of a series motor is based on an hour's run, with a rise 
of 75° C. above the surrounding air, and as conditions of bad track, bad 
bearings, or poor alignment of track wheels may be met with, in factory 
operation 1% times the above result should be taken as the proper size 
motor for the bridge. 

48 



754 Electric Cranes. 



For the trolley the power required would be 

(L + w) X P X S . 

2 ! — '„ n = horse-power, 

33000 * 

which will be used for speed and gear reductions ; but 1% times this shouL 
be used for size of motor. 

For hoist work we cannot have so large a margin of power, as the 
variation from full load to no load may imply a possible dangerous increase 
of speed, and unless the crane is to be subjected to its maximum load 
continuously, or is to be worked where the temperature of the surrounding 
air will be high, it is safe to use the size by assuming 1 horse-power per 10 
foot-tons per minute of hoisting. This is nearly equal to assuming the 
useful work done as 60 per cent, of the power consumed. 

As an illustration, let us take a crane of 50-ton capacity ; lifting speed 
of hoist, 15 feet per minute ; bridge to be 70 feet span and to run 200 feet 
per minute with load ; trolley to travel 100 feet per minute with full load. 
On the foregoing assumption the bridge would weigh 50 tons and require 
40 pounds per ton for traction, and the trolley would weigh 20 tons and 
require 35 pounds per ton for traction. 

Mr. Wales also gives formulas for the power required for driving the 
rollers in rolling-mill tables. 

The power required by roller tables in mill work varies greatly, as they 
are subjected to tight bearings and lack of oil to a greater extent than elec- 
tric cranes ; and as there will be from 2^ to 3 bearings to each roller, and 
many rollers per table, the chances for trouble are greatly multiplied. 

For the average conditions of mill tables, where each roller is driven by 
a mitre gear from a common line-shaft and with usual mill lubrication, 
the following empirical formulae, derived from the test of 20 tables, repre- 
sent about the power required : 

WXDXSXN , 

950^ = horse-power, 

where w = weight of roller, in pounds, the load to be carried on table 
being considered as uniformly distributed over all rollers, 1-iVto each. 

D = diameter of bearings, in inches ; 
S = speed of table, in revolutions per minute, of rollers ; 
N = number of rollers in table. 

The same 1% times power required for size of motors should be taken 
as for crane bridges. 

This takes no account of diameter of roller used, which would of course 
have some effect on the power required to move the load to be handled, 
and would also show some fly-wheel effect when starting, but still it will 
check fairly well with tables now in use under existing conditions, two 
examples of Avhich are given here : 

JV=18; 

W= 1000 pounds; 
D = 4% inches ; 
S = 200 revolutions per minute. 
Diameter of roller, 10 inches. 

1600 X 4 % X 200 X 18 „ _ _ 

950 000 ~ = 27 ' 2 hOTSe -P° wer - 

From actual test under working conditions, this table required 28.8 
horse-power, or the nearest Westinghouse motor being No. 38,-50 horse- 
power.— this type should be used. As a matterof fact, this table is equipped 
with a 80 horse-power motor, and is the source of continual annovance 
from over-load. 

2V=16; 

W 1000 pounds; 

D = 3 inches; ( 

S = 110 revolutions per minute. 
1000X3X115X16 _ B _ 

950000 = 5.8 horse-power. 

By actual test 5.5 horse-power was required. 



* 



Electric Driving. 755 



Choice of Motors and System. 

In discussing the selection of electric motors for driving machinery, Mr. 
P. R. Moses, writing in the Engineering Magazine for September, 1901, says : 
, " The best system in general will be that which will be free from break- 
down, able to stand hard usage and frequent sudden overloads, simple and 
safe to handle, with parts standard and available. It should be uniform 
and applicable to all the requirements liable to arise in the work contem- 
plated, the speed of the motors should be variable at will of the operator, 
and in some cases, like hoisting, should vary inversely with the load to 
prevent undue use of power. The motors should start with small currents 
and should have high efficiencies at average loads. The first cost should 
be as low as possible, and the number of parts a minimum. 

"The alternating two- or three-phase system at low pressure (500 to 220 
volts) meets the first few conditions slightly better than the direct-current 
system of the same voltages. This system consists of a polyphase generator 
composed of a stationary and a revolving part, an exciter— sometimes 
revolving on shaft, sometimes belted to shaft — for delivering the current 
required to magnetize the fields, a system of distributing wires and motors 
frequently built without brushes, but sometimes, where adjustable speed 
and good dynamo regulation are required, with brushes and collector rings. 
The system is simpler than the direct-current system, in that no current 
has to be delivered to any moving part of the motors. In the direct-current 
system, current must be delivered to the rotating armatures of the motors 
through brushes of carbon and commutators made up of copper bars held 
firmly, clamped by a collar, with mica between the bars and between the 
bars and collar. This commutator is the chief difference between the 
motors of the polyphase alternating systems and the motors of the direct- 
current systems. The connections to the commutator and the commutator 
itself are the only parts of the motors in which trouble is liable to arise, 
with careful construction ; and although probably a hundred thousand are 
in use daily, and the manufacture has been carefully studied, trouble does 
arise,— generally on account of accumulation of grease or dirt, allowing 
the current to jump from the copper bars to the iron frame of the machines, 
or from breaking of connections between winding and commutator lugs, 
caused by frequent stopping and starting, combined with overheating and 
slow cooiing. This alternation of heating and cooling causes the copper 
to become brittle and— unless the connections are made flexible— to 
break off. 

"The advantages of simplicity, durability, and freedom from break- 
down, therefore, are with the alternating polyphase motors, — more espe- 
cially of the brushless type ; but, unfortunately for the polyphase system 
at the present day, all the other requirements are much more easily and 
i better met by the direct-current motor. 

" The alternating- current system is not yet fully standardized, but is 
constantly being perfected and broadened in its scope. Its parts are ob- 
tainable from but two or three first-class companies ; it is not applicable 
I yet to charging storage batteries, to railroad work, or to hoisting, although 
I it has been used for both the last; the speed of motors is not adjustable 
unless the brush and collector-ring type is used ; the starting currents 
under load are large, causing more or less fluctuation in lights ; and the 
first cost of dynamos and motors is between 25 per cent, and 35 per cent. 
I higher than that of direct-current apparatus of the same capacity. There- 
fore, unless the value of adjustable speeds, 25 per cent, less first cost, higher 
average efficiency, etc., are balanced by the possibility of commutator 
troubles, the direct-current system is at present preferable and advisable 
ifor ordinary cases of factory transmission. In such cases the polyphase 
ialternating current's value is confined to the transmission of power from 
distant sources at higher pressures. In special cases the alternating-current 
system may prove advisable even for medium distances. One of these 
instances is in hat, candy, or similar factories, where electricity is largely 
used for heating, as well as for power and for light ; the ease with which 
in alternating current can be transformed — i.e., small quantity at high 
pressure changed to large quantity at low pressure, or vice versa — gives this 
system the preference, as quantity and not pressure is the essential feature 
)f electric heating. Works where naphtha or other explosive gas is used 
require a spar kl ess motor, and the alternating motor is the only one fitted 
lor this use. Machine shops with a number of small, scattered machine 



756 Electric Driving. 



tools may be better suited by one system or the other, depending on the 
question as to the advisability of direct connection of tools, the amount of 
probable overload, and the question of speed variation. In mills, large 
machine shops, and factories of all kinds, where the power to transform is 
not valuable, there is no benefit to be derived from the polyphase current 
sufficient to offset the disadvantages mentioned. . 

"As to the best pressure to be used for transmitting direct current, the ' 
answer cannot be so definite. The advantages of a high transmitting press- 
ure, such as 440 to 500 volts, are low first cost and small bulk of wiring, 
switches, and controlling apparatus. The disadvantages are increased 
liability to ground, danger of shock, increased number and decreased size 
of field armature wires and of commutator bars. The liability to ground 
may be guarded against in the construction, but the other difficulties are 
inherent, and are sufficiently objectionable to make the 500-volt motor and 
system inadvisable, except for such purposes as electric railroads, where 
power has to be transmitted a long distance. The 220 to 250-volt motor has 
only half as many commutator segments, armature conductors, and con- 
nections as the 500-volt motor ; and the 110 to 125-volt motor but one-quarter 
the number. The sizes of these parts increase in proportion to the decrease 
in number ; hence, the lower the voltage of a motor, the more substantial 
and the stronger mechanically. The lower voltage offers the advantage, 
too, of decreased tendency of the current to jump to the frame or from 
one wire to another. The winding of the fields of the 120-volt motor is 
made up of less than one-half the number of turns used in the 240-volt 
motor, and while the wire has twice the area, it does not occupy twice the 
space ; on this account the field-coils of the low-voltage motor do not heat 
as much as the higher-voltage motor. On the other hand, the resistances 
used to control the speed and starting current, the switches, fuses, and the 
wiring are much larger and heavier for the 120-volt motor than for the 
240-volt motor, and for motors of large size this is a decided disadvantage, as 
it interferes with ease of regulation and control, and increases the first cost. 

1 ' For small motors, 5 to 50 horse-power, for transmission over short dis- 
tances (from 200 to 400 feet), and for fluctuating work, such as elevators, 
presses, cranes, punches, etc., the 125-volt 2-wire system seems preferable. 
For large motors, transmission over comparatively long distances, or for 
steady work, where mechanical strength is not essential and where the 
motor will receive attention, as in silk mills, carpet works, etc., or for work 
where minimum weight and size are important, the 240-volt system, or one 
of higher pressure, is usually the best fitted for the purpose. 

"It is my opinion that at present prices the 125-volt direct-current 
2-wire system is the preferable one in most instances, and little or no trouble 
with the motors or dynamos is experienced at this voltage. Where the 
distance through which the power is to be transmitted is such as to make 
the cost of wires for carrying the current too great, the 240-volt direct- 
current 3-wire system should be used. Where there are special features, 
such as those heretofore mentioned, or where the power is transmitted a 
distance too great for the 240-volt direct-current svstem, the three-phase 
alternating-current system, 60 to 70 cycles per minute, becomes necessary 
and jrires thoroughly satisfactory results. 

"The greatest drawback to the alternating-current svstem to-day is its 
excessive cost, which brings no equivalent advantage. In the course of a 
few years this difference will disappear, and the system will probably be 
0860 for all situations demanding a higher pressure than 125 volts." 

Speed Variation. 

The question of the control of speed with electric driving of machine 
tools is an importanl one, especially as the use of modern high-speed tool 
Bteels involves the correct speeding for each diameter and material in the 
lathe, boring mill, or other machine. This subject was thoroughly dis- 
cussed at a meeting of the Engineer's Club of St. Louis, Januarv 7, 1903, 
and sonic abstract of the j .oints there made represent the latest opinions. , 
Mr. \v. a. Layman say- : * 

"The individual drive system may be generally classified under three 
headings: 

" Rheostatic control svstems, 

" Multi-voltage control, 

" Special systems for special tools. 



Electric Driving. 



757 



' ' In the rheostatic control system the motor is of the well-known shunt 
type, supplied from a constant-potential system of distribution. Speed 
variation above the normal speed of the motor is secured by the introduc- 
tion of resistance into the motor shunt-field circuit ; speed variation below 
j normal is secured by the introduction of resistance into the armature 
w circuit. 

"The disadvantages of the system are its inefficiency when armature 
resistance is made use of for speed reduction, and variation of speed on a 
given armature resistance with variation of load. To overcome both dis- 
advantages, motors have been designed capable of a very wide variation 
in speed by variation of field resistance. 

" The multiple-voltage control is regarded with favor by many. 

"The Westinghouse and the General Electric Companies use a 3-wire 
system, as shown in the illustration. 




Westinghouse 3-wire System for Variable Speed Control. 



"The usual direct-current generator is provided with a set of collector 
rings, these collector rings being connected to the armature winding in 
such a way as to establish an exact two-phase relation between the poten- 
tials of the two pairs of collector rings. By means of choking coils, con- 
nected as shown, the neutral wire of the 3-wire system is exactly and 
constantly maintained, irrespective of load, at zero potential relative to 
the outside wires. 

"In connection with this 3-wire system the individual tool is equipped 
with a standard 250- volt shunt motor, and speed variation is secured in 
two ways: first, by running the armature either on 250 volts (normal speed 
: condition) ; and second, by running it on 125 volts (half normal speed 
condition). For any speed desired between normal and half normal, 
shunt field resistance is introduced. 

"The shunt motor is capable of 100 per cent, speed variation by varia- 
, tion of shunt resistance when the armature is on half voltage (and corre- 
spondingly at half load). If speed above normal full speed is required, 
shunt resistance is again introduced. 

"The Bullock Electric Manufacturing Company advocates a system as 



758 



Electkic Driving. 



shown in the illustration. A generator, standard in every respect, is sup- 
plemented by a small motor-generator set, the design of which is such that 
a 4- wire system of distribution is established, providing for six different 
voltages upon which the motor armature may be operated without the use - 
of armature resistance. The form of motor used is the standard shunt-^ft 
wound type. Without the use of field resistance six speeds may be se- 
cured, corresponding in ratio to the ratio of the voltages supplied by the 
4-wire distribution system. By means of shunt resistance any speed in- 
termediate to that possible with the several armature voltages may be 
secured. The motor-generator set is so proportioned as to take care of the 




no Volt 
Generator 



jut 



H T T B — i 



IF 



i 



I 



Motor and Double 
Commutator Generator 



Bullock Electric Manufacturing Company's Multiple-voltage System. 



unbalanced load. This system is also adaptable to 3-wire distribution, 
where less speed variation is required ; and in the event of 3-wire distribu- 
tion an increased amount of field regulation is introduced. This 3-wire 
distribution differs from the Westinghouse and General Electric systems, 
in that the voltages on the two sides of the intermediate wire differ, thus 
giving three pressures instead of two." 

In the same discussion Mr. W. Cooper said, in considering the fact that 
many tools are not designed to stand the high speeds that modern tool 
st.cls will permit, "that even if the machines will not stand the maximum 
speed oi the tool, they may yet be operated at the highest speeds of which 
fchey are capable. 

"There is no reason why a machine tool that is adapted to do a certain 
work should not do this work at two or three times the speed. The reason 
for this seems obvious, in the fact that the strains on a machine are due 
entirely to the torque required to make a given cut. With this given cut 
the speed may be 1 acres s< K 1 three or four times without producing any r 
greater strains on the machine itself , because the torque remains constant. 
However, the horse-power will increase directly in proportion as the in- 
crease in speed : and right here we have a. factor that limits the ordinary 
belted machine tool, the belts will not pull the load. For instance, 
suppose that we have a given machine running with a belt on the largest 



Electric Driving. 759 



step of the cone pulley on the machine, and taking a certain cut ; assume 
the cutting speed to be 20 feet a minute. If it is desired to increase this 
cutting speed to, say, 80 feet per minute, it is found necessary to put the 
belt on the smallest step on the cone on the machine. We at once en- 
counter the difficulty that the belt will not begin to pull the cut. This is 
also true of the various mechanical speed-changing devices that have been 
introduced. Thus it will be seen that machine tools that were designed 
on the lines of the cutting speed of 20 feet per minute are not adapted at 
all to cutting speed of 80 feet per minute. However, they are not limited 
by the strength of the tool, but by the pulling power. Under these con- 
ditions it is only necessary to increase the pulling power of this machine 
to make it do four times the actual work that it formerly did. 

" This can readily be accomplished by the use of the electric motor, so 
that the limit lies in the stiffness of the bed or frame of the machine to 
carry the increased load without springing or chatter." 

In a paper read before the Iron and Steel Institute, in 1903, by Mr. D. 
Selby-Bigge, the power required to operate electrically a great variety of 
machine tools was tabulated. Although in many instances the depth of 
cut and the speed of cutting surfaces is not given, yet it is understood that 
these records represent average practice with ordinary tool steels, and that 
for lathes, boring mills, etc., the power given in the tables would have to 
be increased in proportion to the increased capacity given by use of the 
modern high-speed steels. 



760 



Electric Driving. 



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Electric Driving. 



(A 

H 

u 
© 





a 






Size 

and 

type of 

motor. 


20 B. 
H.P. 

20 B. 
H.P. 


"8 

u 

i 
■a 

w 


H~ o 


14.9 E.H. 
P., with 
fluctua- 
tions to 
19 E. H. 
P. 

Fluctua- 
tion to 
about 18 
E. H. P. 


Mo 

< a- 


9.5 E. H. 
P. 

Average 
full 
load, 
11.2 E. 
H.P. 


Average 

machines 

light. 


W 
W 

CO 




Shafting 

and all 

machines 

light. 


Shafting 
alone, 1.75 
E. H. P. 

All 

machines 
light. 

62 E. H. P. 

Shafting 
alone, 1.5 
E. H. P. 


u 
o 
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a 
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2-9" centre lathes. 
1-12" centre lathes. 
1-3' chuck lathe. 

1 slotting machine. 

1 dril ling machine. 
1 emery wheel. 

Fan. 

All machines, 
pots, and drying 
machine contin- 
uous, and others 
intermittent. 


o 
o 

a 


OB 

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u 

i 

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3 

o 
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2-9 // centre lathes. 
1-12" centre lathes. 
1-3' centre chuck lathe. 
1 planing machine, 9-foot 
stroke. 

1 slotting machine. 

1 milling machine. 

1 shaping machine. 

1 punch and shears machine. 

2-2%" vertical drills. 

1 emery wheel. 

1 cold saw (hack). 

1 fan, supplying 9 smiths' 

Shafting: 20' X 3" C shaft; 
five others to 10 feet long. 

4 large galvanizing pots, each 
for 20 tons of metal. 

2 drying machines, attached. 
1 large sheet-stretching ma- 
chine. 

1 large corrugating machine 
(press). 

2 circular shearing machines, 
counter-shafting, and belts. 


Works 

depart- 
ment or 

machine. 


«. / \ j 


tuo ti 


No. 1. 
Gal- 
van- 
izing- 
house 
motor. 



Electric Driving. 



765 






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Electric Driving. 



Tests on Cranes. 



Motor 

Quick=motion, 

Taken August 





Size and type of 
travelling crane. 


Work done by 


Actual tests on crane. 


Works 


crane. 


Motion. 


E. H. P. absorbed. 
Crane light. 




Aver- 
age. 


Maxi- 
mum. 






Starting 

effort. 


Running 
Power. 


No. 1 crane. 
Plate mills 
loading. 

No. 2 crane. 
Plate mills 
floor. 

No. 3 crane 
(new). 
Plate mills 
loading. 


5-ton 3-motor 
crane. Works 
in exposed 
position, both 
in and out of 
shop. 

5-ton 3-motor 
crane. Works 
under cover. 

6-ton 3-motor 
crane. Built 
and erected 
by Dowlais 
Cardiff 
Works. 
Works in ex- 
posed posi- 
tion, both in 
and out of 
shop. 


lto3 
tons. 

lto2 
tons. 

lto3 
tons. 


4K 
tons. 

4 tons. 

4% to 5 
tons. 


Lifting. 

Traversing. 

Travelling. 

Lifting. 

Traversing. 

Travelling. 

Lifting. 

Traversing. 

Travelling. 


13.4 
14.8 
23.6 

18.0 
20.6 

28.5 

22.3 
11.3 
29.5 


.50 

7.70 

11.80 

9.60 
9.35 
11.3 

9.5 

6.2 

12.0 






Note.— The starting efforts given above can be regarded only as ap- 
proximate, being merely momentary, and volts drop being disregarded. 
The cranes above are on 220 volts circuit. One longitudinal trolley wire 



Electric Driving. 



767 



Tests. 

Overhead Travelling. 

18, 1902. 



Figures taken August 18, 1902. 


Size 












E.H.P. 


absorbed. 


Speeds and loads 


of motor. 




Crane loaded. 


during test. 


H. P., brake 

or 

electrical. 


Remarks. 












Starting 


Running 


Actual 


Approximate 






effort. 


power. 


load. 


speeds. 






29.1 


18.3 




43 feet per 


22 E.H.P. 


This crane has the 






1 3 tons J 
[ 6 cwt. 1 


minute. 




motors fixed on the 


16.8 


9.6 


125 feet per 


15 E. H. P. 


main girders above 






minute. 




end carriage. Drives 


29.4 


12.6 




150 feet per 
minute. 


22 E. H. P. 


by square shaft and 
gear. Water start- 
ing and regulating 
switch, and metallic 
lever controlling 
switch. 


27.8 


12.8 


[ 


Speeds 


[22 E. H. P. 


As above. 


24.5 


10.3 


[•3 tonsJ 


approxi- 
mately as 


jl5 E.H.P. 




34.0 


12.4 


I 


above. 


122 E. H. P. 




29.4 


19.2 


1 , 


60 feet per 


20 B. H. P. 


This crane has travel 






1 3 tons 1 
[ 6 cwt. 1 


minute. 




motor on main 


14.9 


7.3 


150 feet per 


20 B. H. P. 


girders and lift and 






minute. 




traverse motors on 


32.8 


13.8 




165 feet per 
minute. 


20 B. H. P. 


the bogie. Gear 
driven. Water start- 
ing and regulating 
switch, and metallic 
lever controlling 
switch. 



only employed, the return being to "earth." The above working-load 
tests present a fair and good heavy average load, and are seldom exceeded 
under actual working conditions in these works. 



768 



Electric Driving. 



Motor Tests. 

Taken August 18, 1902. 



Description 
of machine. 


"Work done 
by machine. 


E.H.P. 

absorbed. 


Type and 

size 
of motor. 


Remarks. 




Light. 


Loaded. 




3-ton skull- 
breaking 
winch. 


Lifts ball 
weighing 3 
tons 8 cwt. 
to height 
of 50 feet at 
speed of 60 
feet per 
minute 
(timed). 


8.5 


17.8 


2-pole open 
type ar- 
mature at 
bottom, 
18E.H.R, 
series- 
wound. 


This winch is of or- 
dinary band pat- 
tern, driven 
through works and 
spur gear, with 
brakes and clutch. 
Water starting and 
regulating switch. 



A table of machine tests in a somewhat different form is appended : 



Condensing Plant. 

One 10-horse-power motor of 220 volts, driving direct-coupled 3-inch 
centrifugal pump, driving also with belt air pump 9% inches in diameter 
by 9-inch stroke, and feed pump 2 inches in diameter by 9-inch stroke. 
Boiler pressure, 200 pounds. 





Revolu- 
tions. 


Amperes. 


Volts. 


Vacuum, 
in 

inches. 


E.H.P. 


Operation. 


Total. 


Actual per 
operation. 


Centrifugal pump 

Centrifugal with air and 
feed pump 


1100 
160 


6 
12 


240 

240 


27 


1.9 

3.8 


1.9 
1.9 







Brass-shop Motor, 5 Horse-power, 240 Volts. 



Operation. 



Revolu- 
tions. 



Volts. 



Amperes. 



E.H.P. 



Total. 



Actual per 
operation. 



Motor and shaft 

Disc grinder, 18-inch emery discs 

running light 

Facing 6%-inch brass valves 

6-inch capstan lathe (light) 

Turning and screwing 1%-inch 

brass bars for %-inch tap bolts. . 
Parting 1%-incn brass bars for 

%-inch tap bolts 



220 



1800 
1800 



250 

246 
246 

248 

248 

248 



7.00 

8.50 

24.00 

9.75 

12.0 

10.0 



2.3 

2.8 
7.9 
3.2 

4.0 

3.3 



2.3 

.5 

5.6 

.9 

1.7 

1.0 



Electric Driving. 



769 



No. 1 Foundry.— Roots Blower, Acme No. M. 



Operation . 


Revolu- 
tions. 


To Its. 


Amperes. 


E.H.P. 


Time, 


Total. 


Average. 


hours. 


Motor and shafting, light 
running 




245 

246 
233 


14 

104 
66 


4.6 

32.7) 
21.7 J 


28.2 




Blowing cupola : 
Maximum 


360 
350 




Minimum 


^A 







Total weight of iron melted, 22 tons 10 hundredweights ; total weight 
of iron melted per hour, 5 tons. 



No. 2 Foundry. — Roots Blower, Acme No. K. 



Operation. 


Revolu- 
tions. 


Volts. 


Amperes. 


E. H. P. 


Time, 


Total. 


Average. 


hours. 


Motor and shafting, light 
running 




232 

237 
225 


9.5 

57.0 
50.0 


2.96 

17.1) 
15.2 j 


15.94 




Blowing cupola : 
Maximum 


430 
394 




Minimum 


3 



Total weight of iron melted, 12 tons ; total weight of iron melted per 
hour, 4 tons. 

Remarks.— No. 1 cupola is capable of melting on an average 7 tons per 
hour. 

Boiler Shop.— Vertical Plate=bending Rolls. 

Length of rolls, 11 feet 7 inches ; diameter of rolls, 1 foot 11 inches ; 
mean size of plates rolled, 20' X 10' 6" X 1" ; maximum size of plates 
rolled, 16' X 11' 5" X V4". 



Operation. 


Amperes. 


Volts. 


E. 
Total. 


E.P. 

Actual per 
operation. 


Time, 
in min- 
utes. 


Motor and shafting, light run- 
ning 


20 
24 

90-68 

30-60 

50 


242 
242 

233 
233 
233 


6.40 
7.70 
Average. 
19.2 
9-18 
15.6 


6.4 
1.3 

12.8 

6-12 

9.2 




Rolls, light running 




Rolling plate, 23' X IV 2" X 
V/i" 


3 


Putting squeeze on plates 

Reversing the rolls 









Air Compressor, Belt Driving from Motor. 


Operation. 


Revolu- 
tions. 


Volts. 


Amperes. 


E.H.P. 


Remarks. 


Motor, shafting, and 
pumps. 

Pumping up to maxi- 
mum pressure. 


175 
170 


230 
230 


22 
70 


6.7 
21.5 


Air compressor, 9 
inches in diame- 
ter, 10-inch stroke. 

Maximum pressure, 
80 pounds. 



49 



770 



Electric Driving. 



Pattern=shop Motor, 15 Horse=power 


220 Volts. 






Revolu- 
tions. 


Am- 
peres. 


Yolts. 


E. H. P. 


Operatiou. 


Total. 


Actual per 
operation. 


Motor and shafting 


170 

800 

Maxi- 
mum. 

Mini- 
mum. 

Average. 

3800 


9.5 

10.5 

40.0 

18.0 
29.0 

12.0 

17.9 


233 

233 

233 

233 
233 

230 

232 


2.9 

3.2 

12.4 

5.6 
9.0 

3.7 

5.5 


2.9 


Circular saw, 2 feet 8 inches in 
diameter, running light 


.3 


Cutting yellow pine 11 inches | 
deep, 7 feet per minute ■{ 

Thickness of machine, 2 feet 6 inch 
bed running light 


9.2 

2.4 

5.8 

8 


Surfacing yellow pine 11 inches 
wide, 13 feet per minute 


1.8 









37 Horse-power Motor. 



Operation. 



Revolu- 
tions. 



Am- 
peres. 



Volts. 



Total, 



xVctual per 
operation. 



Motor and shafting 

Circular saw, 3 feet in diameter, 
running light 

Sawing yellow pine 11 inches deep, 
20 feet per minute 

Circular saw, 33 inches in diame- 
ter, running light 

Cross-cut lignura-vitse, 9% inches 
deep by 18 inches long 



1200 



12 
21 
71 
26 
41 



230 
230 
230 
230 
230 



3.7 
6.4 

21.5 
8.0 

12.6 



3.7 
2.7 
15.1 
4.3 
4.6 



30-Ton Cranes. Boiler Shop, 25 Horse-power Motor, 220 Volts. 



Operation. 



Am- 
peres. 



Volts. 



E.H.P. 



Total. 



Actual per 
operation 



Time, 
in min- 
utes. 



Feet. 



Motor and shafting and belts 
(light) 

Heaving (light) 

Cross travel (light) 

Longitudinal travel (light) .., 

Longitudinal and cross travel 
(light) 

Travelling long 

Weight, 16 tons, heaving 

Cross travel 

Longitudinal travel 



12.0 
16.0 
15.0 
16.0 

18.0 
16.2 
36.0 
30.0 
18.0 



230 
230 
230 
230 

230 
230 
230 
230 
230 



4.3 
4.9 

4.8 
4.9 

5.5 
4.9 
11.1 
9.2 
5.5 



4.30 
.60 
.50 
.60 

1.20 
.68 

6.2 

4.4 
.6 



90 
5 



90 



Cost of Powee. 



771 



THE COST OF POWER. 

In a series of articles in the Journal of the Franklin Institute, October, 
November, December, 1901, Mr. Clyde D. Gray gives an extended discussion 
of the cost of power under various conditions, and from these papers the 
following abstract is made : ' 



Water Power. 

The costs of water-power plants are widely different, depending upon 
the location, size, and extent of the hydraulic works needed, length of 
penstock and flume, and many other things that differ in the various local- 
ities. Below are given some figures in regard to the costs of plants. These 
are low-head plants fitted with turbine wheels, and are used principally 
for mill or factory purposes. The costs do not include costs of dam unless 
so specified, but include everything else in the plant. The horse-power 
basis upon which they are figured is the horse-power delivered at the wheel 
shaft. 

WATER=PLANT COSTS. 



Place. 


Cost 

per 

D.H.P. 


Authority. 


Lawrence, Massachusetts 

Manchester, New Hampshire . . 

Lowell, Massachusetts, 13 feet 

head 


$68.67) 
66.00] 

110.00~ 

57.00 
63.00 * 

67.50^ 

57.75") 

34.20 1 
37.50 [ 

24.00 J 

67.33 

100.00) 

120.00 j 

108.25 


Manning, A.S.M.E., Vol. X., p. 
499. 


Lowell, Massachusetts, 18 feet 
head 


C. T. Main, A. S. M. E., Vol. XL, p. 

108. 


Lawrence, Massachusetts 

Lawrence, Massachusetts, 1000 
horse- power 


Concord, New Hampshire (with 
dam) 




Augusta, Georgia 

Columbia, South Carolina 

Caratonk Falls, Maine (with 
dam) 


Webber, A. S. M. E., Vol. XVII., p. 
41. 


Omaha, Nebraska (estimate) . . 
Zurich (with dam) 


Eng. Mag., Vol. VII., p. 409. 


Paderna. Italy (with dam) — 
Big Cottonwood, 3000 horse- 
power 


Eng. Mag., February, 1900. 
Eng. News, October 1, 1896. 




Average without dam (exclud- 
ing Lowell, $110.00) 


$53.41 
79.55 









It is probable that the cost of such plants will be from $40.00 to $60.00, 
excluding the cost of dam, but including all other parts ; and when the 
dam is included that it will be from $60.00 to $100.00. Webber, in Iron Age, 
February and March, 1893, says that water-power plants can be put in for 
$100.00 per horse-power; and Stilwell, in A.I. E. E., Vol. X., p. 484, says 
that the cost may be as low as $65.00. 

The cost of water power per horse-power year is variable, depending, as 
it does, upon the first cost of plant; and hence no very good average can 
be found. The following table may serve to show the costs in some cases 
that have been reported. 



772 



Cost of Power. 



COST OF WATER POWER. 



Place. 


Cost per 
H.P. 
year. 


Authority. 


Lawrence, Massachusetts 

Canada (lowest) 


$13.70 

6.25 
16.10 

22.62 

19.13 

8.64) 

n.05 y 

9.50) 

8.08 

11.25 

13.00 

5.42 


C. T. Main, A.S.M.E., Vol. XIIL, 

p. 140. 
Mever, Sci. Am., February 9, 1882. 
Eng. News, October 1, 1896. 

Manning, A. S. M. E., Vol. X., p. 48. 

Main, A. S. M. E., Vol. XIIL, p. 140. 
Webber, A.S.M. E., Vol. XVIL, p. 


Cottonwood. 


Lawrence, Massachusetts, 1000 
horse-power 


Lawrence, Massachusetts, 500 
horse-power 


Concord, New Hampshire. — 
Augusta, Georgia 


Columbia, South Carolina 

Omaha, Nebraska (estimate). . 
Norway (electrolytic work) . . . 


41. 

Eng. Mag., Vol. VII., p. 409. 
Chem. Ind., Vol. XXIII., p. 121. 
Emery, A. 1. E. E., Vol. XII., p. 358. 




Webber, W. 0., Eng. Mag., Vol. 




XV., p. 926. 




$10.72 







From the above table it may be seen that the cost per horse-power year 
is $10.72. Webber gives it as $10.00 to $12.00 {Iron Age, February and 
March, 1893) ; and Conant, in an article in the Street Railway Journal for 
October, 1898, gives the cost as ranging from $10.40 to $22.40. A fair average 
may be taken as varving from $10.00 to $15.00. 



Steam Power. 

SUMMARY OF BOILER TESTS. 

Water Evaporation per Pound of Fuel. 



Authority. 



Kent (Christie), A.S.M. E., Vol. 

XVIlI.,p. 365 

Barrus, horizontal tubular 

Barrus, horizontal tubular, low flue 

temperature 

Barrus, horizontal tubular, high flue 

temperature 

Barrus, horizontal tubular 

Average from Gray's Tables, W. T 

Average from Gray's Tables, tubular. . 

Average of all the above 



No. of 
tests. 



95 
16 



5 

10 
37 
23 



192 



Water 
evaporated, 
in pounds. 



11.11 
10.76] 

10.40 I 

ll.OOj 
11.59 
10.80) 
10.40/ 



10.86 



Kind of fuel. 



All kinds. 

Anthracite. 

Cumberland. 
All kinds. 



Cost of Power. 



773 



SUMMARY OF ENGINE TESTS. 

Pounds of Water per Horse-power Hour. 



Authority. 


Auto- 
matic. 


Corliss. 


Simple, Non-condensing. 

Carpenter (Sibley theses, Sib. Jour., Vol. XIV., p. 228) .... 
L. Bell, " Electrical Transmission of Power" 


34.3 
33.0 
33.0 
31.5 
36.0 
33.4 


28.3 
28.0 


Hutton, " Mechanical Engineering of Power-plants" 

Thurston & Carpenter, A I. E. E., Vol. X., p. 297 


29.0 
28.6 


Davis, C. H., Eng. Maq., Vol. XII., p. 942 


31.5 


Average of Gray's Tables 


28.9 






Average of all the above 


33.8 


29.0 






Compound, Non=condensing. 

Carpenter 


32.3 
24.0 
26.0 
24.5 
27.0 
23.6 




Bell 


22.0 


Hutton 


24.0 


Thurston & Carpenter 




Davis 


26.0 


Gray's Tables 








Average 


26.2 


24.0 






Simple, Condensing. 

Bell 


25.0 
22.0 


21.0 


Hutton , 


20.0 


Thurston & Carpenter 


23.0 


Thurston, Eng. Mag., Vol. VII., p. 844 




25.0 


Davis , 


31.0 

22.2 


26.5 


Gray's Tables - 


20.2 






Average 


23.1 


22.6 






Compound, Condensing. 

Carpenter 


22.7 
20.8 
20.0 
18.8 
22.5 


18.3 


Bell 


18.0 


Hutton 


18.0 


Thurston & Carpenter 


17.2 


Davis 


23.0 


Thurston 


18.0 


Grav's Tables 


19.8 


15.7 






Average 


20.6 


17.7 






Bell 


17.0 
17.0 


13.0 




16 


Thurston & Carpenter 


14.6 






14.0 


Gray's Tables 




13.3 








Average 


17.0 


14 2 







774 



Cost of Powek. 



STEAJVUPLANT COSTS. 

The cost of steam plants varies greatly with the locality, size, kind of 
machinery, boilers, and many other items. The table given herewith 
shows good approximations to the costs of various constructions. 



TABLE OF PLANT COSTS. 



Authority. 




Remarks. 



Manning, A.S.M.E., Vol. X., p. 
48 

Field, C. J., A. S. M. E., Vol. XVI. 
p. 504 

Webber, A. S. M. E., Vol. XVII., p. 
41 



Dean, A.S.M.E., Vol. XIX., p, 
301 

Rathwell, A.I.M.E., Vol. XVII., 
p. 555 

Western Elec., March 16, 1901 

Carpenter, Sib. Jour., Vol. XIV.. 
p. 298 



Elec. World, February 2, 1901, p. 214. 



Thurston, Eng. Mag., Vol. VII., p. 
841 



$68.26 

52.50 

50.00 
65.00 
54.71 

59.51 
60.35 
70.00 
70.00 

57.00 

60.50 



40.00 
60.00 
28.60 

30.20 
30.00 

30.00 
33.25 
28.50 

38.00 

45.00 
53.00 
62.00 



Total — engine, boilers, stack, 

500 horse-power plant. 
Total— engine, boilers, stack, 

1000 horse-power plant. 
Steam plant complete. 
Plant complete. 
High speed, condensing, from 

Emery's tables for 550. 
Low speed, from Emery. 
Compound low, from Emery. 
Triple compound, from Emery. 
Simple Corliss, condensing, 

best, 1000 horse-power. 
Compound, condensing, best, 

1000 horse-power. 
Actual cost of a yarn mill, 

1132 horse-power. 

Engines and boilers. 

Complete plant. 

Simple slide valve, non-con- 
densing.* 

Corliss, non-condensing. 

Compound slide valve, non- 
condensing. 

Compound, non-condensing. 

Compound Corliss, condensing. 

Estimates on plant for South 
Africa. 

Engines, boilers, and piping, 
simple, condensing. 

Compound, condensing. 

Triple, condensing. 

Quadruple, condensing. 



Field, in A. S. M. E., Vol. XVI., p. 504, gives the average cost of steam 
plants as ranging from $50.00 to $55.00 per horse-power, and Professor 
Ryan, in an article in the Engineering Magazine, Vol. VII., p. 733, says that 
the cost of steam plants with high-speed engines is about $50.00, and that 
for slow-speed Corliss engines ranges from $65.00 to $75.00. This is ex- 
clusive of the cost of the buildings. 

The costs of electric plants are dependent upon the cost of engines 
and boilers, and their cost is usually a constant quantity, for the cost ot 
dynamos is nearly constant per kilowatt plus the cost of engine plant. 
The cost of dynamos and other electrical apparatus may be assumed as 
ranging from $20.00 to $35.00, including switchboard; hence, the cost of 
complete plants for electric lighting and power may be assumed to be 
$75.00 to $100.00, according to circumstances. 



♦ The costs under this are for engines, boilers, and pining alone, exclusive of 
cost of building. 



Cost of Power. 



775 



COST OF STEAM POWER. 

The following table is a condensation of a more detailed one in Mr. 
Gray's paper, and gives practically the same result. The costs are for the 
total of fixed and operative charges, in cents, per horse-power hour. 



Authority. 



Cost per H. P. 
hour, cents. 



Emery, A. I. E. E., for 3080 hours per annum 

Emery, A. I. E. E., for 7090 hours per annum 

Emery, Eng. Mag., for 3080 hours per annum 

Webber, 650 horse-power, for 3080 hours per annum 
Webber, 1050 horse-power, for 3080 hours per annum 

Hale, for 2985 hours per annum 

Main, for 3080 hours per annum 

Foster, for 3080 hours per annum 

Gray's Table 

Average of all 



.784 
.617 
.856 
.720 
.646 
.557 
.637 
.824 
.720 



707 



Dr. Louis Bell, in his book, "Electrical Transmission of Power," gives 
as the cost for 10-hour day, full load, with large compound-condensing 
engines, 0.8 to 1.0 cent per horse-power hour, and for simple engines, 1.5 
to 2.5 cents: while, if the load is partial and intermittent, these figures 
become 1.0 to 1.5 and 3.0 to 4.0 cents, respectively. 

The cost of engines varies considerably with the class, but the following 
table is a very good approximation for the different kinds : 

Simple slide-valve engine $7.00 to $£10.00 

Simple Corliss or low-speed type 11.00 to 13.00 

Compound slide-valve 12.00 to 15.00 

Compound Corliss 18.00 to 23.00 

High-speed automatic 10.00 to 13.00 

Low-speed automatic 15.00 to 17.00 

In addition to this is the price of boilers, which is approximately $10.00 
to $12.00 for the plain tubular and about $15.00 for the water-tube type, and 
the cost of pumps, which is about $2.00 for a non-condensing and $4.00 for 
a condensing plant, including heaters. 



Gas Power. 

The following tables contain some tests of different gas engines, using 
various kinds of gas. Some of these are small and others large, although 
there are but few tests of the larger sizes, from the fact that they have not 
been on the market until recently. The amount of gas used per horse- 
power is in some cases based upon the indicated, and in others upon the 
developed or brake horse-power. This is indicated by an (I.) or (B.) 
placed after the column. 

Using Natural Gas. 



Kind. 


H.P. 


Cubic feet 
per H.P. 


Authority. 


Westinghouse 


J 621 
1 67 


9.3(1.) 
10.4 (B.) 


Miller & Gladden, Sib. Jour., 
June, 1900. 




Lond. Eng., January 4, 1901. 



The Westinghouse Company will guarantee a gas consumption of 12 
cubic feet of gas per B. H. P. on their small engines, and as low as 10 cubic 
feet of gas per B. H. P. for the larger sizes. The Standard Automatic Gas 
Engine Company guarantees less than 15 cubic feet per B. H. P. 



776 



Cost of Power. 



Using Coal Gas. 



Kind. 



H.P. 



Cubic feet 
per H. P. 



Authority. 



Westinghouse . 
Springfield 



9 

12 
10 



Campbell 

Otto-Crossley 

Clerk 

Atkinson, differential 

Atkinson, cycle 

Griffin 

Forward 

Simplex 

Wells Bros 

Premier 



17 

9 



6 

16 

6 



Railway plant . 



[12 

118 

61 

50 

31 



Average on B. H. P. 



14.5 
16.5 
15.5 
16.6 
15.4 

24.1 

30.4 
25.7 
22.5 
28.5 
22 5 
20.4 
27.8 
21.0 
19.7 
17.0 
21.0 



(I.) 

(I.) 

(I.) 
(B.) 
(B.) 

(B.) 
(B.) 



(I.) 
(B.) 



22.9 



Budd & Moody, Sibley thesis. 
Spier & Keely, Sibley thesis. 
Perry. 

Lond. Eng., January 4, 1901. 
Donkin, Eng. Mag., December, 
1900. 



London Elec. Eng., January 25, 
1901. 



Hill & Brocksmit thesis. 



This average is rather higher than can be expected of the best modern 
engines, for these are all rather small units. About 17 or 18 cubic feet 
may be expected of the average modern engine of moderate size. 







Using Producer Gas. 


Kind. 


H.P. 


Cubic feet 
per H. P. 


Coal, in 
pounds. 


Authorit}'. 


Crossley 

Koerting 


142 
349 
377 

)170 
J220 
{ 59 


65.7 (I.) 
83.2 (B.) 
60.1 (I.) 


.92 

"liij 

.88 
1.30) 
1.30 j 
Less 
than 
2.00 

.80 
1.03 

.95) 

.76/ 
1.60 
1.03) 

.81/ 
1.41) 

.76 
1.06 f 

.86 J 
1.25 


Lond. Eng., January 4, 1901. 
Mond. gas, January 4, 1901. 


Diesel 








Dowson gas, January 4, 1901. 
Witz, Dowson gas. 

Dowson gas. 

Lond. Eng., November 30, 1894. 

Richmond, Eng. Mag., Vol. 

X., p. 853. 
Spangler, Cass. Mag., Vol. IX., 

p. 47. 
Power Quarterly, 1901. 
Adams, St. By. Jour., June, 

1900. 

Trans. I. C. E., Vol. V., p. 73. 

Adams, Eng. Mag., Vol. XVI., 
p. 513. 


Simplex 

Otto 


86.8 (B.) 
77.7 (B.) 


Simplex 

Crossley 

Otto 


220 
280 






(I.) 

fa 

(I.) 
I. 
I. 
I. 
109.0 (B.) 


Crossley 




Le Tombe 




Crossley-Otto . . 

Atkinson 

Stockport 


100 

320 

f 28 

\118 

22 

76 


Average of those on I. II. P 1.04 





Average cubic feet on B. II. P., 82, or on I. H. P., 62.9. These values are 
only approximate, as the data are not very complete. 



Cost of Power. 



Using BIast=furnace Gas. 



Kind. 


H. P. 


Cubic feet 
per H. P. 


Authority. 


Cockerill 

Otto type 


(182.0 . 

< 650.0 

(725.0 

79.5 

175.0 


116.5 (B.)l 

135.7 (B.) 1 

101.0(1.) f 

79.4(1.) J 

145.2 (B.)\ 
140.0 r 
112.9 (B.)1 

102.3 (B.) j 
90.0 (B.) 

111.4 (B.)) 
101.0 (B.)j* 
133.0 (B.)i 
231.0 (B.) y 

91.0(1.) j 


Donkin, Eng. Mag., December, 
1900. 


Wishaw, England. 


Booth, Cass. Mag., March 9, 
1901. 


Cockerill 


/ 661.0 

(715.0 

15.0 

575.0 

" 177.0 ' 

90.0 

100.0 


Hubert, test at factory. 

"Colliery Guardian," quoted 
in Eng. Mag. , Vol. XV. , p. 140. 


Acme 


Cockerill 


Cockerill, with blower 
Frieden 


Eng. Mag., Vol. XIX., p. 587. 




Schutte. 


Average 




115.1 











COST OF GAS POWER. 

The following table gives some figures on the cost of power produced 
by different-sized engines using various fuels. The kind of gas used is 
indicated by the capital letters in parentheses, as well as whether the 
horse-power is based upon the indicated or the brake. 



Kind. Place. 



Cubic 

feet of 

gas. 



Coal per 
H.P. 



Cost, cents. 



Authority. 



Ordinary . 



17 
20 
23 



1 Schwabing . 
Otto 



Ordinary . 
Clausthal . 



200 
250 



Average engine. 

Glasgow 

Ordinary 

Average 



20 



Crossley . . , 

Le Tombe . 

Otto 

Charon — 
Crossley . . . 



1.00 



1.50 

'i'io' 



1.73 
.93 



1.25 



( 2.05 ) 
< per y 
(K. W.j 

1.60 

1.00 

1.06 

1.23 



1.00 (B. P.) 1 
1.02 (B. C.) J 

.87 (B. p.n 
1.50 (B. P.) 
1.50 (B. P.) \ 
3.00 (B. C.) I 

.16 (I. P.) 



.56 

2.00 (B. P.) 
(B.P.) 

.90 (B. P.) 

.50 (B.N.)l 
2.00 (B. C.) J 
(B.B.) 

2.00 (B. C.) 

2.40(1. C.) 



(B.P.) 
(B.P.) 
(B. P.) 



Elec. Eng., January 25, 
1901. 

Guy, " EJectric Light and 
Power." 

Cost of fuel alone, Robin- 
son, "Gas and Petro- 
leum Engine." 

Eberle, Eng. Mag., Vol. 
XIV., p. 687. 

Elec. World, 1897, p. 822. 

Elec. World, Vol. XXXVI., 
p. 457. 

Cassier's, 9. 

Kerr, Cass. Mag., Vol. 
XVIIL, p. 425. 

Fuel alone, Cass. Mag., 
Vol. XVIIL, p. 425. 

Bolton, A.S.M.E., Vol. 
XX., p. 873. 

Krone, Dg., Elec. World, 
1900, p. 443. 

Eng. Mag., Vol. XV., p. 
295. 

Power Quarterly, October, 
1900, 



Cost of Power. 



Cost of Gas Power.— Continued. 



Kind. Place. 



Cubic 
feet of 



Cost, cents. 



Authority. 



Oil engine 

Gasoline engine 
Diesel 



Ordinary 

Blast furnace . 



1.74 (B.) 

1.50 (B.) 1 

2.00 (B.) J 
.66 (B.B.) I 
.83 (B. P.) J 

3.10 (B. C.) 



Guy, " Electric Light and Power." 
Kerr, Cass. Mag., Vol. XVIII., p. 
425. 

Meyer, Set. Am., February 9, 1901. 

West. Elec, February 23, 1901. 
Dg. Elec. World, January 19, 1901. 



The letters in the parentheses are read as follows : The first one refers 
to brake or indicated horse-power, and the second to the kind of gas used, 
either natural, producer, coal, or blast-furnace. 

The costs of gas-engine plants are not very different from those of 
steam plants. Mr. N. W. Perry, in A.J. E. E., 1894, says that the cost of 
producers or generators is about $11.00 per horse-power, which is less than 
that of steam boilers. He also gives an estimate by Dawson on a plant to 
have an output of 400 kilowatts, occupying a floor-space 27 X 54 feet on 
one level, and costing, complete, about $10.38 per horse-power. 

Electric Power. 

While electricity can be considered only as a secondary source of power, 
requiring itself to be generated from some prime mover, some data as to 
costs will be found acceptable. 

The following table gives some figures on the cost of generating electric 
power. The costs are based upon the kilowatt-hour, which is the practical 
unit used to designate power, it being equal to 1.34 horse-power hours. 
The cost is based, in most cases, upon the power delivered to the feeders 
that carry the current to the point of application. The cost for lighting 
and for power is usually different, as the power used for motors does not 
need such careful regulation of the pressure, and again the amount is 
usually large as compared with the amount sold for lighting, so that the 
cost to produce is less per unit, and the amount is not so variable ; hence, 
the machines can be run at better efficiency. 

The cost also depends upon the load factor of the plant, — that is, upon 
the ratio of the average to the maximum output of the station, and the 
cost increases as this factor decreases. 

COST OF ELECTRIC POWER. 



Place and use 



Cost per 
K. W. 
hour, 
cents. 



Authority. 



Cheltenham, England, lighting.. 

Dundee, England, lighting 

London, price sold, lighting 

London, price sold, motors 

Estimated operating expense 

Lighting, 0.5 load factor 

Manufacturing works, large 

Manufacturing works, fairly con- 
stant 

Manufacturing works, small, con- 
stant 

Ordinary works, varying load . . . 

Small works, varying load 




London, E. Rev., January 4, 

1901. 
London, E. Eng., January 4, 

1901. 



Humphrey, Lond. Eng., Janu- 
ary 4, 1901. These seven are 
for the operating expenses 
only. 



Cost of Power. 



779 



Cost of Electric Power.— Continued. 



» Place and use. 


Cost per 
K. W. 

hour, 
cents. 


Authority. 


Dudley, England, selling price. , . . 

Average for United Kingdom 

American, practice, range 3.00 to. . 

Met. Electric Railway, Chicago, 
operating expenses 


( 4.00 

\io.oo 

5.34 
7.50 

.88 

2.56 
3.56 \ 

.90/ 
1.00) 

.50 y 
2.00 j 

} i.oo{ 

.40 
8.00 

.62 
1.10 \ 

4.00 j 

n 

•41 j 

1.57 
.50 


Elec. World, February 2, 1901, 

for tramways. 
Other uses. 
Garcke, Elec. World, January 

26, 1901. 
Bolton, A. S.M.E., Vol. XX., 

p. 873. 

E. B., February 15, 1901. 
B. I. E. E. 


Glasgow, Scotland, operating ex- 
penses 


Lighting, to get to customer 

Railway, operating expenses 

Railway, large, at bus bars 

Railway, large, operating expenses 
Niagara power in Buffalo 


Field, Cass. Mag., March, 1896. 

Kennelly, Cass. Mag., Vol. 
XIII., p. 531. 


Railway, estimated, 33 per cent, 
load factor 


Con ant, St. By. Jour., Vol. 

XIV., p. 621. 
Conant, St. By. Jour., Vol. 

XIV., p. 71. 
Editorial, St. By. Jour., Vol. 

XIV., p. 92. 


Kansas City, operating expenses. . 
Average at board 


Brooklyn Heights, operating ex- 
penses 


Denver Railway 




Denver, motor work 


Elec. World, February 26, 1901. 


Kansas City Railway, operating 
expenses, 1899 




Kansas City Railway, operating 


W. E., October 20, 1900. 


Met. Street Railwav, New York, 
1898 


St. By. Jour., November, 1898. 
Bell, " El. Trans, of Power." 


Estimate at bus bars 







From the above table it may be seen that the cost of generating power 
is extremely variable in the different cases, depending upon the purpose 
for which it is used, the load factor, the cost of fuel, and the size of plant. 
For the case of large plants run by compound condensing engines, with 
generators directly connected, operating under fairly good load factors, it 
I may be assumed that the cost of power per kilowatt at the bus-bars is not 
far from 1 cent, and it may be less with careful attendance. For water- 
power plants this figure may be lowered. The cost of distribution is so 
jvariable that no attempt has been made to estimate it, and it can only be 
! approximated for specific cases. 

The cost of electrical machinery depends upon the price of steel and 
copper to a large extent, and so is variable ; but it may be assumed to 
range from $15.00 to $25.00 per kilowatt output for generators, motors, or 
rotaries of the medium or large sizes. The price per unit increases as the 
size decreases, as they are less efficient and require more material and 
,more labor in manufacture. 



780 Works Management. 



WORKS MANAGEMENT. 

The operative management of engineering establishments is necessarily 
governed largely by the character of the product, but there are certah^ 
basic principles which may be stated in a general form. 

The object of good management is the production of good work at a 
minimum cost. Good work involves good tools and skilful mechanics; 
but good tools are costly, and skilful mechanics demand high wages. At 
the same time, it is fully established that poor and antiquated machinery 
and cheap labor are both unprofitable things. The problems of good man- 
agement, therefore, may be divided into the successful use of tools and the 
successful handling of men. 

There are two ways to reduce the cost of a manufactured product,— one 
being to cut down wages and capital charge, and the other to increase the 
output. In other words, the value of a fraction may be diminished either 
by diminishing the numerator or by increasing the denominator. The 
latter is recognized as being the best and most satisfactory method. 

In order to increase the output of the workmen two things are neces- 
sary, — one, to systematize the operations ; the other, to provide an incen- 
tive to the men. 

Systematization of shop methods involves the principle of employing 
a limited number of highly-skilled and highly-paid mechanics to keep the 
tools in order, to maintain them at a high efficiency, and to devise im- 
proved methods, while the tools themselves are attended by a much lower 
grade of labor, costing less, and at the same time competent to perform the 
limited duties assigned to them. This also includes the use of the most 
efficient handling appliances and the proper arrangement of machines, so 
that machines are kept supplied with rough parts, while finished parts are 
promptly transferred to the next machines in orderly sequence. 

The incentive to the men involves the use of some system of wage 
adjustment by which the man's earnings depend to a greater or less extent 
upon his output. The daily or hourly wage system, in which the pay 
depends wholly upon the time, has been found satisfactory in the mod- 
erate-sized shop, in which comparatively few men are employed, and 
where the owner or superintendent can keep his eyes upon everything 
and establish personal relations with all the men. The great difficulty in 
the extension of the system to large establishments lies in the fact that* the 
day's output is determined by the men themselves, and naturally tends to 
a minimum. 

In order to provide an incentive to the workman, piece=work has long 
been used, but, except in rare instances, it has not proved satisfactory. 
The reason for this is readily seen. After a piece price has been put upon 
a certain article there is undoubtedly a direct incentive to the workman to 
do as many pieces as he can, since the more he does the more he makes. 
As soon as he has thus established a new rate for himself the employer 
compares the wages the man is receiving with his former pay, and comes 
to the conclusion that the man is making too much. The piece rate is 
accordingly cut, and soon the establishment reaches a sort of equilibrium 
in which the man does only about enough to make his piece wages equal 
the current day rate. The incentive is thus only temporary, and the 
method becomes unsatisfactory to employer and employee alike. 

The Premium Plan is intended to obviate this defect in piece-work. 
Instead of fixing a price upon the piece, a time is fixed within which it 
can be completed. If the work is finished just in this time, or takes a 
longer time, the workman is paid the regular hourly wages. If, however, 
he finishes the job in less than the allotted time he is credited with a cer- 
tain portion of the time saved, usually receiving one-half the time. Thus, 
if the time fixed for a piece of work is six hours, and the man does it in 
four hours, there are two hours saved, and he receives one hour's addi- 
tional wages. The essential element in the premium system lies in the 
fact that the time for a piece, when once fixed, is never changed unless 
some alteration is made in t lie piece affecting the work upon it, or unless 
some new tool or method is furnished by the employer to aid in acceler- 
ating the work. The premium system, with various modifications, is in 
successful use in many large establishments. 

The Bonus System differs from the premium system, in that a definite 
sum of money is allotted as an extra payment for completing a job within 



Works Management. 781 

an allotted time limit. The job is analyzed into its elementary operations 
and the actual time in which they can be performed, and a time-card con- 
taining this information is given to the man with the job. If he gets the 
work out in less time he gets a bonus, the amount of which is fixed accord- 
ing to the value of the job, and if not, he gets his time wages, anyhow. 

It will be seen that all these methods involve the determination by some 
one else of the time in which the job should be done, and this is one of the 
essentials of successful shop management. If the time is left to the judg- 
ment or choice of the man, the employer is relinquishing one of the most 
essential elements of his control, and success may be endangered. In all 
methods it is important to be liberal with the men. Any attempt to 
squeeze them to a minimum wage rate is to invite failure. The object of 
every improved system should be not to reduce wages but to increase 
output. 

The importance of this will be seen when it is understood that the fixed 
charges of an establishment must be paid out of the product, regardless of 
the wage cost, and hence it is important that a maximum output be 
attained to bear the establishment charges. 



Cost Keeping. 

The subject of cost keeping is closely allied to that of general works 
1 management, and, while details must differ in various establishments, 
i there are certain fundamental principles underlying all successful systems. 

The following general outline of a cost-keeping system is condensed 
■ from a chapter by Mr. J. Newton Gunn, forming a portion of "The Com- 
i plete Cost Keeper," by H. L. Arnold, published by the Engineering Maga- 
zine, New York and London : 
5 " The first consideration is to see that the plant is properly divided as to 
1 its various departments, — that is to say, that each foreman has as many 
' men as he can profitably handle and no more, that no one foreman has 

* charge of two or more classes of work w 7 hich are in no way related, and 
i that the disposition of the work in the building is not such as to preclude 

* the possibility of the foreman in charge giving to the work the best super- 
vision and direction possible. 

J "Having determined so much, the next step is to obtain a complete 
j statement of all the operations performed throughout the establishment. 
" These are then divided into two main classes, —first, operations which 

* are performed so directly upon the work that it is possible to charge the 
labor and material expended to a given production-order; and second, 

i those operations w r hich are so general in their character as to be necessarily 
(• treated as indirect expense, together with the cost of operations which are 

of the nature of expenses incidental to the operation and maintenance of 
I the plant rather than to any particular class of work. 

i "In the particular plant, which is the example selected for presentation 
a here, the w r ork is of such a character and the performances are of such 
i duration that in the majority of cases the direct operations are chargeable 
J to specific production-orders, and in every case it is possible for the work- 
I men to record their own time and performances. Where one workman 

performs several minor operations on a single production-order these 
. operations are recorded on his time-card, but he is not required to indi- 
i vidualize the actual time spent on each of the minor operations. The rule 
i which governs in determining w r hether he shall indicate the time down to 

the least operation, or record the time on a group of operations, is that the 
| time shall be separately recoided for all operations which could not be 
. performed under one contract if the work were on a piece-work basis. It 

may be noted here that prior to the installation of the system taken as an 
I example no piece-work w r as performed in the factory. Under the system 
' adopted the indirect operations are classified w T holly by the results which 
j are obtained ; thus, sweeping belongs to the group of operations incidental 
a to the care of the plant, while oiling of shafts and care of belting pertains 
'to the production and use of power. For all these indirect operations 
1 fixed or standing order-numbers are provided, and all the indirect work 
I performed is charged to one or another of these standing order-numbers, 
i unless a specific production- or plant-order has previously been given 

therefor. 



782 Works Management. 



"In order to reduce the mechanical effort in the making of the time- 
card, specially devised time-cards are prepared for each department, the 
margin of the time-card containing the principal operations performed in 
that department, so that the record to be made on the card consists of the 
date, the workman's number, the workman's name, quantity and descrip- ( ; 
tion of the work in hand, a check-mark or cross opposite the operation ' 
performed by the workman, a check-mark indicating the time the work 
was commenced, and another showing the time the work was finished. 
The record of the work performed as indicated on these time-cards is 
checked at the end of each day by the foreman or sub-foreman in charge, 
and the cards are then immediately transferred to the office of the pay- 
master. 

" The same methods are used in securing the original records of material 
disbursed in producing an order. Forms termed Material Cards serve to 
record the entire history of all material from the time it enters the factory 
until it becomes finished product, 'and has all of its costs properly recorded 
upon the cost-cards. These material cards are differentiated in color to 
unmistakably indicate to the workman and clerks the different classes of 
records. Thus, a white material-received card is used for recording all 
material when it enters the factory, no matter whether this is raw material 
or parts purchased ready to assemble into a complete machine. A buff 
material-delivered card shows that material has been transferred from one 
of the storerooms to some department, or from one department to another. 
A salmon-colored materials-returned card shows that material is to be 
credited to the order-number or account which appears thereon, as the 
material is returned, either from a department to the storeroom or from 
one department to another department from which it may have been 
received. A blue material-requisition card indicates a requisition made 
by a foreman or a storekeeper for the purchase of material by the pur- 
chasing department. 

11 The schedule of machine details records those parts of any one com- 
pleted unit of factory product which are carried in stock in the storeroom 
ready for use by the assembling department, each schedule being individu- 
ally numbered', and a single schedule representing all the parts necessary 
to complete a machine of a certain size and type. Duplicates of each of 
these schedules are in possession of each of the following officials : super- 
intendent, cost clerk, stores-ledger clerk, storekeeper, and foreman of 
assembling department. These schedules enable the foreman of — for 
instance— the assembling department to use a single materials-delivered 
card calling for all parts on, say, schedule number seventeen, which he 
sends to the storekeeper, thus saving the writing and possible errors which 
would be incidental to the reproduction in detail of the individual items 
covered by the entire schedule every time a machine was to be assembled 
or two or more were to be assembled. These material cards always bear 
either the production-order number or plant-order number to which they 
are related, or, if the material is solely chargeable to expense, must bear 
the standing-order number which indicates the division of the expense 
account for which they are disbursed. It may be said here that one of the 
most important precedents to accurate factory accounting is absolute 
uniformity of nomenclature throughout the factory. Any variation in 
naming leads to doubt, and opens the way to inaccurate returns. 

44 As an example of organization the following, representing a successful 
modern plant, is given : 

"The chief of the entire office and factory staff is the President and 
Treasurer, who is Acting General Manager, arid is responsible only to the 
Directors of the Company. The next in authority is the General Factory 
Manager, who bears the entire responsibility of 'production, and who is 
subordinate to the President alone. The General Factory Manager's staff 
consists of a Cosl clerk, who is also Paymaster, and a Purchasing Clerk, 
who is also Production-order Clerk. The Production-order Clerk originates 
product ion-orders on the factory under requisitions from the Superin- 
tendent. 

"The general factory manager also has a stores-ledger clerk, who is , 
responsible Eor all records of material up to the time these records are 
turned over to the cost clerk; a stenographer; a superintendent, who 
supplements the general factory manager in all his work, but is more 
especially responsible for the direct supervision of the foreman. The 
drawing-room is in charge of the constructing engineer and chief draughts- 



Works Management. 783 

man. who has charge of the subordinate draughtsmen, and who undertakes 
the designing of new machinery and the remodelling of old, under the 
immediate direction of the superintendent. The chief draughtsman is also 
responsible for the revision of the schedules of machine details, which of 
„ course must be made to conform to the drawings in his possession. 

"Each department has a foreman in charge, who supplies the workmen 
with pads of time-cards and pads of material-order cards. All stores are 
kept by a storekeeper, who, with one assistant, has in charge all raw 
material, or rough stores, and also all semi-finished material or parts, as 
well as finished parts and machines. Heavy materials, such as rough 
castings and finished machines, which from their bulk and the expense 
attendant upon handling them cannot economically go into the storeroom, 
are delivered to, and kept in, the part of the factory where they are to be 
used ; the nearest foreman is made responsible for their care and disburse- 
ment. 

"The president has as his executive force a stenographer, a filing clerk, 
a bookkeeper, and an order clerk, who handles the shipping orders and 
does the billing, as well as renders general assistance to the bookkeeper. 
Directly responsible to the president is the general manager of the Sales 
Department, who has a number of salesmen responsible in turn to him. 
A branch office, which is purely a department of the selling organization, 
is responsible, in so far as its selling functions go, to the general manager 
of the sales department, but for its finances it is directly responsible to the 
President-treasurer. 

"A complete shop telephone system gives communication from the 
superintendent's office to all the foremen and to the stores department. 

" The only bound book kept is a ledger, which contains the commercial 
accounts, and from which the general statement of the business is made 
up. This book is simply a general ledger containing personal accounts 
and customers' accounts. There are no other books employed in the 
accounting organization, every other record being made either on cards or 
on loose sheets. These cards and sheets are provided for as to their storage 
either in card-cabinets or in binders, or in filing boxes. The same record 
at different stages of its work may be in any one of the three forms of 
receptacle specified. 

"Origination of Production=order. 

"A production-order originates either from a specific shipping order, 
which demands a specific machine, or else from the depletion of the stock 
of a given machine or machine detail ; stock shortage is immediately 
reported by the stores-ledger clerk, as he has a schedule showing the maxi- 
mum and minimum limits not only of the parts of machines but of all 
complete machines. The schedule limits of stock are assumed, at first, 
and revised as experience may dictate. 

" The moderate size of the concern under consideration precludes any 
necessity for having a formal record of the instructions that the superin- 
tendent gives to the production-order clerk for the creation of a production- 
order; these instructions are in every case verbal, the superintendent 
vitalizing the production-order as soon as made, and assuming the respon- 
sibility by affixing his signature thereto. 

"Other orders, termed ' plant-orders,' are issued by the production-order 
clerk. These cover all labor and material expended for the betterment of 
the plant or for experimental purposes ; in the case of the production-order 
it is expected that a direct profit will be realized from all work ordered to 
be performed, while plant-orders are expected to be indirectly profitable. 
Both production-orders and plant-orders are carbon-sheet copies in dupli- 
cate, the original going to the foreman who is to do the initial work, the 
duplicate remaining at the desk of the production-order clerk, where it is 
filed by family,— that is, by class of machine and number of machine, or 
class of work, — until the return of the original order with the foreman's 
signature gives notice that the order has been filled. 

"Both production- and plant-orders, before being sent to the foreman of 
the proper production department, go to the chief draughtsman, who sup- 
plements each order with the drawings needed for its production, and the 
drawings and order then go together to the department foreman. The 
foreman, on receipt of this order with its drawings, starts the work, mak- 



784 Works Management. 

ing immediate provision for identification of the work throughout his 
department, either bv tags showing the production-order number in case 
of large pieces, by painting the production-order number oh the piece, or, 
in case of a large number of small pieces, by adding to the receptacle 
containing the parts a memorandum-slip bearing the production-order 
number. 

' ' The material-delivered card, signed by any foreman, obtains from the 
st( >rekeeper the stores needful for the completion of an order. At any time 
during the progress of the work, or at its completion, any excess of mate- 
rial may be returned, recorded on a ' material-returned' card. All these 
material-delivered and material-returned cards are identified by having 
the production- or plant-order number upon them. 

"There is no limitation as to the amount of material which a foreman 
may draw for the completion of any one order, but as often as four times 
each day it is the duty of the assistant storekeeper to return all material 
cards to the stores-ledger clerk, on whom the responsibility is placed of 
seeing that the proper limits of material required for the completion of any 
order are not exceeded. 

"The Sub=production=order. 

"The sub-production-order is produced in duplicate by manifolding by 
any foreman who may require the product of any other department. The 
original goes to the department making the required product, and the 
duplicate to the office of the superintendent. Both original and duplicate 
bear the production-order number under which the foreman using the 
sub-production-order number is operating. On the completion of the re- 
quired details by the department receiving the sub-production-order, the 
order is returned with the work to the originating foreman, who checks 
the sub-production-order, certifies its completion, and forwards it at once 
to the office of the superintendent. 

"A plant-order is issued by the superintendent at the request of any 
foreman, or as his own judgment may dictate. Plant-orders cover all work 
in the nature of repairs or betterments to the plant or building, all special 
tools and special experiments, and, in fact, any work to be performed 
which will eventually become an addition to the plant or an expense- 
charge to the business. These orders are made in duplicate by manifold- 
ing, the original going to the foreman of the department required to 
produce the work, while the duplicate remains in the office of the super- 
intendent, plant-orders being handled precisely as production -orders are 
handled. 

•• There is one case in which it is necessary to make an exception to the 
rule requiring all production-orders to be issued by the superintendent. 
This occurs in the general machine shop, where it sometimes happens that 
certain machines are left idle. The foreman of that department is sup- 
plied with a pad of production-orders, and has authority to issue to himself 
an order for the undertaking of such work as will keep his machines busy, 
Bending the duplicate order immediately to the office of the superintend- 
ent, on whom rests the responsibility of approving the order or of commu- 
nicating with the foreman and stopping the work, if for any reason that 
particular work is not justifiable in the judgment of the superintendent. 
In no case, however, is work stopped until the particular operation in 
progress is completed. 

"The production- and sub-production-orders, with the plant-orders, 
account tor all direct labor and material and a portion of the expense- 
labor, hut it is necessary that the general factory manager, superintendent, 
draughting department, the foremen, the general laborers, and men em- 
ployed in the production of power, heat, and light shall have some definite 
means by which to indicate the services which they have rendered to the 
company. To this end a communication is addressed to all the heads of 
departments, and through them brought to the notice of the employees, 
which communication is a standing order under which all employees of 
the company are to work. The standing order directs that whenever work- 
men perform any ol the operations or render any of the services enumer- 
ated on the list accompanying the order, the labor-time, together with any 
material that may be drawn from the stores department necessary to com- 
plete the service, shall be charged to some one of the various numbers set 



Works Management. 785 

opposite the items enumerated. These numbers are called standing-order 
numbers, and are specific orders, to be treated in the accounting as are the 
production- or plant-orders. 

"For convenience, and as an aid to all persons concerned in becoming 
familiar with these numbers, the numbers from 1 to 499 relate to the plant 
J and expenses incidental to production. The numbers from 500 to 999 specify 
a general classification of the product, and relate to such operations upon 
the product as testing, which operation, though it may be performed — as in 
the case of a dynamo— upon a specific machine, gives information so gen- 
eral in its character as to be of service to the entire class to which this 
machine belongs, and therefore is an item of cost which may more justly 
be spread over the entire related product than charged to the individual 
machine under test. 

"The plant-orders commence with 1000 and continue in numerical 
sequence, each number being preceded by the letter 'P.' Production- 
orders commence at 1000 and run in sequence, but have no distinguishing 
letter. 

"The numbers from 1 to 499 are subdivided into five general groups, 
denned as follows : 

"From 1 to 99, additions to or betterments of the 'permanent invest- 
ment,'— buildings and plant ; 100 to 199 report depreciations of the perma- 
nent investment and cost of up-keep of plant ; 200 to 299 carry the time-cost 
of foremen, superintendents, draughtsmen, and all other indirect labor, 
excepting 'engineers' (American) or 'engine drivers' (English) ; 300 to 399 
cover all labor and material used in the generation and distribution of 
power, heat, and light; 400 to 499 include all alterations, errors, and 
changes, also fixed charges, such as taxes and insurances." 

General Expense. 

In addition to the determination of the material and direct wage charges 
upon a job there is the item of indirect charges, usually called general 
expense. This includes interest and depreciation on plant, cost of motive 
power, lighting, insurance, taxes, etc., as well as the wages and salary 
expense of men whose work is not chargeable to any direct job, but must 
be borne by the entire output. This must be included in the actual shop 
cost ; and while the total of such charges can usually be determined with 
a fair degree of accuracy, there are various methods of distributing it over 
the different jobs. 

The commonest method of charging general expense is to make it a 
-percentage of the ivage charge. 

Thus, if the grand total of all the expenditures which cannot be prop- 
erly charged directly to shop orders be summed up for the year, it will be 
found to bear a certain relation to the total of the direct wages for the 
same period. In many cases the indirect charges will be found to be as 
much as the direct wages ; in other instances they will reach 80 per cent, 
of the wages, etc. This relation having been determined, it is only neces- 
sary to add to the cost of materials and labor an amount equal to the 
expense percentage of the labor to have the entire shop cost. Thus, if the 
material on a job cost $12.00, and the direct labor charges amounted to 
$80.00, and the expense percentage had been determined to be 100 per 
cent., the amount added for general expense would be 80 X .70 = $56.00, 
and the total shop cost would be 12 + 80 + 56 = $148.00. 

This method has the advantage of simplicity, but it is not always satis- 
factory, since it assumes that the expense charge bears a direct relation to 
the labor charge. But it is evident that the expense goes on just as heavily 
for the time of a cheap workman as it does for a highly-paid one, and so 
this method saddles the jobs upon which skilled mechanics are working 
I with more than their fair share of expense, and does not put enough upon 
< those done by cheaper men. 

Another and better method of distributing general expense is the so- 
j called "hourly burden." 

In this system the expense is made to depend upon time, instead of 
wages. Thus, if the total expense for a year be divided by the total num- 
ber of hours during which all the workmen were occupied during that 
year, we shall obtain an expense charge per hour for every man's time. 

A shop containing 100 men, each man working 3000 hours per year, will 

50 



786 



Works Management. 



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os os os os os os o. _-. r. -. / / / / s x /: x x x x x i- i- t- 1— i- i- t- i— 



ri 10 o t- x os o 



Works Management. 787 



have 100 X 3000 = 300,000 hours among which to divide the general expense. 
If the total expense charges for the year amount to $75,000, the burden per 

... , 7,500,000 
h0UrWlllbe -300^00- = 25centS - 
»-.. To find the expense charge against any job by this method the number 
of hours' time put on it will be multiplied by 0.25, regardless of the rate of 
wages paid to the men. This plan is very satisfactory in many cases, but 
it lias been criticised in one respect. The expense properly chargeable to 
some important and expensive tools, it is maintained, should be greater 
than that charged against work done with tools which cost less to buy and 
to operate; hence, in many shops there is what is termed a machine rate for 
various machines, this rate being computed — more or less arbitrarily— from 
the importance and cost of the machine. It is impossible to provide for all 
the general expense by machine rates, because the various tools are not in 
continual operation, but it has been proposed by Mr. Hamilton Church to 
combine the two methods, using machine rates for the important tools, 
and having a supplementary hourly burden to take care of the balance of 
the expense charges. 

The method to be selected depends largely upon the character of the 
work. When there is much uniformity in the methods and products the 
hourly burden is satisfactory, and when there is a diversified product the 
establishment may often be divided into departments, each with a fairly 
uniform product, and each with its own hourly burden. The choice of 
: method must therefore be made by the works manager according to the 
I conditions of operation. 

The question of depreciation is one which demands attention, and it is 
now considered upon an entirely different basis from that which formerly 
obtained. At one time the amount charged off from the inventory value 
of a machine tool was based upon its durability, and the original cost was 
divided by the length of time such a tool might reasonably be supposed 
to last. 

It is now well understood, however, that a machine may become super- 
annuated in a few years, and while it is still in perfect condition, simply 
i by the appearance of some improved machine or method of doing the 
' same work, the improved method rendering a cheaper product possible 
and causing the old machine to be unsuited for competition. 

Since the time which may elapse before a machine becomes superannu- 
ated is a very uncertain quantity, it is most necessary that the maximum 
output of all new machines be gotten from them as soon as practicable 
after they have been put into operation ; and it is good management to 
, place as high a depreciation rate upon a machine as it can reasonably 
! stand, and drive the tool as hard as possible, so that it may pay for itself in 
a few years. If no improved device or method appears the machine will 
| still be available, while if a new and improved tool comes out the old 
machine may, and indeed must, be promptly scrapped to make way for 
the new one/ 

If this method were more closely followed there would be fewer super- 
j annuated establishments; and a shop which uses its tools up instead of 
I nursing and coddling them is dealing more fairly by its stockholders than 
J is the old-time plant. 

The deduction for depreciation may be made in the form of a percentage 
of the cost of the machine or tool, and in such case the table opposite 
■ will be found convenient. This method is defective, however, in that the 
i depreciation never equals the full original cost, and hence there is always 
'. some value left to the tool. Another plan is to charge off one- tenth, one- 
I fifth, or one-third of the cost of the machine, as the case may be, and so 
| have the entire inventory value wiped out at the expiration of ten, five, or 

• three years, after which it may at any time be scrapped without compunc- 

• tion. 

Valuable works upon the subject of works management and cost keep- 
ing are the following: J. Slater Lewis's "Commercial Organization of 
I Factories," Arnold's "Complete Cost Keeper," Arnold's "Factory Manager 
and Accountant," Garcke & Fells's "Factory Accounts," Matheson's 
"Depreciation of Factories," and Metcalfe's " Cost of Manufactures." 



APPENDIX 



i 



ALUMINUM. 

In various modern structures aluminum or some of its alloys are used 
when lightness is of importance, and the following information, furnished 
by the Pittsburg Reduction Company, will be found useful : 

The low specific gravity of aluminum is one of its most striking proper- 
ties, being 2.56 in ordinary castings of pure aluminum, and 2.68 in the 
compressed and worked. 

Specific Gravity at 62° F. of Aluminum and Aluminum 

Alloys. 

Aluminum commercially pure, cast 2.56 

Nickel-aluminum alloy ingots, for rolling 2.72 

Nickel-aluminum casting alloy 2.85 

Special casting alloy, cast * 3.00 

Aluminum commercially pure, as rolled, sheets, and wire 2.68 

•Aluminum commercially pure, annealed 2.66 

Nickel-aluminum alloy, as rolled, sheets, and wire 2.76 

Nickel-aluminum alloy, sheets annealed, ... = . . . 2.74 

Weight. 

Using these specific gravities, assuming water at 62° F. and at standard 
barometric height as 62.355 pounds per cubic foot. 

Sheet of cast-aluminum, 12 inches square and 1 

inch thick, weighs 13.3024 pounds. 

Sheet of rolled aluminum, 12 inches square and 1 

inch thick, weighs 13.9259 pounds. 

Bar of cast-aluminum, 1 inch square and 12 inches 

long, weighs 1.1085 pounds. 

Bar of rolled aluminum, 1 inch square and 12 

inches long, weighs 1.1605 pounds. 

Bar of aluminum, cast, 1 inch round and 12 inches 

long, weighs .8706 pound. 

Bar of rolled aluminum, 1 inch round and 12 inches 

long, weighs .9114 pound. 

The weight per cubic inch of pure cast-aluminum is .0920 pound. 

The weight per cubic inch of pure rolled alumi- 
num, annealed, is .0970 pound. 

The weight per cubic foot of pure cast-aluminum is 159.6288 pounds. 

The weight per cubic foot of pure rolled aluminum is 167.1114 pounds. 

Strength of Aluminum. 

The tensile, crushing, and transverse tests of aluminum vary consider- 
ably with different conditions of hardness, due to cold working ; also by 
the amount of work that has been put upon the metal, the character of 
the section, amount of hardening ingredients, etc. Cast-aluminum has 
about an equal strength to cast-iron in tension, but under compression it 
is comparatively weak. The following is a table giving the average results 
of many tests of aluminum of 99 per cent, purity : 

{castings 8,500 pounds, 
sheet 12,500 to 25,000 pounds, 
wire 16,000 to 33,000 pounds, 
bars 14,000 to 23,000 pounds. 

{castings 18,000 pounds, 
sheet 24,000 to 40,000 pounds, 
wire 30,000 to 55,000 pounds, 
bars 28,000 to 40,000 pounds. 

{castings 15 per cent, 
sheet 20 to 30 per cent, 
wire 40 to 60 per cent, 
bars 30 to 40 per cent. 

791 



792 Aluminum. 



Elastic limit per square inch under compression in 
cast cylindrical short columns, with length twice 

the diameter 3,500 pounds. 

Ultimate strength per square inch under compression 
in cast cylindrical short columns, with length twice 

the diameter 12,000 pounds. 

The modulus of elasticity of cast-aluminum is about 11,500,000. 
Aluminum in castings can readily be strained to the unit stress of 1500 
pounds per square inch in compression, and to 5000 pounds per square inch 
in tension. It is rather an open metal in its texture ; and for cylinders, to 
stand pressure, an increase in thickness over the ordinary formulse should 
be given to allow for its porosity. 

NickeUaluminum. 

In order to obtain a greater strength than is possessed by pure alumi- 
num, and at the same time to retain as much as possible the advantage of 
the low specific gravity, an alloy containing from 2 to 5 per cent, of nickel 
and copper is made, this having a specific gravity of about 2.85, as compared 
with 2.56 for pure aluminum. 

The following table gives the average results of many tests of nickel- 
aluminum : 

Elastic limit ner sauare inch in f castin S s • • 8 > 500 to 12 * 000 Pounds. 

tliei™ P square mcn m < sheet 21,000 to 30,000 pounds. 

tension (bars 18,500 to 25,000 pounds. 

Ultimate strength r>cr sonar* f castin g s • • 18,000 to 28 000 pounds. 

V^h in tSfSSn q i sneet 35 » 000 to 50 < 000 rounds. 

men in tension ^ barg 30,000 to 45,000 pounds. 

( castings 6 to 8 per cent. 

Per cent, of reduction of area . .< sheet 12 to 20 per cent. 

(bars 12 to 15 per cent. 

Elastic limit per square inch ur.der compres- 
sion in short columns, with length twice 

the diameter 6,000 to 10,000 pounds. 

Ultimate strength per square inch under 
compression in short columns, with length 
twice the diameter 1,600 to 24,000 pounds. 

Aluminum for Structural Purposes. 

In the use of aluminum for structural purposes, a great deal depends 
upon the specific purpose to which it is desired to apply the metal, so as to 
know just what is the proper grade that should be used; but, generally 
speaking, for purposes where aluminum is brought into tension,— such as 
in sheets or in rolled shapes, as angles, beams, etc.,— an ultimate tensile 
strength of from 32,000 to 40,000 pounds per square inch may be reckoned 
upon, and using a safety factor of 4 gives an allowable working strain oi 
from 8000 to 10,000 pounds. This, of course, is not for pure metal, but for 
the stronger alloys. 

The ultimate "tensile strength of pure metal in plates and shapes may 
be taken at from 24,000 to 28,000 pounds. With the same safety factor of 4 
it gives an allowable working strain of from 6000 to 7000 pounds. 

For the alloys of east-aluminum in tension the ultimate strength may 
betaken at from ls.000 to 28,000 pounds per square inch. Using a safety 
factor here of 5, as aluminum castings are quite uniform and solid, a work- 
ing strain Is obtained of from 3600 to 5600 pounds per square inch. 

It is difficult to give a value for the ultimate strength of pure cast- 
aluminum in tension, for the reason that while the ordinary pure alumi- 
num will run about 16,000 pounds per square inch, this can be increased 
very considerably by cold working, and in some cases to as much as 24,000 
pounds per square inch. Using a safety factor of 4 gives an allowable 
working strain of from 3200 to 4800 pounds. 

In compression, the alloys of aluminum in rolled plates and structural 
shapes— such as struts, columns, etc.— have an ultimate tensile strength of 
from 26,000 to 34,000 pounds per square inch, which, using a safety factdr 
of 1, gives an allowable working strain of from 6500 to 8500 pounds pp" 
square inch. 



Aluminum. 



793 



Pure aluminum sheets and structural shapes in compression have an 
ultimate tensile strength of from 20,000 to 24,000 pounds per square inch, 
which, with a safety factor of 4, gives an allowable working strain of from 
5000 to 6000 pounds per square inch. 

Castings of aluminum in compression can be taken at 16,000 pounds per 
square inch for pure aluminum, and from this to 24,000 pounds per square 
inch for the alloys. Using again a safety factor of 5, an allowable working 
strain is given of from 3200 to 4600 pounds per square inch. But the pure 
metal should not be used in castings, except for electrical purposes, as it is 
similar to pure copper in being difficult to cast, and is soft, comparatively 
weak, and has a large shrinkage. In its stead, alloys with from 5 to 20 per 
cent, of copper, nickel, or other hardeners should be used. 

The alloys of aluminum in rivets and similar shapes in shear have an 
ultimate shearing strength of from 24,000 to 27,000 pounds, which, using 
here a safety factor of 6, gives an allowable working strain of from 4000 to 
4500 pounds per square inch. 

The ratios of the ultimate shearing strength to the ultimate tensile 
strength for double-riveted joints is about 60 per cent., and for single- 
riveted joints the ratio is about 70 per cent. The ratio for steel is about 75 
per cent. 

In bearing, the ultimate value of the alloys of aluminum is from 32,000 
to 40,000 pounds per square inch, which, using a safety factor of 4, gives an 
allowable working strain of from 8000 to 10,000 pounds. 



Electrical Properties of Aluminum. 

The electrical conductivity of silver being taken as 100, that of pure 
aluminum is about 63. 

Aluminum is practically non-magnetic, and may therefore be used for 
many purposes in electrical work, where a magnetic metal would be use- 
less. At the same time, its electrical conductivity is excellent, as the fol- 
lowing electrical conductivities of various metals will show. Aluminum 
may therefore in the future be largely used in the windings of field 
magnets on dynamos, where weight is an object, and, in general, for 
switches, brushes, brush-holders, and apparatus where its non-tarnishing 
and non-corrosive qualities render it specially valuable. 

As is the case with other metals of good electrical conductivity, the 
conducting power of aluminum is greatly decreased as the result of the 
presence of alloying metals. Pure aluminum has a much higher relative 
conductivity to pure copper than is ordinarily given in the books, occa- 
sioned by the considerable impurities in the aluminum that has been in 
the past tested for its relative electrical conductivity. 

The following tests, made by Mr. Charles F. Scott and Professor J. W. 
Richards, will serve to show the relative conductivity of various samples: 



Sample. 


Ohms per 
1000 feet, 
.01 inch di- 
ameter, at 
15° C. 


Percentage 
of conduc- 
tivity at 
15° C. 


Percentage of 

variation 
between 15° 
C. and 80° C. 


Pure copper wire 


101.83 
161.40 
163.80 
181.30 
174.10 

185.10 


100.00 
63.09 
62.17 
56.17 

58.48 

55.01 
64.50 


.388 


No. 1, 99% per cent, pure aluminum . 
No. 2, 99 per cent, pure aluminum . . 
No. 3, 98 per cent, pure aluminum . . 
No. 4, XB, nickel-aluminum alloy . . 
No. 5, XCWC, copper-zinc-alumi- 
num allov 


.385 
.385 
.360 
.361 

.359 


Result of Professor J. W. Richards on 


.300 






l 



; Taking into account the relative specific gravity as well as the relative 
conductivity, it has been computed that when the price of aluminum per 
pound is double that of copper their values lor electrical conductors are 
equal. 



794 Locomotives. 



LOCOMOTIVE DATA. 

The following formulas are those of the Baldwin Locomotive Works, 
Philadelphia, and have shown themselves reliable in the practice of that 
well-known establishment. 



Speed Resistance, Locomotive and Train. 

R = resistance, in pounds, per ton of 2000 pounds ; 
y= speed, in miles, per hour. 

This formula represents the resistance for sustained speed, and the ele- 
ment of acceleration is not taken into consideration. It is deduced from 
the results obtained by comparison of a large number of indicator cards 
taken at various speeds. 

Grade Resistance. 

The resistance for a straight grade of 1 foot per mile is 
0.3788 pound per ton. 
If 

G = grade, in feet, per mile ; 

T= weight of train, in tons (2000 pounds) ; 

R = resistance, in pounds. 

# = 0.3788 GT. 



Curve Resistance. 

Taking the curve as expressed in degrees of deflection from a tangent 
measured from stations 100 feet apart, the resistance of curves may be 
expressed as in proportion to the number of degrees in the curve. The 
resistance naturally varies with the construction of the road-bed, speed 
of train, and other conditions of service, so that no general rule can be 
expected to apply to all cases. Approximately, with moderate speed and 
under ordinary conditions, the resistance mav be computed on the basis 
that each degree of curvature is equal to a straight grade of 1% feet per 
mile. 

The following formula corresponds to this allowance : 

Let A - angle of curve, in degrees ; 

T weight of train, in tons ; 
R = resistance, in pounds. 

R = 0.5682 AT. 

Acceleration Resistance. 

Tin- resistance opposed to the acceleration of a train from any speed 
to any higher speed may be computed by the following formula : 

Let fi the resistance, in pounds; 

T weight of train, in tons (2000 pounds) ; 

V Initial speed, In miles, per hour ; 

V accelerated speed 

R = 0.0132 ( V* — V-) T. 



Locomotives. 795 



Thus, for a weight of 1 ton and an acceleration from 30 miles per hour to 
50 miles, we have 

0.0132 (502 _ 3 2) = 

0.0132 (2500 — 900) = 

0.0132 X 1600 = 21 pounds, 

and this, multiplied by the weight of the train, in tons, will give the total 
resistance due to the acceleration. 



Tractive Power. 

Let 

d = diameter of cylinder ; 
I = length of stroke ; 
D = diameter of driving-wheels, in inches ; 

T == tractive power, in pounds, per pound of mean effective press- 
ure in cylinder. 

D ' 

The mean effective pressure may be taken as equal to 85 per cent, of the 
boiler pressure. 

The tractive power of a locomotive multiplied by the speed, in miles, 
per hour, divided by 375, gives the horse-power. 



THE POWERING OF STEAMSHIPS. 

The most reliable method of determining the power required to propel 
a vessel at a given speed is to use a model of the hull in a testing tank, 
and this should be done in all important designs. 

The following tables (pages 796-799), originally prepared by Nystrom, 
will be found to agree closely with the results attained by modern steam- 
ships in actual service, and may be used when experimental data are 
lacking. 

The average powering of modern steamships is about 1 horse-power per 
ton of displacement, while for the fast liners it reaches 2 horse-power per 
ton and over. 



796 



Powering of Steamships. 







Steamship Performance. 








Displace- 
ment, 
in tons. 






Knots, or 


nautical miles 


per hour. 




1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


T 


H 


H 


H 


H 


H 


H 


H 


H 


II 


H 


1 


.004 


.035 


.118 


.280 


.55 


.949 


1.50 


2.24 


3.20 


4.38 


2 


.007 


.055 


.190 


.444 


.87 


1.51 


2.40 


3.55 


5.08 


6.96 


3 


.009 


.075 


.248 


.598 


1.14 


1.98 


3.12 


4.79 


6.91 


9.12 


4 


.010 


.084 


.300 


.673 


1.40 


2.40 


3.80 


5.39 


8.06 


11.1 


5 


.012 


.102 


.348 


.818 


1.52 


2.78 


4.40 


6.55 


9.36 


12.2 


6 


.014 


.115 


.390 


.924 


1.81 


3.12 


4.96 


7.39 


10.6 


14.5 


7 


.016 


.128 


.435 


1.025 


2.01 


3.48 


5.50 


8.20 


11.7 


16.1 


8 


.017 


.138 


.479 


1.125 


2.20 


3.80 


6.01 


8.96 


12.8 


17.5 


9 


.019 


.151 


.501 


1.211 


2.38 


4.12 


6.51 


9.69 


13.8 


19.0 


10 


.020 


.161 


.552 


1.30 


2.54 


4.42 


6.98 


10.4 


14.9 


20.3 


11 


.022 


.175 


.590 


1.40 


2.72 


4.70 


7.46 


11.1 


15.9 


21.8 


12 


.023 


.185 


.624 


1.48 


2.88 


4.99 


7.90 


11.8 


16.8 


23.0 


13 


.024 


.195 


.654 


1.56 


3.04 


5.25 


8.33 


12.5 


17.7 


24.3 


14 


.024 


.198 


.690 


1.62 


3.18 


5.52 


8.75 


13.0 


18.6 


25.4 


15 


.026 


.213 


.725 


1.70 


3.32 


5.80 


9.20 


13.6 


19.5 


26.6 


16 


.028 


.223 


.780 


1.78 


3.49 


6.04 


9.55 


14.2 


20.4 


27.9 


17 


.029 


.236 


.785 


1.89 


3.64 


6.28 


9.95 


15.0 


21.2 


29.1 


18 


.030 


.242 


.815 


1.94 


3.78 


6.52 


10.3 


15.5 


22.0 


30.2 


19 


.031 


.250 


.850 


2.00 


3.90 


6.80 


10.7 


16.0 


22.8 


31.2 


20 


.032 


.258 


.875 


2.06 


4.02 


7.00 


11.1 


16.5 


23.0 


32.2 


25 


.038 


.300 


1.015 


2.40 


4.14 


8.12 


12.9 


19.2 


24.2 


33.1 


30 


.042 


.338 


1.14 


2.70 


5.30 


9.18 


14.6 


21.6 


31.0 


42.4 


35 


.047 


.375 


1.26 


3.00 


5.89 


10.1 


16.2 


24.0 


34.2 


47.1 


40 


.050 


.409 


1.39 


3.27 


6.41 


11.1 


17.6 


26.2 


37.5 


51.3 


45 


.056 


.445 


1.50 


3.56 


6.95 


12.0 


19.0 


28.5 


40.5 


55.6 


50 


.056 


.474 


1.61 


3.79 


7.44 


12.9 


20.5 


30.3 


43.2 


59.5 


55 


.062 


.501 


1.72 


4.06 


7.95 


13.8 


21.8 


32.5 


46.2 


63.6 


60 


.067 


.538 


1.80 


4.30 


8.41 


14.4 


23.1 


34.4 


49.1 


67.3 


65 


.071 


.570 


1.90 


4.56 


8.88 


15.1 


24.4 


36.5 


51.8 


71.0 


70 


.074 


.597 


2.02 


4.77 


9.36 


16.2 


25.5 


38.2 


54.4 


74.9 


75 


.078 


.625 


2.12 


5.00 


9.77 


16.9 


26.8 


40.0 


56.8 


78.0 


80 


.081 


.650 


2.20 


5.20 


10.2 


17.6 


28.0 


41.6 


58.1 


81.6 


85 


.085 


.680 


2.30 


5.44 


10.6 


18.4 


29.2 


43.5 


62.0 


85.0 


90 


.088 


.705 


2.38 


5.64 


11.0 


19.1 


30.5 


45.2 


64.5 


88.4 


95 


.088 


.710 


2.49 


5.68 


11.4 


19.9 


31.3 


47.0 


66.6 


91.5 


100 


.094 


.755 


2.56 


6.04 


11.8 


20.5 


32.4 


48.4 


68.5 


94.5 


110 


.101 


.810 


2.73 


6.48 


12.6 


21.9 


34.6 


51.8 


73.2 


101.0 


125 


.109 


.877 


2.98 


7.02 


13.7 


23.8 


37.5 


56.2 


80.0 


110.0 


150 


.124 


.99 


3.38 


7.72 


15.5 


27.0 


42.8 


61.7 


90.5 


124.0 


175 


.138 


1.10 


3.72 


8.81 


17.2 


29.8 


47.2 


70.5 


100.0 


138.0 


200 


.150 


1.20 


4.06 


9.6 


18.8 


32.5 


51.5 


76.9 


110.0 


150.0 


225 


.162 


1.30 


4.39 


10.4 


20.2 


35.1 


56.0 


83.3 


118.0 


162.0 


250 


.175 


1.40 


4.70 


11.2 


21.9 


37.6 


59.8 


89.2 


127.0 


175.0 


275 


.188 


1.50 


5.04 


11.9 


23.2 


40.3 


63.8 


95.2 


136.0 


186.0 


300 


.196 


1.57 


5.31 


12.6 


24.5 


42.5 


67.5 


100.0 


142.0 


196.0 


325 


.201 


1.66 


5.63 


13.3 


26.0 


45.0 


71.2 


106.0 


152.0 


208.0 


350 


.220 


1.75 


5.91 


14.0 


27.4 


47.3 


75.0 


112.0 


159.0 


219.0 


375 


.228 


1.82 


6.12 


14.6 


28.6 


49.0 


78.4 


117.0 


166.0 


229.0 


400 


.240 


1.91 


6.42 


15.3 


29.8 


51.4 


81.7 


122.0 


172.0 


238.0 


450 


.250 


2.06 


6.98 


16.5 


32.2 


55.8 


88.5 


132.0 


188.0 


258.0 


500 


.276 


2.21 


7.45 


17.7 


34.6 


59.6 


94.3 


141.0 


200.0 


276.0 


550 


.295 


2.35 


7.98 


18.9 


36.9 


63.8 


101.0 


151.0 


215.0 


295.0 


600 


.312 


2.50 


8.40 


20.0 


39.0 


67.2 


107.0 


160.0 


226.0 


313.0 


650 


.330 


2.64 


8.90 


21.1 


41.2 


71.2 


113.0 


169.0 


240.0 


329.0 


700 


.348 


2.78 


9.32 


22.2 


43.3 


74.6 


119.0 


177.0 


250.0 


337.0 


750 


.362 


2.90 


9.80 


23.2 


45.2 


78.4 


124.0 


186.0 


264.0 


352.0 


800 


.380 


:\.o:, 


10.2 


24.2 


47.3 


81.5 


130.0 


194.0 


274.0 


378.0 


850 


.394 


3.15 


10.6 


25.2 


49.2 


85.0 


135.0 


202.0 5 


288.0 


394.0 


900 


.410 


3.28 


11.0 


26.2 


51.1 


88.1 


140.0 


210.0 


296.0 


409.0 


950 


.422 


3.41 


11.4 


27.3 


53.1 


91.8 


146.0 


218.0 


310.0 445.0 



Powering of Steamships. 



797 



Steamship Performance. 







Knots, or 


nautical miles 


per hour. 


Displace- 


















ment, 
in tons. 


11 


12 


13 


14 


15 


16 


17 


18 


19 


20 


H 


H 


H 


If 


H 


H 


H 


H 


H 


H 


T 


5.85 


7.59 


9.63 


12.0 


14.8 


17.9 


21.6 


25.6 


30.1 


35.1 


1 


9.28 


12.0 


15.3 


19.1 


23.5 


28.4 


34.2 


40.6 


47.8 


54.7 


2 


12.2 


15.8 


20.0 


25.0 


30.8 


38.3 


44.8 


53.3 


62.6 


73.0 


3 


14.8 


19.2 


24.4 


30.3 


37.4 


43.1 


54.3 


64.5 


75.9 


88.4 


4 


17.2 


22.2 


28.3 


35.2 


43.4 


52.4 


63.0 


74.9 


88.0 


97.8 


5 


19.4 


25.1 


31.9 


39.7 


49.0 


59.1 


71.1 


84.5 


99.2 


116.0 


6 


21.4 


27.8 


35.3 


44.0 


54.0 


65.5 


79.0 


93.7 


110.0 


128.0 


7 


23.4 


30.4 


38.6 


48.1 


59.3 


68.7 


86.2 


102.0 


121.0 


140.0 


8 


25.3 


32.9 


41.8 


52.1 


64.0 


77.5 


93.2 


110.0 


130.0 


152.0 


9 


27.2 


35.3 


44.8 


55.8 


68.8 


83.2 


100.0 


119.0 


140.0 


163.0 


10 


29.0 


37.6 


47.8 


59.7 


73.5 


89.0 


107.0 


127.0 


150.0 


174.0 


11 


30.7 


39.9 


50.6 


63.2 


77.7 


94.4 


113.0 


134.0 


158.0 


184.0 


12 


32.4 


42.0 


53.3 


66.6 


82.0 


99.6 


120.0 


142.0 


167.0 


194.0 


13 


34.0 


44.2 


56.0 


70.0 


86.0 


105.0 


126.0 


149.0 


176.0 


203.0 


14 


35.6 


46.3 


58.7 


73.5 


90.0 


109.0 


131.0 


156.0 


183.0 


213.0 


15 


37.2 


48.3 


61.3 


76.5 


94.0 


114.0 


137.0 


163.0 


192.0 


223.0 


16 


38.7 


50.2 


63.8 


79.6 


98.0 


120.0 


143.0 


170.0 


200.0 


233.0 


17 


40.2 


52.2 


66.2 


82.7 


102.0 


124.0 


148.0 


176.0 


207.0 


242.0 


18 


41.7 


54.0 


68.7 


85.8 


106.0 


128.0 


154.0 


182.0 


215.0 


250.0 


19 


43.2 


56.0 


71.0 


88.9 


111.0 


132.0 


159.0 


189.0 


222.0 


258.0 


20 


50.0 


65.0 


82.5 


103.0 


127.0 


154.0 


184.0 


194.0 


258.0 


265.0 


25 


56.5 


73.4 


93.2 


117.0 


143.0 


173.0 


208.0 


248.0 


291.0 


339.0 


30 


62.6 


81.3 


103.0 


130.0 


159.0 


192.0 


230.0 


274.0 


322.0 


377.0 


35 


68.4 


88.8 


113.0 


141.0 


173.0 


209.0 


252.0 


300.0 


350.0 


410.0 


40 


74.0 


96.2 


122.0 


152,0 


188.0 


228.0 


273.0 


324.0 


382.0 


445.0 


45 


79.4 


103.0 


131.0 


164.0 


201.0 


242.0 


293.0 


346.0 


410.0 


476.0 


50 


84.6 


110.0 


140.0 


174.0 


215.0 


260.0 


312.0 


370.0 


437.0 


509.0 


55 


90.0 


117.0 


149.0 


185.0 


226.0 


285.0 


330.0 


393.0 


464.0 


538.0 


60 


94.7 


123.0 


156.0 


195.0 


240.0 


292.0 


349.0 


414.0 


488.0 


568.0 


65 


99.6 


130.0 


164.0 


206.0 


252.0 


306.0 


367.0 


437.0 


512.0 


599.0 


70 


104.0 


135.0 


171.0 


214.0 


264.0 


320.0 


383.0 


455.0 


536.0 


624.0 


75 


109.0 


141.0 


180.0 


224.0 


276.0 


333.0 


400.0 


467.0 


561.0 


653.0 


80 


113.0 


147.0 


187.0 


234.0 


287.0 


348.0 


417.0 


496.0 


584.0 


680.0 


85 


118.0 


153.0 


194.0 


243.0 


298.0 


362.0 


433.0 


516.0 


607.0 


707.0 


90 


122.0 


158.0 


201.0 


251.0 


309.0 


376.0 


448.0 


533.0 


629.0 


732.0 


95 


126.0 


164.0 


207.0 


259.0 


318.0 


387.0 


464.0 


551.0 


648.0 


756.0 


100 


135.0 


175.0 


222.0 


277.0 


340.0 


414.0 


495.0 


588.0 


693.0 


807.0 


110 


146.0 


190.0 


241.0 


300.0 


370.0 


450.0 


539.0 


640.0 


753.0 


878.0 


125 


165.0 


215.0 


273.0 


342.0 


420.0 


494.0 


609.0 


724.0 


852.0 


992.0 


150 


183.0 


238.0 


302.0 


378.0 


464.0 


564.0 


675.0 


802.0 


946.0 


1100.0 


175 


200.0 


260.0 


330.0 


412.0 


506.0 


615.0 


737.0 


875.0 


1027.0 


1201.0 


200 


217.0 


281.0 


358.0 


447.0 


548.0 


666.0 


800.0 


947.0 


1118.0 


1300.0 


225 


232.0 


301.0 


384.0 


478.0 


588.0 


714.0 


855.0 


1016.0 


1200.0 


1400.0 


250 


248.0 


322.0 


409.0 


510.0 


627.0 


762.0 


912.0 


1087.0 


1286.0 


1490.0 


275 


262.0 


340.0 


432.0 


540.0 


662.0 


806.0 


966.0 


1146.0 


1347.0 


1573.0 


300 


277.0 


360.0 


457.0 


570.0 


700.0 


852.0 


1010.0 


1213.0 


1428.0 


1665.0 


325 


290.0 


378.0 


480.0 


600.0 


737.0 


896.0 


1073.0 


1276.0 


1500.0 


1750.0 


350 


305.0 


395.0 


502.0 


627.0 


770.0 


936.0 


1122.0 


1332.0 


1570.0 


1830.0 


375 


317.0 


412.0 


522.0 


654.0 


803.0 


976.0 


1170.0 


1402.0 


1632.0 


1907.0 


400 


343.0 


446.0 


567.0 


708.0 


870.0 


1060.0 


1265.0 


1500.0 


1770.0 


2065.0 


450 


368.0 


478.0 


607.0 


759.0 


932.0 


1131.0 


1358.0 


1611.0 


1896.0 


2213.0 


500 


393.0 


510.0 


648.0 


810.0 


995.0 


1210.0 


1450.0 


1720.0 


2025.0 


2362.0 


550 


415.0 


540.0 


684.0 


856.0 


1036.0 


1280.0 


1532.0 


1820.0 


2140.0 


2500.0 


600 


440.0 


570.0 


724.0 


905.0 


1111.0 


1350.0 


1618.0 


1923.0 


2265.0 


2636.0 


650 


460.0 


599.0 


759.0 


938.0 


1166.0 


1417.0 


1700.0 


2016.0 


2373.0 


2770.0 


700 


483.0 


627.0 


797.0 


995.0 


1220.0 


1485.0 


1780.0 


2113.0 


2490.0 


2900.0 


750 


503.0 


654.0 


830.0 


1038.0 


1274.0 


1548.0 


1857.0 


2206.0 


2593.0 


3026.0 


800 


525.0 


680.0 


866.0 


1080.0 


1330.0 


1620.0 


1935.0 


2300.0 


2710.0 


3152.0 


850 


545.0 


708.0 


898.0 


1123.0 


1380.0 


1675.0 


2009.0 


2385.0 


2803.0 


3274.0 


900 


565.0 


734.0 


933.0 


1170.0 


1430.0 


1740.0 


2080.0 


2478.0 


2920.0 


3400.0 


950 



798 



Powering of Steamships. 



Steamship Performance. 



Displace- 






Knots, or nautical miles per hour. 






ment, 
in tons. 




















1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


T 


H 


H 


H 


H 


H 


H 


H 


H 


H 


H 


1000 


.438 


3.50 


11.8 


28.0 


54.9 


94.6 


150 


225 


318 


439 


1100 


.456 


3.75 


12.5 


30.0 


58.4 


100 


160 


239 


338 


467 


1200 


.500 


4.00 


13.4 


32.0 


62.0 


107 


170 


254 


359 


495 


1300 


.515 


4.12 


14.0 


33.0 


65.3 


112 


179 


267 


378 


523 


1400 


.548 


4.38 


14.9 


35.0 


68.7 


119 


189 


281 


398 


549 


1500 


.562 


4.50 


15.5 


36.0 


71.9 


124 


197 


295 


417 


575 


1600 


.578 


4.62 


16.2 


37.0 


75.0 


130 


206 


307 


435 


600 


1700 


.594 


4.75 


16.9 


38.0 


78.1 


135 


215 


320 


453 


625 


1800 


.625 


5.00 


17.5 


40.0 


81.2 


140 


224 


332 


470 


649 


1900 


.634 


5.25 


18.1 


42.0 


84.2 


145 


231 


345 


488 


673 


2000 


.700 


5.60 


18.8 


44.0 


87.0 


150 


239 


356 


504 


696 


2100 


.719 


5.75 


19.4 


46.0 


90.0 


155 


247 


369 


521 


720 


2200 


.735 


5.88 


20.0 


47.0 


92.7 


160 


255 


380 


537 


741 


2300 


.765 


6.12 


20.6 


49.0 


95.6 


165 


262 


391 


554 


764 


2400 


.788 


6.28 


21.1 


50.2 


98.4 


170 


270 


402 


569 


786 


2500 


.805 


6.44 


21.8 


51.5 


101.0 


174 


277 


414 


585 


808 


2600 


.828 


6.62 


22.4 


53.0 


104.0 


179 


285 


424 


600 


826 


2700 


.851 


6.81 


23.0 


54.5 


106.0 


184 


292 


436 


616 


850 


2800 


.872 


6.98 


23.5 


55.8 


109.0 


188 


299 


446 


631 


871 


2900 


.876 


7.12 


24.0 


57.1 


111.0 


192 


306 


457 


646 


893 


3000 


.909 


7.35 


24.6 


58.8 


114.0 


197 


313 


467 


660 


913 


3100 


.931 


7.45 


25.1 


59.8 


117.0 


201 


320 


478 


676 


933 


3200 


.952 


7.62 


25.6 


61.0 


119.0 


205 


327 


488 


690 


952 


3300 


.972 


7.78 


26.1 


62.2 


121.0 


209 


334 


498 


704 


972 


3400 


.992 


7.94 


26.8 


63.5 


124.0 


214 


340 


508 


718 


992 


3500 


1.01 


8.10 


27.2 


64.8 


127.0 


218 


347 


518 


733 


1010 


3600 


1.03 


8.25 


27.8 


66.0 


129.0 


222 


354 


528 


746 


1025 


3700 


1.05 


8.39 


28.2 


67.1 


131.0 


226 


360 


538 


759 


1049 


3800 


1.08 


8.60 


28.7 


68.5 


133.0 


230 


367 


548 


774 


1070 


3900 


1.09 


8.70 


28.9 


69.6 


135.0 


234 


373 


558 


787 


1087 


4000 


1.11 


8.85 


29.9 


70.8 


138.0 


238 


380 


567 


801 


1105 


4100 


1.13 


9.01 


30.4 


71.1 


140.0 


242 


386 


577 


814 


1122 


4200 


1.14 


9.14 


30.9 


73.1 


142.0 


246 


392 


586 


827 


1141 


4300 


1.16 


9.30 


31.4 


74.4 


145.0 


250 


398 


595 


840 


1160 


4400 


1.18 


9.42 


31.9 


75.5 


147.0 


254 


404 


604 


853 


1179 


4500 


1.19 


9.56 


32.4 


76.5 


150.0 


258 


410 


613 


866 


1198 


4600 


1.22 


9.72 


32.8 


77.7 


152.0 


261 


416 


622 


879 


1216 


4700 


1.23 


9.86 


33.4 


78.9 


154.0 


266 


422 


631 


891 


1232 


4800 


1.25 


10.0 


33.9 


80.0 


156.0 


270 


428 


640 


904 


1248 


4900 


1.28 


10.1 


34.4 


81.1 


158.0 


274 


434 


649 


916 


1265 


5000 


1.30 


10.3 


34.8 


82.7 


160.0 


277 


440 


65S 


929 


1282 


5250 


1.32 


10.6 


35.6 


85.0 


165.0 


283 


455 


670 


959 


1324 


5500 


1.36 


10.9 


36.4 


87.5 


171.0 


290 


469 


700 


990 


1367 


5750 


1.40 


11.2 


37.5 


90.0 


176.0 


298 


483 


721 


1024 


1408 


6000 


1.42 


11.4 


38.0 


92.8 


181.0 


303 


497 


742 


1050 


1448 


6250 


1.47 


11.9 


40.2 


95.2 


188.0 


322 


512 


762 


1065 


1488 


6500 


L.52 


12.2 


41.2 


97.8 


191.0 


330 


526 


782 


1078 


1526 


6750 


1.56 


12.5 


42.4 


100.0 


196.0 


339 


540 


802 


1123 


1567 


*/)00 


L.60 


12.9 


43.2 


L03.0 


202.0 


346 


554 


822 


1174 


1616 


7250 


1.64 


13.1 


44.4 


105.0 


205.0 


355 


566 


842 


1198 


1644 


7500 


1.68 


18.5 


45.5 


108.0 


210.0 


364 


579 


861 


1226 


1682 


7750 


1.72 


13.8 


46.5 


110.0 


215.0 


372 


599 


879 


1253 


1719 


8000 


1.75 


14.0 


17. 1 


112.0 


220.0 


379 


603 


899 


1280 


1757 


8250 


1.7s 


14.2 


48.4 


115.0 


224.0 


387 


615 


918 


1306 


1793 


8500 


l.M 


1 1.5 


49.4 


116.0 


229.0 


395 


628 


929 


1333 


1829 


8750 


1.84 


14.9 


50.0 


119.0 


233.0 


403 


640 


955 


1354 


1865 


9000 


L.88 


L5.2 


51.1 


122.0 


238.0 


411 


653 


973 


1385 


1902 


9250 


1.92 


15.1 


52.2 


124.0 


212.0 


418 


668 


991 


1411 


1937 


9500 


1.95 


15.6 


53.2 


126.0 


216.0 


426 


683 


1008 


1437 


1972 


10000 


2.05 


16.4 


55.1 


131.0 


256.0 


441 


714 


1044 


1488 


2042 



Powering of Steamships. 



799 



Steamship Performance. 







Knots, or 


nautical miles 


per hour. 






Displace- 
ment, 
in tons. 


11 


12 


13 


14 


15 


16 


17 


18 


19 


20 


H 


H 


H 


H 


H 


H 


H 


H 


H 


H 


T 


585 


759 


963 


1206 


1480 


1798 


2157 


2560 


3008 


3514 


1000 


622 


806 


1024 


1284 


1574 


1913 


2295 


2723 


3203 


3736 


1100 


660 


858 


1090 


1360 


1670 


2030 


2435 


2390 


3400 


3907 


1200 


696 


903 


1147 


1432 


1758 


2136 


2564 


3043 


3576 


4178 


1300 


732 


950 


1204 


1508 


1850 


2248 


2697 


3200 


3762 


4394 


1400 


766 


995 


1264 


1580 


1938 


2355 


2825 


3252 


3943 


4605 


1500 


800 


1038 


1317 


1648 


2020 


2458 


2948 


3500 


4113 


4803 


1600 


833 


1083 


1374 


1718 


2107 


2561 


3072 


3646 


4286 


5006 


1700 


864 


1123 


1422 


1784 


2188 


2660 


3140 


3785 


4448 


5195 


1800 


897 


1166 


1479 


1850 


2270 


2760 


3310 


3928 


4615 


5390 


1900 


927 


1205 


1527 


1913 


2345 


2854 


3420 


4060 


4770 


5570 


2000 


958 


1247 


1582 


1979 


2382 


2948 


3535 


4195 


4935 


5762 


2100 


988 


1284 


1628 


2037 


2500 


3038 


3642 


4325 


5084 


5935 


2200 


1017 


1324 


1680 


2102 


2578 


3134 


3755 


4460 


5241 


6120 


2300 


1047 


1360 


1723 


2160 


2646 


3220 


3860 


4580 


5386 


6290 


2400 


1077 


1400 


1777 


2222 


2725 


3313 


3970 


4715 


5542 


6470 


2500 


1102 


1435 


1820 


2280 


2796 


3400 


4075 


4835 


5655 


6637 


2600 


1131 


1473 


1870 


2338 


2868 


3486 


4180 


4960 


5832 


6813 


2700 


1160 


1508 


1911 


2395 


2935 


3568 


4280 


5076 


5970 


6970 


2800 


1189 


1545 


1960 


2452 


3010 


3655 


4385 


5200 


6115 


7142 


2900 


1215 


1582 


2000 


2508 


3075 


3740 


4485 


5318 


6255 


7300 


3000 


1242 


1614 


2048 


2565 


3145 


3822 


4585 


5440 


6394 


7470 


3100 


1268 


1648 


2092 


2616 


3210 


3905 


4680 


5550 


6525 


7622 


3200 


1296 


1683 


2134 


2671 


3280 


3985 


4775 


5670 


6666 


7781 


3300 


1320 


1717 


2178 


2725 


3343 


4063 


4870 


5784 


6784 


7936 


3400 


1347 


1750 


2220 


2779 


3408 


4143 


4965 


5893 


6936 


8090 


3500 


1373 


1783 


2264 


2830 


3475 


4222 


5060 


6010 


7061 


8250 


3600 


1398 


1815 


2303 


2881 


3534 


4300 


5155 


6115 


7184 


8400 


3700 


1422 


1848 


2348 


2941 


3606 


4385 


5250 


6238 


7333 


8563 


3800 


1446 


1880 


2385 


2986 


3660 


4453 


5340 


6336 


7444 


8696 


3900 


1473 


1912 


2427 


3038 


3725 


4530 


5430 


6444 


7580 


8847 


4000 


1497 


1944 


2468 


3086 


3785 


4610 


5520 


6550 


7700 


8988 


4100 


1520 


1975 


2507 


3137 


3850 


4680 


5610 


6655 


7830 


9141 


4200 


1545 


2008 


2546 


3186 


3910 


4750 


5700 


6761 


7950 


9285 


4300 


1568 


2037 


2585 


3238 


3970 


4825 


5790 


6865 


8072 


9432 


4400 


1593 


2070 


2624 


3286 


4025 


4900 


5875 


6970 


8195 


9572 


4500 


1614 


2100 


2664 


3333 


4087 


4975 


5960 


7070 


8320 


9710 


4600 


1639 


2130 


2702 


3382 


4145 


5040 


6045 


7172 


8437 


9850 


4700 


1663 


2160 


2740 


3431 


4202 


5112 


6130 


7275 


8555 


9990 


4800 


1686 


2190 


2779 


3478 


4260 


5193 


6215 


7375 


8673 


10120 


4900 


1708 


2220 


2817 


3525 


4321 


5253 


6300 


7475 


8792 


10250 


5000 


1760 


2293 


2909 


3640 


4414 


5426 


6507 


7723 


9081 


10601 


5250 


1822 


2365 


3000 


3755 


4608 


5600 


6715 


7972 


9370 


10953 


5500 


1876 


2436 


3090 


3868 


4744 


5767 


6917 


8204 


9652 


11269 


5750 


1930 


2507 


3180 


3981 


4880 


5935 


7120 


8436 


9935 


11586 


6000 


1982 


2574 


3261 


4094 


5013 


6096 


7313 


8519 


10203 


11902 


6250 


2035 


2642 


3352 


4207 


5146 


6258 


7505 


8603 


10472 


12218 


6500 


2088 


2710 


3438 


4320 


5281 


6419 


7698 


8986 


10741 


12534 


6750 


2141 


2778 


3524 


4434 


5417 


6580 


7892 


9370 


11010 


12851 


7000 


2191 


2842 


3606 


4531 


5542 


6733 


8076 


9587 


11265 


13152 


7250 


2241 


2907 


3688 


4629 


5668 


6886 


8260 


9805 


11521 


13453 


7500 


2290 


2971 


3770 


4726 


5794 


7039 


8445 


10022 


11776 


13754 


7750 


2340 


3036 


3852 


4824 


5920 


7192 


8628 


10240 


12032 


14056 


8000 


2488 


3098 


3931 


4923 


6042 


7340 


8806 


10451 


12280 


14345 


8250 


2636 


3161 


4011 


5023 


6164 


7488 


8984 


10662 


12528 


14634 


8500 


2784 


3223 


4095 


5123 


6286 


7637 


9162 


10823 


12776 


14922 


8750 


2933 


3286 


4170 


5222 


6408 


7785 


9340 


11084 


13024 


15211 


9000 


3080 


3346 


4247 


5343 


6516 


7926 


9512 


11289 


13364 


15493 


9250 


3222 


3407 


4324 5465 


6645 


8068 


9685 


11494 


13505 


15775 


9500 


3370 


3529 


4478 


5708 


6882 


8351 


10030 


11904 


13987 


16340 


10000 



800 



Electric Power. 



Conversion of Horse=power into Kilowatts and 
Kilowatts into Horse=power. 



No. 


Kilowatts 
to horse- 
power. 


Horse- 
power to 
kilowatts. 


No. 


Kilowatts 
to horse- 
power. 


Horse- 
power to 
kilowatts. 


No. 


Kilowatts 
to horse- 
power. 


Horse- 
power to 
kilowatts. 


1 


1.341 


.746 


38 


50.943 


28.3 


75 


100.545 


55.9 


2 


2.681 


1.49 


39 


52.283 


29.1 


76 


101.886 


56.7 


3 


4.022 


2.24 


40 


53.624 


29.8 


77 


103.226 


57.4 


4 


5.363 


2.98 


41 


54.965 


30.6 


78 


104.567 


58.2 


5 


6.703 


3.73 


42 


56.305 


31.3 


79 


105.907 


58.9 


6 


8.044 


4.48 


43 


57.646 


32.1 


80 


107.248 


59.7 


7 


9.384 


5.22 


44 


58.986 


32.8 


81 


108.588 


60.4 


8 


10.725 


5.97 


45 


60.327 


33.6 


82 


109.929 


61.2 


9 


12.065 


6.71 


46 


61.667 


34.3 


83 


111.269 


61.9 


10 


13.406 


7.46 


47 


63.008 


35.1 


84 


112.610 


. 62.7 


11 


14.747 


8.21 


48" 


64.349 


35.8 


85 


113.951 


63.4 


12 


16.087 


8.95 


49 


65.689 


36.5 


86 


115.292 


64.2 


13 


17.428 


9.7 


50 


67.030 


37.3 


87 


116.632 


64.9 


14 


18.768 


10.4 


5L 


68.371 


38.0 


88 


117.973 


65.6 


15 


20.109 


11.2 


52 


69.711 


38.8 


89 


119.313 


66.4 


16 


21.450 


11.9 


53 


71.052 


39.5 


90 


120.654 


67.1 


17 


22.790 


12.7 


54 


72.392 


40.3 


91 


121.995 


67.9 


18 


24.131 


13.4 


55 


73.733 


41.0 


92 


123.335 


68.6 


19 


25.471 


14.2 


56 


75.074 


41.8 


93 


124.676 


69.4 


20 


26.812 


14.9. 


57 


76.414 


42.5 


94 


126.016 


70.1 


21 


28.153 


15.7 


58 


77.755 


43.3 


95 


127.357 


70.9 


22 


29.493 


16.4 


59 


79.095 


44.0 


96 


128.698 


71.6 


23 


30.834 


17.2 


60 


80.436 


44.8 


97 


130.038 


72.4 


24 


32.174 


17.9 


61 


81.777 


45.5 


98 


131.379 


73.1 


25 


33.515 


18.6 


62 


83.117 


46.2 


99 


132.719 


73.8 


26 


34.856 


19.4 


63 


84.458 


47.0 


100 


134.06 


74.6 


27 


36.196 


20.1 


64 


85.798 


47.7 


200 


268.12 


149.0 


28 


37.537 


20.9 


65 


87.139 


48.5 


300 


402.18 


224.0 


29 


38.877 


21.6 


66 


88.480 


49.2 


400 


536.24 


298.0 


30 


40.218 


22.4 


67 


89.820 


50.0 


500 


670.30 


373.0 


31 


41.559 


23.1 


68 


91.161 


50.7 


600 


804.36 


448.0 


32 


42.899 


23.9 


69 


92.501 


51.5 


700 


938.42 


522.0 


33 


44.240 


24.6 


70 


93.842 


52.2 


800 


1072.48 


597.0 


34 


45.580 


25.4 


71 


95.183 


53.0 


900 


1206.54 


671.0 


35 


46.921 


26.1 


72 


96.523 


53.7 


1000 


1340.60 


746.0 


36 


48.261 
49.602 


26.9 
27.6 


73 
74 


97.864 
99.204 


54.5 
55.2 








37 

















Electric Heating. 



801 



Unit Equivalents for EIectric=heating Problems. 



1 horse- 
power 
hour = 



f 1000 watt hours. 
1.34 horse-power hours. 
2,656,400 foot-pounds. 
3,600,000 joules. 
3440 heat units. 
366,848 kilogrammetres. 
0.229 pound of coal oxi- 
1 kilowatt J dized with perfect ef- 
hour = * ficiency. 

3 pounds of water evap- 
orated at 212° F. 
22.9 pounds of water 
raised from 62° to 
212° F. 
8 cents at usual rates 
[ for electric heating. 

0.746 kilowatt hour. 

1,980.000 foot-pounds. 

2580 heat units. 

273,740 kilogrammetres. 

0.172 pound of coal oxi- 
dized with perfect ef- 
ficiency. 

225 pounds of water 
evaporated at 212° F. 

17.2 pounds of water 
raised from 62° to 
212° F. 

6 cents at usual rates 
for electric heating. 

f 1000 watts. 

1.34 horse-power. 

2,656.400 foot-pounds 
per hour. 

4424 foot-pounds per 
minute. 

73.73 foot-pounds per 
second. 

3440 heat units per hour. 
) 573 heat units per min- 
ute. 

9.55 heat units per sec- 
ond. 

0.229 pound of coal oxi- 
dized per hour. 

3 pounds of water evap- 
orated per hour at 
212° F. 

f 746 watts. 
0.746 kilowatts. 
33,000 foot-pounds per 

minute. 
550 foot-pounds per sec- 
ond. 
2580 heat units per hour. 
1 horse- J 43 heat units per min- 
power = j ute. 

j 0.71 heat unit per sec- 
ond. 
0.172 pound of coal oxi- 
dized per hour. 
2.25 pounds of water 
evaporated per hour 
at 212° F. 



. kilo- 
watt = 



i joule" = 



1 foot- 
pound = 



1 watt = 



1 heat 
unit = 



1 watt per 
square 
inch = 



r 1 watt second. 
0.00000278 kilowatt 

hour. 
0.102 kilogrammetre. 
0.00094 heat unit. 
0.73 foot-pound. 



1.36 joules. 

0.1383 kilogrammetre. 

0.000000377 kilowatt 

hour. 
0.000291 heat unit. 
0.0000005 horse-power 

hour. 



r 1 joule per second. 

00134 horse-power. 

0.001 kilowatt. 

3.44 heat units per 
hour. 

0.73 foot-pound per sec- 
ond. 

0.003 pound of water 
evaporated per hour. 

44.24 foot-pounds per 
minute. 



8.26 thermal units per 
square foot per min- 
ute. 

120° F. above surround- 
ing air (japanned 
cast-iron surface). 

66° C. above surround- 
ing air (japanned 
cast-iron surface). 



1.048 watt seconds. 
778 foot-pounds. 
0.252 caloric (kg. d.). 
108 kilogrammetres. 
0.000291 kilowatt hour. 
0.000388 horse-power 

hour. 
0.0000667 pound of coal, 

oxidized. 
0.00087 pound of water 

evaporated at 212° F. 



1 heat unit "} 
per 

square 
foot per 
minute = 



0.021 watt per square 

inch. 
0.0174 kilowatt. 
0.0232 horse-power. 



1 kilo- 
gramme- 
tre = 



7.23 foot-pounds. 
0.00000366 horse-power 

hour. 
0.00000272 kilowatt 

hour. 
, 0.0092 heat unit. 



51 



INDEX 



Absolute zero, 489. 

Accelerated circular motion, 272. 

motion, 259, 271. 
Acceleration, 234. 

resistance of trains, 794. 
Accumulator, efficiency of, 570. 

hydraulic, 569. 
Acetic acid, specific heat of, 495. 
Acid in feed water, sulphuric, 636. 
Adiabatic expansion, 666. 
Advance, angle of, 678. 
Advantages of electric driving, 749. 
Air, 500. 

compression and expansion of, 501- 
503. 

compressor efficiencies, 506. 

compressor, electric power re- 
quired for, 769. 

discharge, coefficients of, 505. 

flow of, 508. 

friction, 510-513. 

friction in pipes, 510-513. 

movement of, 509. 

pressure, 610. 

pressure and temperature of, 501. 

pressure and volume of, 502. 

pressures and water-heads, 610. 

pump, size of, 685. 

required for combustion, 607. 

required for motors, 506. 

specific head of, 496. 

transmission through pipes, 504, 
508. 

velocity of escape, 504. 

vessels for pumps, 554. 

volume and temperature of, 501. 

volume and weight of, 500. 
• volumes and velocities, 608, 609. 

work required to compress, 505. 



Alcohol, coefficient of expansion of 
486. 

latent heat of, 496. 

specific heat of, 495. 
Algebra, 76. 

Allan's link motion, 680. 
Allen valve, 682. 

valve, Zeuner diagram for, 682. 
Alloys, fusing-points of, 489. 

percentage of metals in, 283. 
Alternating systems, electric, 755. 
Altitude, barometric determination 
of, 514-518. 

effect on air compressors, 506. 

table, metric, 515. 
Aluminum, 791-793. 

coefficient of expansion of, 487. 

conductivity of, 793. 

for structural purposes, 792. 

fusing-point of, 489. 

heat transmission through, 497. 

specific gravity of, 284. 

specific heat of, 495. 

strength of, 791. 

weight of, 791. 
American boiler specifications, 628- 
632. 

coals, 578. 
A. S. M. E. code for gas and oil en- 
gines, 690-698. 

rules for boiler trials, 595. 

steam-engine code, 656. 
Ammonia, latent heat of, 496. 
Analogies, electrical, 700. 
Analyses of boiler feed water, 638. 

of boiler scale, 637. 
Analysis of coal, 599. 

of flue gases, 600. 

of indicator diagrams, 663. 
Angle of advance, 678. 

803 



804 



Index. 



Angle of oscillation, 276. 


Average strength of materials, 


411- 


sections, moment of inertia of, 357. 


415. 




valves, 335. 


Aveyron coals, heating value of, 575. 


Angles and bends, resistance in, 567. 


Avoirdupois weight, 59. 




constructions with, 106. 


Axis, neutral, 351. 




elements of, 380-390. 


Axle, crank, 441. 




Angstrom's valve gear, 680. 






Angular belting, 477. 


B 




functions, 132-223. 


Back-gear ratios, 475. 




functions, logarithmic, 179-223. 


Band brake, 662. 




Antimony, coefficient of expansion 


Banki motor, 688. 




of, 486. 


Bar iron, metric weight of, 292, 


293. 


f using-point of, 489. 


Barometric determination of 


alti- 


heat transmission through, 497. 


tude, 514-518. 




specific gravity of, 284. 


tables, 515-518. 




specific heat of, 495. 


tables, British, 517. 




Apothecaries' weight, 60. 


tables, metric, 515. 




Apparent efficiency of electrical ap- 


Base circle for gear teeth, 451. 




paratus, 748. 


Basic open-hearth steel, 349. 




Arc lamps, 727. 


Batteries, storage, 708. 




lamps, installation of, 712. 


Baume hydrometer, 565. 




lighting, series, 712. 


Bavarian coals, heating value of, 575. 


lights on low-potential circuits, 717. 


Bazin formula, diagram for, 535 


536. 


Arcs of belting contact, 469. 


formula for water flow, 535. 




of circles, 117-121. 


Beams, bending moments for, 


364- 


Area of chimneys, 605. 


367. 




Areas for safety valves, Board of 


deflections of, 364-367. 




Trade, 634. 


double-trussed, 248. 




of circles, 110-116. 


elements of, 360, 361. 




of flues, 608, 609. 


framed, 255-257. 




of plane figures, 127, 128. 


multiple-trussed, 249, 250. 




of safety valves, 633-635. 


simple-trussed, 247. 




Arithmetical progression, 77. 


strength of wooden, 408, 409. 




Arkansas coal, heating value of, 574. 


timber, 405. 




Armored cable, electric, 722. 


triple-trussed, 249. 




Arm sections, transformation of, 440. 


Beam sections, moment of inertia 


Arms, lever, 439. 


of, 356. 




of gear-wheels, 464. 


Bearings, 434. 




proportions of, 439. 


collar, 428. 




Arresters, lightning, 706. 


pressure on engine, 675. 




Ashes, 599. 


thrust, 428. 




Ash, percentage in coal, 578. 


Beau de Rochas cycle, 687. 




Asphalted pipe, water flow in, 529. 


Belgian coals, heating value of, 576. 


Atmospheric pressure, 514. 


Belting, 467. 




Augusta, Ga., water-power costs, 


angular, 477. 




771, 772. 


arcs of contact of, 469. 




Austrian coals, heating value of, 576. 


centrifugal force of, 468. 




Automatic cutouts, 713. 


horse-power of, 468. 




engines, performance of, 676. 


rules, 468. 




Auxiliaries, steam used by, 660. 


speed of, 467. 




Auxiliary circles for bevel gears, 459. 


tension of, 468. 





Index. 



805 



Belts and pulleys, 467. 


Boiler dimensions, Philadelphia 


arrangements of, 476. 


rules, 621. 


creep of, 472. 


domes, 631. 


crossed, 476. 


drums, 631. 


guide pulleys for, 478. 


efficiency, 592. 


half-crossed, 477. 


efficiency, calculation of, 600. 


leading-off angle of, 477. 


factors of safety, 631. 


open, 476, 


feed water, 635. 


pull of, 478. 


feed water, analyses of, 638. 


quarter-twist, 477. 


furnace flues, 615. 


slack side of, 478. 


furnaces, corrugated, 615, 618 


stiffness of, 472. 


furnaces, stiffened, 615. 


width of, 468. 


furnaces, strength of, 615. 


Belt transmission, 467. 


hanging, 632. 


transmission, losses in, 472. 


heat balance, 600. 


Bending, 346-351. 


horse-power, 592. 


boiler plates, 629. 


horse-power, Centennial standard, 


moments for beams, 364-367. 


593. 


Bends, pipe, 333. 


horse-power, commercial, 593. 


Bevel-gear cutting machines, 459. 


incrustation, 635. 


Bevel gears, 458. 


incrustation, causes of, 636. 


gears, auxiliary circles for, 459. 


man-holes, 631. 


gears, rims of, 463. 


materials, 611-616, 628. 


Binomial coefficients, 76. 


performances, 772. 


theorem, 76. 


plate rolls, power required for, 769. 


Bismuth, coefficient of expansion of, 


plates, bending, 629. 


486. 


plates, flanging, 629. 


specific gravity of, 284. 


plates for flanging, 615. 


specific heat of, 495. 


plates, forming, 629. 


Bisulphide of carbon, latent heat of, 


plate, tensile strength of, 613. 


496. 


proportions, 592. 


Blast-furnace gas, 581. 


proportions, metric, 627. 


gas-power cost, 777. 


riveting, 419, 630. 


Blowers, power required for foundry, 


riveting, pitch of, 611. 


769. 


riveting, rules for, 611. 


Board measure, 321. 


rivets, 629. 


measure, table. -of, 322. 


saddles, 631. 


of Trade areas for safety valves, 


scale, analyses of, 637. 


634. 


shells, factor of safety of, 613. 


of Trade boiler specifications, 611- 


shells, materials for, 613. 


616. 


shells, proportions of, 613. 


of Trade unit, 700. 


specifications, American, 628-632. 


Bohemian coals, heating value of, 


stay bolts, 614, 629, 631. 


577. 


stay girders, 616. 


Boiler and engine test, complete, 


stays, 614, 616. 


658. 


stays, loads on, 616. 


braces, 629-631. 


supporting, 632. 


braces, factor of safety of, 631. 


surfaces, flat, 630. 


bracing, 614. 


test, hydrostatic, 632. 


calking, 630. 


tests, 772. 


details, 611. 


trials, A. S. M. E. rules, 595. 



806 



Index. 



Boiler trials, conditions of, 597. 


Boxes, resistance, 706. 


trials, duration of, 596. 


Boyle's law, 514. 


trials, records of, 598. 


Braces, boiler, 629, 631. 


trials, report of, 601. 


Bracing, boiler, 614. 


trials, starting and stopping, 597. 


Brake, band, 662. 


troubles due to water, 636. 


horse-power, 662. 


tube holes, 630. 


rope, 662. 


tube plates, 617. 


water, 662. 


tubes, 629. 


Brass, coefficient of expansion of, 


tubes, draught area through, 604. 


486, 487. 


tube setting, 631. 


fusing-point of, 489. 


tubes, extra strong, 311. 


heat transmission through, 497. 


tubes, lap-welded, 310, 326. 


specific gravity of, 284. 


tubes, standard, 619. 


specific heat of, 495. 


workmanship, 629. 


wire, weight of, 313. 


Boilers, 592-638. 


working tools, power required for, 


causes of corrosion in, 637. 


768. 


Cornish, 603. 


Brasses, connecting-rod, 445. 


cylinder, 603. 


Breast w T ater-wheel, 545-547. 


evaporation in, 592. 


Brick, coefficient of expansion of, 


evaporation tests of, 772. 


486. 


evaporative performance of, 593. 


conduits, water flow in, 529. 


evaporative power of, 604. 


strength of, 415. 


examination of, 595. 


Bridge rivets, weight of, 301. 


flow of gases in, 593. 


trusses, 257, 258. 


grate surface of, 603, 604. 


British coals, heating value of, 574. 


heating surface of, 603, 604. 


measures of volume, 61. 


locomotive, 603. 


steam table, 583-585. 


prevention of corrosion in, 637. 


thermal unit, 490, 572. 


prevention of incrustation in, 636. 


Brittleness, 282. 


prevention of priming in, 637. 


Bronze, coefficient of expansion of, 


proportions of, 603. 


487. 


return tubular, 603. 


fusing-point of, 489. 


scale in, 635. 


specific gravity of, 284. 


Scotch, 603. 


Brotherhood hydraulic motor, 570. 


steam, 592-638. 


Brownlee's formula for discharge of 


water-tube, 603. 


steam, 589. 


working pressures for, 622-626. 


Brown's valve gear, 680. 


Bolts, 420. 


Brunswick coals, heating value of, 


for steam cylinders, 420. 


574. 


proportions of stay, 620. 


B. T. U.,490. 


stresses on, 420. 


to calories, table, 491. 


weight of, 302, 303. 


Bulging, resistance to, 614. 


Bolt threads, special, 421. 


Bumped heads for boilers, 630, 631. 


Bomb, calorimetric, 57:;. 


Burden, hourly, 785. 


Bonus plan of wages, 780. 


Bureau Veritas boiler specifications, 


Boom, framed, 256. 


611-616. 


Boring mill, power required for. 


Bus bars, 705. 


752. 


Butterfly valves, 335. 


Bosnian coals, heating value of, 


Butt-joint riveting, 418. 


577. 


Butt straps, boiler, 613. 



Index. 



807 



c 


Cast-iron pipe, standard dimensions 


Cables, electrical, 701. 


of, 308, 309. 


Calcium carbonate in feed water, 


pipe, water flow in, 529. 


635. 


pipe, weight of, 299. 


sulphate in feed water, 635. 


spheres, weight of, 297, 298. 


Calculus, differential and integral, 


Catenary, 238-241. 


230-233. 


Causes of boiler incrustation, 636. 


Calibration of indicator springs, 661. 


of corrosion in boilers, 637. 


Calking, boiler, 630. 


of priming in boilers, 637. 


Calorie, 490, 572. 


Cements, strength of, 415. 


Calories to B. T. U., table, 492. 


Centigrade expansion coefficients, 


to foot-pounds, table, 493. 


487. 


to thermal units, table, 492, 


thermometer, 482. 


Calorific power, 572. 


to Fahrenheit tables, 484. 


power, computation of, 572. 


Central condensers, 686. 


power of blast-furnace gas, 581. 


forces, 274. 


power of coal gas, 581. 


Centre of gravity, 242-246. 


power of gas fuels, 581. 


of gravity experimentally deter- 


power of natural gas, 580. 


mined, 244. 


power of producer gas, 580, 581. 


of gravity graphically determined, 


power of water gas, 581. 


243. 


tests of coal, 599. 


of gyration, 272. 


values of fuels, table, 573. 


of hydrostatic pressure, 566. 


Calorimeter, throttling, 591. 


of oscillation, 276. 


Calorimetric bomb, 573. 


Centrifugal force, 274, 275. 


Canals, flow of water in, 535-538. 


force of belting, 468. 


Cantilever beams, 255. 


force of Manila rope, 479. 


Capacity of wires, carrying, 711. 


pumps, 568. 


steam-engine, 656. 


Centripetal force, 274. 


Caratonk Falls water-power plant 


Chains, 315. 


costs, 771. 


crane, 315. 


Carbonate of lime, solubility of, 


Chain sheaves, 315. 


637. 


Channel sections, moment of inertia 


of magnesia, solubility of, 637. 


of, 356. 


Carbon, coefficient of expansion of, 


Channels, elements of, 362, 363. 


487. 


flow of water in open, 535. 


Carbonic anhydride, specific heat 


Charging machine, power required 


of, 496. 


for, 765. 


oxide, specific heat of, 496. 


Charleroi coals, heating value of, 576. 


Car-houses, wiring of, 717. 


Check valves, 335. 


Carnot cycle, 667. 


Cheval-vapeur, 638. 


Carrying capacity of wires, 711. 


Chezy formula for water, 529. 


Car-wiring, 717. 


Chimney area, 605. 


Cast-iron columns, 369. 


dimensions, table of, 606. 


columns, safe loads for, 391, 392. 


flues, 608, 609. 


expansion of, 488. 


formulas, 605. 


pipe, 307-310. 


formulas, metric, 606. 


pipe connections, 310. 


Chimneys, 605. 


pipe fittings, 331. 


height of, 665. 


■ pipe, metric system, 307. 


proportions of, 605. 


pipe, metric weight of, 300. 


Chlorine, specific heat of, 496. 



808 



Index. 



Choice of electric motors, 755. 

of electric systems, 755. 
Choke coil for lightning arrester, 706. 
Chords, table of, 107, 117-121 . 
Circle, 109. 

formulas, 109. 

ring, area of, 128. 

sector, centre of gravity of, 245. 

segment, centre of gravity of, 245. 

zone, surface of, 130. 
Circles, arcs of, 117-121. 

areas of, 110-116. 

circumferences of, 110-116. 

segments of, 117-121. 

tables of, 110-116. 
Circuit-breakers, electric, 724. 

for railway power plants, 707. 

general rules for, 712. 
Circular arc, centre of gravity of, 
246. 

mils, 701. • 

motion, accelerated, 272. 

pitch tables, 455. 
Circulating water, 685. 
Circumferences of circles, 110-116. 
Circumferential and diametral pitch, 
450. 

pitch, 449. 
Clearance and expansion ratio, 646. 

in steam engines, 646, 676. 
Clutch, cone, 438. 
Coal, analysis of, 599. 

and oil, comparative evaporation 
of, 580. 

and oil, relative value of, 580. 

calorific tests of, 599. 

determination of character of, 595. 

determination of moisture in, 598. 

gas, calorific power of, 581. 

gas power cost, 776. 

measurement in engine tests, 660. 

mine water, 638. 

percentage of ash in, 578. 

sampling, 598. 

specific gravity of, 284. 

unit, standard, 656. 
Coals, American, 578. 

heating values of, 574-577. 
Code, A. S. M. E. steam-engine, 656. 

National Electric, 704-733. 
Coefficients, binomial, 76. 



Coefficients for deflection, 359. 

for heat transmission, 496. 

for pipe flow, 529. 

for safe load, 359. 

of air discharge, 505. 

of discharge, 527. 

of expansion, 486, 487. 

of friction, 281. 

of radiation, 497. 

of wear, 462. 
Coils, reactive, 727. 
Coinage, fineness of, 62. 
Cold saw, power required for, 760. 
Collar bearings, 428. 

bearings, friction in, 281. 
Columbia, S. C, water-power costs, 

771, 772. 
Columns, 368. 

cast-iron, 369. 

elements of Z-bar, 378. 

safe loads for cast-iron, 391, 392. 
Combustible, distribution of heating 

value of, 601. 
Combustion, air required for, 607. 

draught pressure for, 607. 

rate of, 607. 

total heat of, 579. 
Commercial cut-off, 664. 

test of steam plant, 657. 
Commutating machines, efficiency 
of, 738. 

machines, electrical, 736. 
Comparative evaporation of coal and 
oil, 580. 

steam-engine economy, 667. 
Complete engine and boiler test, 658. 
Compound engines, cylinder ratios 
for, 648. 

engines, performance of, 675, 773. 

expansion, 648. 

interest, 57. 

interest table, 57. 

non-condensing engines, perform- 
ance of, 773. 

pendulum, 276. 

shapes, moment of inertia of, 357. 

shapes, radius of gyration of, 358. 

steam engine, 648. 
Compressed air, delivery through 
pipes, 508. 

air required for motors, 506. 



Index. 



809 



Compressed air, velocity of escape 


Constant-potential electric systems, 


of, 504. 


713. 


Compression, 346, 350. 


Construction, materials of, 282. 


and expansion of air, 501-503. 


Contact, arcs of belting, 469. 


curve, 663. 


Contents of pipes, 543, 544. 


in gas engines, 687. 


Contraction of gases, 489. 


Compressor efficiencies, 506. 


Conversion of calories into thermal 


Computation of calorific power, 572. 


units, 572. 


Concord, N. H., water-power costs, 


of fractions, 24. 


771, 772. 


of mils into centimetres, 704. 


Condensing engines, performance of, 


of thermal units into calories, 572. 


675. 


of thermometer scales, 482. 


Condensation, cylinder, 639, 648. 


tables for heat units, 491, 492. 


initial, 648. 


Copper, coefficient of expansion of, 


Condenser plant, electrically driven, 


486, 487. 


768. 


fusing-point of, 489. 


Condensers, 684. 


heat transmission through, 497. 


central, 686. 


pipe, metric weight of, 328. 


electric, 727. 


radiation from, 497. 


gain due to, 684. 


specific gravity of, 284. 


size of, 684. 


specific heat of, 495. 


Condenser, Wheeler surface, 686. 


weight of sheet, 294, 295. 


Worthington j et, 685. 


wire, 702. 


Condensing engines, performance of, 


wire, strands of, 700. 


773. 


wire table, 702. 


water, 684. 


wire, weight of, 313. 


Conditions of boiler trials, 597. 


Cord friction on pulleys, 469. 


Conductivity, Matthiessen's stand- 


polygon, 235. 


ard of, 702. 


Corliss engines, performance of, 675. 


Conductors, flexible cord, 717-721. 


Cornish boiler, 603. 


rules for underground, 711. 


boiler, heating surface of, 604. 


station, 705. 


Corrosion in boilers, causes of, 637. 


Conduits, electric, 723. 


in boilers, prevention of, 637. 


interior, 716. 


Corrugated furnace, Fox, 615. 


Conduit wire, 722. 


furnace, Purves-Brown, 615. 


Cone, centre of gravity of, 245. 


furnaces, 615-618. 


clutch, 438. 


iron, 320. 


coupling, 437. 


iron, weight of, 321. 


pulleys, 472. 


Cosecants, logarithmic, 179-223. 


pulleys, crossed belts for, 472. 


natural, 134-178. 


surface of, 130. 


Cosines, logarithmic, 179-223. 


volume of, 131. 


natural, 134-178. 


Conic frustum, centre of gravity of, 


Cost keeping, 781. 


246. 


of electric power, 778, 779. 


frustum, volume of, 131. 


of gas power, 775-777. 


Connecting rod, 442. 


of power, 771. 


rod ends, 443. 


of steam plants, 774. 


rod, marine type, 444. 


of steam power, 772-775. 


rod stub end, 445. 


of water power, operative, 772. 


Connections, riveted, 416. 


of water-power plants, 771. 


Constant-current electric systems, 712. 


Cotangents, logarithmic, 179-223. 



810 



Index. 



Cotangents, natural, 134-178. 
Cotter, dimensions of, 423. 
Cotters for cross-heads, 447. 
Cotton-rope transmission, 481. 
Cottonwood, water-power costs of, 

771, 772. 
Counters, speed, 663. 
Couplings, 436. 
Crane chains, 315. 
Cranes, electric, 753. 

power required for, 753, 761, 766, 
767. 
Crank arm, graphical analysis of, 
442. 

axle, 441. 

graphical analysis of, 442. 

pins, dimensions of, 676. 

pins, pressure on, 281, 675. 
Cranks, 441. 

proportions of, 441. 
Creep of belts, 472. 
Crossed belts, 476. 

belts for cone pulleys, 472. 
Cross-head, 447. 

pin, dimensions of, 676. 

pin, pressure on, 676. 

pressure on, 676. 

proportions, 447. 
Cross-section of pipes, changes in, 

567. 
Crushing, 346, 350. 
Cube roots, table of, 25-55. 
Cubes, table of, 25-55. 
Cubic feet, contents of cylinders in, 

543, 544. 
Current-wheel, 546. 
Curve, compression, 663. 

expansion, 639. 

hyperbolic, 648. 

isothermal, 647. 

polytropic, 648. 

resistance on railways, 794. 
Curves for Fourneyron turbine, 551. 

for Jonval turbine, 552. 

thermal efficiency of, 668. 
Cut-off, commercial, 664. 

most economical point of, 646. 

point of, 663. 
Cutouts, automatic, 713. 

electric, 724. 

general rules for, 712. 



Cycle, Beau de Rochas, 687. 

Carnot, 667. 

Rankine, 667. 
Cycloid, 126. 

Cycloidal curves, 126, 127. 
Cylinder boiler, 603. 

boiler, heating surface of, 604. 

condensation, 639, 648. 

proportions, steam, 647. 

ratios for compound engines, 648. 

ratios for quadruple-expansion en- 
gines, 649. 

ratios for triple-expansion engines, 
649. 
Cylinders, bolts for steam, 420. 

contents of, 543, 544. 

thick, 398. 

volume of, 131. 
Cylindrical boilers, working press- 
ures for, 622-626. 
Cylindric surface, centre of gravity 
of, 246. 

D 

Dalmatian coals, heating value of, 

577. 
Dam, pressure on, 527. 
Data and results of duty trials, 562. 

and results of engine tests, 670. 

and results of evaporative test, 
602. 

for pumping-engine trial, 560. 
Deane steam pumps, 558. 
Decimals and fractions, 24. 
Deck beam sections, moment of in- 
ertia of, 356. 
Deflection, coefficient for, 359. 

of beams, 364-367. 

of shafts, 432. 

of springs, 399-102. 
Delivery of air through pipes, 508. 

of pumps, 555-557. 
Density, 282. 

and volume of water, 524. 

of water, 519. 
Depreciation table, 786. 
Derbyshire coal, heating value of, 

574. 
Design, machine, 416. 
Details of steam boilers, 611. 
Detectors, ground, 707. 






Index. 



811 



Diagram factor, indicator, 665. 

for Bazin's formula, 535, 536. 

for cone pulleys, 473. 

for Kutter's formula, 538. 

for multiple-expansion engines, 
649. 

gas-engine, 688. 

polytropic, 648. 

temperature-entropy, 666. 
Diagrams, analysis of indicator, 663. 

indicator, 650. 
Diametral and circumferential pitch, 
450. 

pitch, 450. 

pitch formulas, 454. 

pitch, lineal value of, 450. 

pitch tables, 456. 
Diamond, specific heat of, 495. 
Dielectric strength, 743. 
Diesel motor, 688. 

Differential and integral calculus, 
230-233. 

calculus formulas, 232. 
Dimensions of chimneys, table, 606. 

of gear teeth, 461. 

of pipe fittings, 333, 334. 

of pumps, 554. 

of slide valve, 681, 682. 
Direct-current electric system, 756. 
Discharge, coefficient of, 527. 

of steam, Brownlee's formula, 589. 

of water from nozzles, 527. 

of water from orifices, 527. 

of water from pipes, 531. 
Distance, 234. 

Distribution of heating value, 601. 
Domes, boiler, 631. 
Double-acting pumps, delivery of, 

555-557. 
Double angles, elements of, 386-390. 

extra strong pipe, 325. 

riveted joints, 611, 612. 

trussed beams, 248. 
Draft area through boiler tubes, 604. 

pressure, 607. 

required for combustion, 607. 

tube, turbine, 552. 
Drilling machines, power required 

for, 762. 
Drill presses, power required for, 751. 
Driving, advantages of electric, 749. 



Driving rig for indicator, 661. 
Drums, boiler, 631. 
Dry measure, U. S. A., 61. 
Ductility, 348. 

of rivet bars, 611. 
Dulong's formula for calorific power, 

572. 
Duplex steam pumps, 558, 559. 
Duration of boiler trials, 596. 

of engine test, 657. 
Duty trials, data and results of, 562. 

trials of pumping engines, 560-563. 
Dynamical formulas, 268-271. 
Dynamics, 268. 
Dynamo rooms, 705. 
Dynamos (see Generators). 



Eccentric bolts, 446. 

imaginary, 681. 

straps, 446. 
Eccentrics, 445. 

Economical length of engine stroke, 
647. 

point of cut-off, 646. 
Economy coils, 717. 

of actual and ideal engines, 667. 

standard of steam engine, 665. 
Efficiency, boiler, 600. 

curves, thermal, 668. 

of electrical apparatus, 738. 

of gearing, 465. 

of heat motors, thermal, 649. 

of hydraulic accumulator, 570. 

of phase-displacing apparatus, 747. 

of Reynolds's pumping engines, 
650. 

of steam engine, maximum, 650. 

of steam engines, thermal, 650. 

standard of steam engine, 665. 
Efflux of steam, velocity of, 589. 
Elastic bodies, impact of, 279. 

limit, 347, 348. 

limit of wood, 405. 
Elasticity, 347. 

modulus of, 347. 
Elbows, resistance to steam flow, 591. 
Electrical analogies, 700. 

and mechanical units, 699. 

apparatus, apparent efficiency of, 
748. 



812 



Index. 



Electrical apparatus, efficiency of, 738. 

apparatus, overload capacity, 747. 

apparatus, pulsation in, 745. 

apparatus, rating of, 746. 

apparatus regulation, 744. 

apparatus, rise of temperature, 741. 

apparatus, variation in, 745. 

induction apparatus, rotary, 737. 

induction apparatus, stationary, 
737. 

insulation, 743. 

machines, commutating, 736. 

machines, synchronous, 737. 

machines, synchronous commuta- 
ting, 737. 

standardization, 736-749. 

units, 699. 

units, equivalents of, 699. 

wiring formulas, 733-736. 

work, inside, 710. 

work, marine, 730-733. 

work, outside, 708-710. 
Electrically-driven condenser plant, 

768. 
Electric cable, 701. 

cable, armored, 722. 

circuit-breakers, 724. 

code, National, 704-733. 

conduits, metal, 723. 

cranes, power required for, 753, 766, 
767. 

cutouts, 724. 

driving, advantages of, 749. 

driving of machine tools, 750-765. 

fittings, general rules for, 719. 

fixture wire, 722. 

fuses, 725. 

gas-lighting, 729. 

generators, location of, 705. 

heaters, 713. 

heating, unit equivalents for, 801. 

lamp sockets, 725. 

lighting fixtures, 716. 

motors, choice of, 755. 

motors, installation of, 707. 

motor tests, 760-770. 

power, 698. 

power, cost of, 778, 779. 

signalling systems, 728. 

stations, care of, 706. 

switches, 724. 



Electric system, direct-current, 756. 

systems, alternating, 755. 

systems, choice of, 755. 

systems, constant-current, 712. 

systems, constant-potential, 713. 

systems, extra high-potential, 718. 

systems, high-potential, 718. 

systems, low-potential* 714. 
Electricity and water flow compared, 

700. 
Elements of angles, 380-390. 

of beams, 360, 361. 

of channels, 362, 363. 

of double angles, 386-390. 

of structural sections, 353. 

of structural shapes, 359. 

of tees, 379. 

of Z-bars, 376, 377. 
Ellipse, the, 122, 123. 
Ellipsoid, volume of, 131. 
Elliptical arc, centre of gravity of, 

256. 
Emery-wheels, power required for, 

752. 
Engine and boiler test, complete, 658. 

capacity, 656. 

compound, 648. 

dimensions, measurement of, 657. 

economy, standard of, 665. 

efficiency, standard of, 665. 

feed- water test of, 672. 

parts, sizes of, 676. 

performance, 653-655, 675. 

plant, examination of, 656. 

proportions, 675. 

shaft, size of, 676. 

speed, 662. 

steam, 638-686. 

test, duration of, 657. 

test, report of, 669. 

test, starting and stopping, 658. 

tests, 655-674. 

tests, forms for, 670. 

tests, heat unit basis for, 655. 

tests, measurement of coal in, 660. 

tests, recording, 663. 

tests, rules for, 656. 
Engines, cylinder ratios for com- 
pound, 648. 

cylinder ratios for quadruple-ex- 
pansion, 649. 



Index. 



813 



Engines, cylinder ratios for 


triple- 


Expansion, volume of, 486, 487. 


expansion, 649. 




Expense, general, 785. 


gas, 687-698. 




Explosion pressure in gas engines, 


heat analysis of, 666. 




689. 


maximum efficiency of, 650. 




Extra heavy flanges, 330. 


measurement of steam consump- 


Extra high-potential electric sys- 


tion, 659. 




tems, 718. 


multiple-expansion, 648. 




Extra strong pipe, 325. 


performance of automatic, 675. 


strong wire rope, 344. 


performance of compound, 


675. 


Eye bars, standard steel, 340. 


performance of condensing 


675. 


bar steel, 404. 


performance of non-cond 

675. 
performance of slide valve, 


3nsing, 


bars, test of steel, 404. 


675. 


F 


pressure on wearing surfaces, 675. 


Factor of safety, boiler, 631. 


thermal efficiency of, 650. 




of safety of boiler braces, 631. 


water consumption of, 651-655. 


of safety of boiler shells, 613. 


Entropy diagram, 666. 




of safety of flat boiler surfaces, 631. 


Epicycloid, 127. 




of safety of riveting, 611. 


Epicycloidal teeth, 451. 




of safety of rivet seams, 631. 


Equations, 78, 79. 




of safety of stay bolts, 631. 


Equilibrium of forces, 234. 




table, 7-23. 


Equivalent eccentric, 681. 




Factors of evaporation, 593. 


evaporation, 593. 




Fahrenheit expansion coefficients, 


Ether, specific heat of, 495. 




486. 


Evaporation equivalent, 593. 




scale, origin of, 482. 


factors, 593. 




thermometer, 482. 


in boilers, 592. 




to centigrade table, 483. 


tests of boilers, 772. 




Falling bodies, 259-266. 


Evaporative performance of 


boilers, 


bodies, British table of, 262, 263. 


593. 




bodies, metric table of, 264-266. 


power of boilers, 604. 




Fastenings, 416. 


power of liquid fuels, 579. 




keyed, 423. 


test, data and results of, 602 




Feathers, shaft, 423. 


Examination of engine plant 


.656. 


Feed pumps, 559. 


of steam boilers, 595. 




water, 635. 


Exhaust pipe, 676. 




water, analyses of, 638. 


Expansion, adiabatic, 666. 




water, boiler, 635. 


and compression of air, 501- 


503. 


water, calcium carbonate in, 635. 


coefficients of, 486, 487. 




water, calcium sulphate in, 635. 


compound, 648. 




water, impurities in, 635. 


curve, 639. 




water, measurement of, 659. 


curve in gas engines, 689. 




water, purification of, 635. 


linear, 486, 487. 




water, sulphuric acid in, 636. 


multiple, 648. 




water, temperature of, 659. 


of gases, 489. 




water test of steam engine, 672. 


of metals, table, 488. 




Fibre stress, 346. 


,of steam, 639. 




Fineness of coinage, 62. 


ratio, .639, 665. 




Fink's link motion, 680. 


ratio and clearance, 646. 




Fire-brick, coefficient of expansion 


superficial, 486, 487. 




of, 486. 



814 



Index. 



Fittings, cast-iron pipe, 331. 


Fractions, conversion of, 24. 


dimensions of pipe, 333, 334. 


reduced to decimals, 24. 


general rules for electric, 719. 


Framed beams, 255-257. 


steel pipe, 332. 


boom, 256. 


Fixtures, electric-lighting, 716. 


structures, statics of, 247. 


Fixture wire, electric, 722. 


Francis turbine, 552. 


Flagging, weight of, 320. 


French anthracite, heating value of, 


Flange joints, 420. 


575. 


Flanges, extra heavy, 330. 


coals, heating value of, 575. 


standard, 329. 


Frequencies and voltages classified, 


Flanging boiler plates, 629. 


746. 


plates for, 615. 


Friction, 280. 


Flat boiler surfaces, 630. 


air, 510-513. 


plates, proportions for, 614. 


coefficients of, 281. 


stayed surfaces, 620. 


journal, 281. 


Flexible cord conductors, 717, 721. 


loss of water head by, 533. 


Floor glass, 320. 


Morin's laws of, 280. 


Flow of air, 508. 


of air in pipes, 510-513. 


of gases in boilers, 593. 


on pulleys, 469. 


of steam, 588. 


Tower's experiments on, 281. 


of steam in pipes, 589, 590. 


Fuels, 572-581. 


of steam, Napier's rule, 589. 


classified, 572. 


of water, 529-541. 


draught pressure for, 607. 


of water and electricity compared, 


evaporative power of liquid, 579. 


700. 


gaseous, 580. 


of water in open channels, 535. 


table of calorific values of, 573. 


of water, resistance of bends, 567. 


total heat of combustion, 579. 


of water, resistance of valves, 567. 


Funicular polygon, 237. 


of water through pipes, 529. 


Furnace flues, 615. 


Flue areas, 608, 609. 


gas-power cost, 777. 


gases, analysis of, 600. 


Furnaces, corrugated, 615, 618. 


Flues, chimney, 608, 609. 


Morison suspension, 616. 


furnace, 615. 


stiffened, 615. 


Fluid, soldering, 730. 


strength of short, 615. 


Fly-wheels, 274, 676. 


Fuses, electric, 725. 


Foot-pounds into calories, table, 494. 


Fusing-points, 489. 


Force, 234, 268. 


Fusion, latent heat of, 496. 


in moving bodies, 271. 




of gravity, 261. 


a 


Forces, equilibrium of, 234. 


Gallons, contents of cylinders in, 


Forestry Division, experiments of U.- 


543, 544. 


S., 405. 


Galvanizing plant, electric driving, 


Forming boiler plates, 629. 


764. 


Form of gear teeth, 450. 


Gas engine, 687-698. 


Forms for testing gas engines, 693. 


engine diagram, 688. 


Foundry blowers, power required 


engine, forms for testing, 693. 


for, 769. 


engine, heat test of, 696. 


Fourneyron turbine, 551. 


engine proportions, 689. 


turbine, curves for, 551. 


engine testing, 690-698. 


Fox corrugated furnace, 615, 617. 


engines, expansion curve in, 689. 


Fractions, 24. 


engines, explosion pressure in, 689. 



Index. 



815 



Gas engines, mean effective pressures 


German rules for safety valves, 635. 


in, 688. 


silver, coefficient of expansion of, 


engines, power of, 689. 


487. 


lighting, electric, 729. 


silver, heat transmission through, 


natural, 580. 


497. 


power, cost of, 775, 777. . 


Girders, boiler stay, 616. 


power cost, "blast-furnace gas, 777. 


Glass, coefficient of expansion of, 


power cost, coal gas, 776. 


486, 487. 


power cost, natural gas, 775. 


fusing-point of, 489. 


power cost, producer gas, 776. 


radiation from, 497. 


Gaseous fuel, 580. 


skylight and floor, 320. 


Gases, analysis of flue, 600. 


specific heat of, 495. 


contraction of, 489. 


window, 316. 


expansion of, 489. 


Globe valve, resistance to steam flow, 


flow in boilers, 593. 


591. 


specific gravity of, 285. 


valves, 335. 


specific heat of, 496. 


Gold, coefficient of expansion of, 


Gate valves, 335. 


486, 487. 


Gauge, sheet-metal, 314. 


fusing-point of, 489. 


Gauges, wire, 312. 


specific gravity of, 284. 


Gay Lussac hydrometer, 565. 


specific heat of, 495. 


Gear cutting machines, bevel, 459. 


Gooch's link motion, 679. 


Gearing, 448-467. 


Governor, 276. 


efficiency of, 465. 


Porter, 276. 


proportions of, 466. 


Grade resistance of trains, 794. 


Gears, bevel, 468. 


Granite, coefficient of expansion of, 


spiral, 459. 


486. 


Gear teeth, dimensions of, 461. 


Graphical analysis of cranks, 442. 


teeth, form of, 450. 


analysis of shafts, 433. 


teeth, interference of, 453. 


statics, 247-258. 


teeth, strength of, 457. 


Graphite, coefficient of expansion of, 


teeth, stresses in, 461. 


487. 


teeth, wear on, 462. 


specific heat of, 495. 


teeth, wooden, 462. 


Grate surface of boilers, 603, 604. 


teeth, working stresses on, 457. 


surface, ratio to heating, 604. 


Gear-wheel parts, proportions of, 462. 


Gravity, centre of, 242-246. 


Gear-wheels, arms of, 464. 


force of, 261. 


cutters for, 450. 


specific, 282. 


hubs of, 465. 


table of specific, 284, 285. 


pitch of, 449. 


Grindstones, power required for, 761. 


proportions of, 460. 


Grooves for rope sheaves, 480. 


radii of, 449. 


Ground detectors, 707. 


rims of, 462. 


return wires, 709. 


General expense, 785. 


Grounding of lightning arresters, 706. 


Generators, insulated frames for, 705. 


of low-potential circuits, 709. 


location of electric, 705. 


Group driving of machine tools, 764. 


Geometrical progression, 77, 78. 


driving of wood-working machin- 


Geometry, 105-132. 


ery, 770. 


German coals, heating value of, 574. 


riveting, 418, 419. 


Lloyds, boiler specifications of, 611- 


Guide pulleys for belts, 478. 


616. 


Guides, pressure on, 448. 



816 



Index. 



Gyration, centre of, 272. 
radius of, 269, 273, 274, 351, 358, 359. 

H 

Half-crossed belts, 477. 

Half sphere, centre of gravity of, 245. 

Hangers, 435. 

Hanging, boiler, 632. 

Hanover coals, heating value of, 575. 

Hard bodies, impact of, 279. 

steel struts, 374, 375. 
Hardness, 282. 
Haulage rope, wire, 343. 
Head of water, 525-528. 

of water, loss by friction, 533. 
Heads and velocities of water, 528. 
Heat, 481. 

analysis of steam engine, 666. 

balance of boiler performance, 600. 

denned, 481. 

emission, 499. 

latent, 496. 

loss through walls, 499. 

mechanical equivalent of, 495. 

motors, thermal efficiency of, 649. 

of combustion of fuels, 579. 

quantity of, 490. 

specific, 495. 

test of gas engine, 696. 

test of steam engine, form for, 670. 

test of steam engine, standard, 658. 

transmission, coefficients for, 497. 

unit basis for engine tests, 655. 

unit conversion tables, 491, 492. 

unit standard for steam engines, 
656. 

units, 490. 

units in water, 520-523. 

units, measurement of, 658. 
Heaters, electric, 713. 
Heating, electric, 801. 

of electrical apparatus, 741. 

pipes, iron, 499. 

surface of boilers, 603, 604. 

surface, ratio to grate, 604. 

surface, relative value of, 603. 

values of coals, 574-577. 
Heavy flanges, extra, 330. 
Height of chimneys, 605. 
Helfenberger hydraulic regulator, 
570. 



Helical springs, 401. 

Hempen rope transmission, 469. 

Hexagon-headed bolts, weight of, 

303. 
High-potential electric systems, 718. 
Hoisting gears, 460. 

power required for, 754. 

rope, standard wire, 342. 
Hollow shafts, 431. 
Hooke's law, 347. 
Horizontal water-wheels, 545. 
Horse-power, 268, 638. 

and kilowatts, 800. 

brake, 662. 

determined by indicator, 651. 

indicated, 661. 

metric, 638. 

of belting, 468. 

of boilers, 592. 

of cotton-rope transmission, 481. 

of Manila-rope transmission, 480. 

of water, 525. 

of water, table for, 542. 

of water-wheels, 545, 546. 
Hot-well temperature, 684. 
Hourly burden, 785. 
Hubs of gear-wheels, 465. 
Hungarian coals, heating value of, 

577. 
Hydraulic accumulator, 569. 

distribution of power, 570. 

distribution, ring system, 570. 

motor, Brotherhood, 570. 

motor, Rigg, 570. 

motors, 570. 

motor, Schmid, 570. 

press, 566. 

radius, 529. 

radius, table for, 530. 

ram, 564. 

regulator, Helfenberger, 570. 

riveted pipe, 327. 

slopes, table for, 530. 

transmission of power, 569. 
Hydraulics, 525-571. 
Hydrogen, specific heat of, 496. 
Hydrometer, 564. 

Baume\ 565. 

Gay Lussac, 565. 

specific gravity, 565. 
Hydrostatic boiler test, 632. 



; 



Index. 



817 



Hydrostatic paradox, 566. 


Initial condensation, 648. 


press, 566. 


pressure, 639. 


pressure, centre of, 566. 


Injector, steam used by, 596. 


Hydrostatics, 565. 


Inside electrical work, 710. 


Hyperbola, the, 126. 


lap of valve, 681. 


Hyperbolic curve, 648. 


Installation costs of steam plants, 


logarithms, 639-643. 


774. 


Hypocycloid, 127. 


costs of water power, 771. 




of electric motors, 707. 


I 


of switchboards, 706. 


Ice, specific heat of, 495. 


of transformers, 708. 


Ideal engine, 667. 


Insulated frames for generators, 705. 


Illinois coals, 578. 


frames for motors, 707. 


coals, heating value of, 574. 


Insulating joints, electric, 727. 


Imaginary eccentric, 681. 


Insulation, 720. 


Impact, 278. 


electrical, 743. 


of elastic bodies, 279. 


resistance, 729, 743. 


of hard bodies, 279. 


resistance, testing, 706. 


water-wheels, 545. 


slow-burning, 720. 


Impurities in feed water, 635. 


tests, 720. 


Incandescent lamps in series cir- 


weather-proof, 721. 


cuits, 713. 


Integral calculus, 230-233. 


lamp sockets, 716. 


calculus formulas, 233. 


Inch, miner's, 541. 


Interest, 56. 


Incrustation, causes of boiler, 636. 


compound, 57. 


in boilers, 635. 


simple, 56. 


in boilers, prevention of, 636 


table, compound, 57. 


Indiana coal, 578. 


Interference of gear teeth, 453. 


coal, heating value of, 574. 


Interior conduits, 716. 


Indicated horse-power, 661. 


Internal-combustion motors, 687-698. 


Indicator, determination of horse- 


Internal gear teeth, 451. 


power by, 651. 


pressure, 397. 


diagram factor, 665. 


Introduction to steam tables, 582. 


diagrams, 650. 


Involute teeth, 451. 


diagrams, analysis of, 663. 


Iron, coefficient of expansion of, 486, 


diagram, typical, 650. 


487. 


driving rig, 661. 


corrugated, 320. 


for determination of steam con- 


fusing-point of, 489. 


sumption, 651. 


heating pipes, 499. 


reducing motion for, 661. 


heat transmission through, 497. 


showing valve action, 651. 


latent heat of, 496. 


springs, testing, 661. 


radiation from, 497. 


steam accounted for, 663. 


shafting, 431. 


steam engine, 650. 


specific gravity of, 284. 


Inductance factor, 748. 


specific heat of, 495. 


Induction apparatus, efficiency of, 


weight of flat rolled, 286-289. 


740. 


weight of round bar, 290, 291. 


apparatus, rotary, 737. 


weight of sheet, 294-296. 


apparatus, stationary, 737. 


weight of square bar, 290, 291. 


Inertia, moment of, 269, 351. 


wire, weight of, 313. 


polar moment of, 393. 


Irregular figures, areas of, 129. 



52 



818 



Index. 



Irregular figures, centre of gravity 

of, 246. 
Isothermal curve, 647. 
Istria coal, heating value of, 577. 

J 

Jacket water, measurement of, 659. 
Jet condenser, Worthington, 685. 
Joints, double-riveted, 612. 

flange, 420. 

insulating electric, 727. 

riveted, 417-419. 

single-riveted, 612. 

treble-riveted, 612. 
Joint, universal, 438. 
Jonval turbine, 552. 

turbine, curves for, 554. 
Journal friction, 281. 

loads, 426. 

proportions, 426. 
Journals, 424-426. 

dimensions of, 425. 

pressures on, 281. 

K 

Kentucky coals, 578. 
Keyed fastenings, 423. 
Keys, dimensions of, 423. 

standard, 424. 
Kilowatt, 638. 

Kilowatts and horse-power, 800. 
Kinematics, 416. 
Knowles steam pumps, 559. 
Kutter's formula, diagram for, 538. 

formula for water flow, 535-538. 



Lag screws, 324. 
Lamps, arc, 727. 

incandescent, in series circuits, 713. 

installation of arc, 712. 
Lamp sockets, electric, 725. 
Lancashire boiler, heating surface 
of, 604. 

coal, heating value of, 574. 
Lap joint riveting, 418. 

of slide valve, 681. 

of valve, 678. 

of valve, inside, 681. 

riveted pipe, water flow in, 529. 

welded boiler tubes, 311, 326. 
Latent heat, 496. 



* 



Latent heat of fusion, 496. 

heat of vaporization, 496. 
Lathe, speed changes of, 476. 

speeds, "lump" in, 476. 
Lathes, power required for, 751-753. 
Latin monetary union, 62. 
Lawrence, Mass., water-power costs, 

771, 772. 
Lead, fusing-point of, 489. 

heat transmission through 

latent heat of, 496. 

pipe, weight of, 319. 

specific gravity of, 284. 

specific heat of, 495. 

weight of sheet, 294, 295. 
Leading-off angle of belts, 477, 
Leakage test of pumping engine, 561. 
Leaks in steam-engine plants, 65' 
Length, metric measures of, 59. 

of belt for cone pulleys, 472. 
Leverage, 258, 267. 
Lever arms, 439. 

safety valves, 632. 
Levers, 439. 

Lightning arresters, 706, 728. 
Lignite, heating value of, 578. 
Lime, solubility of carbonate of, 637. 

solubility of phosphate of, 637. 

solubility of sulphate of, 637. 
Linear expansion of cast-iron, 488. 
Link motion, Allan's, 680. 

motion, Fink's, 680. 

motion, Gooch's, 679. 

motions, 679, 680. 

motion, Stephenson's, 679. 

motion, Walschaert's, 680. 
Liquid fuels, evaporative power of, 
579. 

measure, U. S. A., 60. 
Liquids, specific gravrty of, 285. 
Lloyd's boiler specifications, 611-616. 

proportions for riveted joints, 612. 

rules for safety valves, 635. 
Loads, maximum, 359. 

on boiler stays, 616. 

on wooden beams, 408, 409. 
Locomotive boiler, heating surface 
of, 604. 

boilers, 603. 

data, 794, 795. 
Locomotives, tractive power of, 795. 



Index. 



819 



Logarithmic angular functions, 179- 


Man-holes, boiler, 631. 


223. 


Manila-rope driving, 479. 


cosecants, 179-223. 


formulas for, 479. 


cosines, 179-223. 


Marble, coefficient of expansion of, 


cotangents, 179-223. 


486. 


secants, 179-223. 


Marine electric work, 730-733. 


sines, 179-223. 


type connecting-rod, 444. 


tangents, 179-223. 


Mariotte's law, 514. 


Logarithms, 79. 


Marshall's valve gear, 680. 


hyperbolic, 639-643. 


Materials, average strength of, 411- 


of numbers, 80. 


415. 


special, 101. 


for boilers, 628. 


table of, 82-101. 


for boiler shells, 613. 


use of, 79. 


for boiler stays, 616. 


Log for engine test, 670. 


for riveting, 611. 


Loire coals, heating value of, 575. 


of engineering, 282. 


Long connecting rods, 413. 


strength of, 345. 


Long-distance transmission, electric, 


ultimate strengths of, 412-415. 


734. 


weight of, 286-310. 


Long furnaces, strength of, 615. 


Mathematics, 5. 


Losses in belt transmission, 472. 


Matthiessen's standard of conduc- 


Loss in steam pipes, entrance, 591. 


tivity, 702. 


of heat by radiation, 497. 


Maximum efficiency of steam en- 


of heat through walls, 499. 


gine, 650. 


of steam pressure in pipes, 590. 


loads, 359. 


of water-head by friction, 533. 


Mean effective pressure, 638, 650. 


Lowell, Mass., water-plant costs, 771, 


effective pressure above atmos- 


772. 


phere, 645. 


Low-potential circuits, grounding, 


effective pressure above vacuum, 


709. 


644. 


electric systems, 714. 


effective pressure, computation of, 


Low temperatures, 490. 


639. 


"Lump" in lathe speeds, 476. 


effective pressure in gas engines, 

688. 
effective pressure, tables of, 644, 


M 


Machine, definition of, 416. 


645. 


design, 416. 


Measurement of coal in engine tests, 


screws, 338. 


660. 


tools, electric power required for, 


of engine dimensions, 657. 


764. 


of feed water, 659. 


tools, group drawing of, 764. 


of heat units, 658. 


tools, motor power for, 750-753. 


of jacket water, 659. 


Magnesia, solubility of carbonate 


of steam consumption, 659. 


of, 637. 


of steam used by auxiliaries, 660. 


Magnesium, specific heat of, 495. 


of water, 538. 


Main bearings, pressure on, 676. 


timber, 321. 


Management of works, 780-787. 


Measures and weights, 58-60 


Manchester, X. H., water-power 


of length, British, 58. 


costs, 771, 772. 


of length, metric, 59. 


Manganese, fusing-point of, 489. 


of length, U.S. A., 58. 


specific gravity of, 284. 


of weight, British, 59. 



820 



Index. 



Measures of weight, metric, 63. 


Moment of inertia, 269, 351, 359. 


of weight, U.S.A., 59. 


of inertia of rectangles, 355. 


Measure, table of board, 322. 


of inertia of standard sections, 356. 


Measuring tanks, 659. 


of inertia, polar, 393. 


Mechanical equivalent of heat, 495. 


of resistance, 351. 


units, electrical equivalents of, 


statical, 242. 


699. 


Moments, static, 267. 


Mechanics, 234. 


Momentum, 269. 


Medium steel, 403. 


Monetary systems, 61, 62. 


steel struts, 372, 373. 


union, Latin, 62. 


Melting-points, 489. 


Monn's laws of friction, 280. 


Mercury, coefficient of expansion of, 


Mons coals, heating value of, 576. 


486. 


Moravian coals, heating value of, 


heat transmission through, 497. 


577. 


latent heat of, 496. 


Morison suspension furnace, 616. 


specific gravity of, 284. 


suspension furnaces, strength of, 


specific heat of, 495. 


618. 


Metal conduits, electric, 723. 


Motion, 259. 


Metals, percentage in alloys, 283. 


accelerated, 259, 271. 


specific gravity of, 284. 


formulas for rotary, 270. 


ultimate strength of, 413, 414. 


retarded, 260, 271. 


Metric areas for safety valves, 635. 


Motor power for machine tools, 750- 


boiler dimensions, 627. 


753. 


conversion tables, 61-75. 


tests, electric, 760-770. 


horse-power, 638. 


Motors, air required for, 506. 


measures of length, 59. 


hydraulic, 570. 


measures of weight, 60. 


installation of electric, 707. 


rules for safety valves, 635. 


insulated frames for electric, 707. 


system, 63-75. 


internal-combustion, 687-698. 


system, steam table, 586, 587. 


switches for electric, 707. 


weight of cast-iron pipe, 300. 


Movement of air, 509. 


weight of copper pipe, 328. 


Moving bodies, force in, 271. 


weight of iron, 292, 293. 


bodies, impact of, 278. 


weight of sheet metal, 295. 


Multiple expansion, 648. 


weight of spheres, 298. 


expansion engines, diagram for, 


weight of wrought-iron pipe, 328. 


649. 


Mils, circular, 701. 


expansion steam engine, 648. 


conversion to centimetres, 704. 


trussed beams, 249. 


Minerals, solubilities of scale-mak- 


trussed roof, 253. 


ing, 637. 


Multiplication table, 5-7. 


Miner's inch, 541. 


Multi-voltage speed control, 756. 


inch, table for, 542. 




Mitre gears, 468. 


N 


Modulus of elasticity, 347. 


Nails, wire, 323. 


section, 351, 359. 


Napier's rule for flow of steam, 589. 


Moisture in coal, determination of, 


National electric code, 704-733. 


598. 


Natural cosecants, 134-178. 


in steam, 591. 


cosines, 134-178. 


in wood, 405. 


cotangents, 134-178. 


Molecular weight of water, 519. 


gas, 580. 


Moment, 269. 


gas, calorific power of, 580. 



Index. 



821 



Natural gas, power costs of, 775. 


Overshot wheel, 545. 


secants, 134-178. 


Overweights allowable on sheared 


sines, 134-178. 


plates, 404. 


tangents, 134-178. 


Oxygen, specific heat of, 496. 


trigonometric functions, 134-178. 




versed sines, 134-178. 


P 


Neutral axis, 351. 


Paderna water-power plant costs, 


Newcastle coal, heating value of, 574. 


771. 


Niagara water-power cost, 772. 


Palladium, coefficient of expansion 


Nickel-aluminum, 791, 792. 


of, 486. 


Nickel, coefficient of expansion of, 


Parabola, 124, 125. 


487. 


area of, 125. 


fusing-point of, 489. 


centre of gravity of, 245. 


Nitrogen, specific heat of, 496. 


length of, 125. 


Non-condensing engines, perform- 


Paraboloid, volume of, 131. 


ance of, 675. 


Paradox, hydrostatic, 566. 


engines, performance of com- 


Pas de Calais coals, heating value of, 


pound, 773. 


575. 


engines, performance of simple, 


Pelton bucket, 548. 


773. 


water-wheel, 545, 548. 


Norway water-power cost, 772. 


water-wheel table, 549, 550. 


Nozzles, discharge from, 527. 


Pendulum, 276, 277. 


Nut locks, 422. 


compound, 276. 


Nuts, standard sleeve, 339. 


length of seconds, 278. 




simple, 276. 





Pennsylvania coals, 578. 


Oblique-angled spherical triangles, 


coals, heating value of, 574. 


227, 228. 


Percentage of ash in coal, 578. 


triangles, 225. 


Performance of simple non-condens- 


Obstructions in pipes, resistance of, 


ing engines, 773. 


567. 


steam engine, 653-655. 


Oil and coal, comparative evapora- 


Perimeter of ellipse, 122. 


tion of, 580. 


value of, 109. 


and coal, relative value of, 580. 


wetted, 529. 


engines, rules for testing, 690-698. 


Phase - displacing apparatus, effi- 


radiation from, 497. 


ciency of, 747. 


Open belts, 476. 


Philadelphia rules for boiler dimen- 


belts for cone pulleys, 473. 


sions, 621. 


channels, flow of water in, 535. 


rules for safety valves, 633. 


hearth basic steel, 349. 


Phosphate of lime, solubility of, 637. 


Operative cost of steam power, 775. 


Physical properties of wood, 406, 407. 


Orifice, discharge of water from, 527. 


Piece-work, 780. 


Oscillation, angle of 276. 


Pillars, strength of wooden, 410. 


centre of, 276. 


Pillow blocks, 434. 


radius of, 269. 


Pinions, 463. 


Outside electrical work, 708-710. 


shrouded, 463. 


lap of valve, 681. 


Pins and nuts, standard, 341. 


wiring, 708. 


Pin steel, 404. 


Overload capacity of electrical appa- 


Pipe bends, 333. 


ratus, 747. 


bends, resistance in, 567. 


Overshot water-wheel, 545, 548. 


cast-iron, 307-310. 



822 



Index. 



Pipe connections, cast-iron, 310. 


Plates for flanging, 615. 


double extra strong, 325. 


overweights on sheared, 404. 


exhaust, 676. 


proportions for flat, 614. 


extra strong, 325. 


Platinum, coefficient of expansion 


fittings, cast-iron, 331. 


of, 486, 487. 


fittings, dimensions of, 333, 334. 


fusing-point of. 489. 


fittings, steel, 332. 


specific gravity of, 284. 


flow, coefficients for, 529. 


specific heat of, 495. 


riveted hydraulic, 327. 


Point of cut-off, 663. 


spiral riveted, 326. 


of release, 663. 


steam, 676. 


Polar moment of inertia, 393. 


threads, U. S. standard, 306. • 


valve diagram, 679. 


threads, Whitworth standard, 306. 


Polygonal roof truss, 253. 


unions, standard, 337. 


Polygon funicular, 237. 


weight of cast-iron, 299. 


Polygons, 107. 


weight of lead, 319. 


table of, 108. 


Pipes, changes in cross-section of, 567. 


Poly tropic curve, 648. 


contents of, 543, 544. 


Poncelet, 638. 


delivery of compressed air through, 


water-wheel, 547. 


508. 


Port areas, 676. 


discharge of water from, 531. 


width, 681. 


flow of steam in, 589, 590. 


Porter governor, 276. 


flow of water through, 529. 


Posts, wooden, 405. 


friction of air in, 510-513. 


Power, 234, 268. 


heating, 499. 


calorific, 572. 


loss of steam pressure in, 590. 


cost of, 771-779. 


pressure of air in, 508. 


factor and inductance factor, 748. 


resistance of obstructions in, 567. 


horse, 638. 


transmission of air through, 504, 


hydraulic transmission of, 569. 


508. 


of water, 525. 


Piston-rod, 676. 


of water-wheels, 545, 546. 


Piston speed, average, 638. 


required for air compressor, 769. 


speed and delivery of pumps, 555- 


required for boring mill, 752. 


557. 


required for brass-working tools, 


Pitch, circumferential, 449. 


768. 


diametral, 450. 


required for charging machine, 


formulas, diametral, 454. 


765. 


of boiler riveting, 611. 


required for cold saw, 760. 


of gear-wheels, 449. 


required for cranes, 701 . 


tables, circular, 455. 


required for drilling machines, 751, 


tables, diametral, 456. 


762. 


Pitches for gear-wheels compared, 


required for electric cranes, 753, 


450. 


766, 767, 770. 


Pivots, proportions of, 427. 


required for emery-wheels, 752. 


Plane figures, areas of, 127, 128. 


required for grindstones, 761. 


Planers, power required for, 750, 760. 


required for hoisting, 751. 


Plate couplings, 437. 


required for lathes, 751-753. 


rolls, power required for, 761, 769. 


required for machine tools, 750- 


springs, 399. 


753, 764. 


tensile strength of boiler, 613. 


required for planers, 750, 760. 


Plates, boiler-tube, 617. 


required for plate rolls, 761, 769. 



Index. 



823 



Power required for punching ma- 


Properties of timber, 405. 


chine, 763. 


of water, 520-523. 


required for roller tables, 754. 


of wood, 406, 407. 


required for shafting, 761. 


Proportions for flat plates, 614. 


required for shears, 760. 


for riveted joints, 418,419, 611, 612. 


required for sheet-metal press, 765. 


for stayed surfaces, 614. 


required for slotter, 752. 


of boiler shells, 613. 


required for tool grinder, 752. 


of chimneys, 605. 


required for wood-working ma- 


of connecting rods, 442. 


chinery, 763, 770. 


of cross-heads, 447. 


table for water, 542. 


of eccentrics, 445. 


transmitted by ropes, 479. 


of gear teeth, 451. 


units of, 269. 


of gear-wheels, 460. 


Powering of steamships, 795-799. 


of pulleys, 471. 


Powers and roots, 24. 


of riveting, 611. 


and roots, table of, 25-55. 


of safety valves, 632. 


Practical engine performances, 675. 


of stay bolts, 620. 


Premature ignition in gas engines, 


of steam engines, 675. 


688. 


of steam-engine cylinders, 647. 


Premium plan of wages, 780. 


Prussian coals, heating value of, 574. 


Press, hydraulic, 566. 


Pulley arms, 471. 


hydrostatic, 566. _^ 


rim, 471. 


Pressure, air, 610. 


Pulleys, 467, 469. 


and temperature of air, 501. 


friction on, 469. 


and volume of air, 502. 


proportions of, 471. 


atmospheric, 514. 


Pull of belts, 478. 


draught, 607. 


Pulsation in electrical apparatus, 745. 


initial, 639. 


Pumping engine, duty of, 560. 


internal, 397. 


engine, leakage test of, 561. 


mean effective, 638, 650. 


engines, standard duty trials of, 


— of air in pipes, 508. 


560-563. 


of water, 525. 


Pumps, 554. 


of water, horizontal, 527. 


air vessels for, 554. 


on dam, 527. 


centrifugal, 568. 


•*-^ tables of mean effective, 644, 645. 


delivery of, 554, 555. 


Pressures for stayed surfaces, 620. 


dimensions of, 554. 


on corrugated furnaces, 617. 


duplex, 559. 


Prevention of corrosion in boilers, 


electric power required for, 763- 


637. 


feed, 559. 


of incrustation in boilers, 636. 


slip of, 554. 


of priming in boilers, 637. 


speed of, 554. 


Priming in boilers, causes of, 637. 


steam, 558, 559. 


in boilers, prevention of, 637. 


Punching machine, power required 


Producer gas, 580. 


for, 763. 


gas, calorific power of, 581. 


Purification of feed water, 635. 


gas, power cost of, 776. 


Purves-Brown corrugated furnace, 


Production order, 783. 


615. 


Progression, arithmetical, 77. 


Purves corrugated furnace, 617. 


geometrical, 77. 


Pyramidic frustum, centre of gravity 


Properties of aluminum, 791. 


of, 246. 


of steam, 582. 


frustum, volume of, 132. 



JA 



824 Index. 



Pyramid, volume of, 131. 
Pyrometer, Le Chatelier, 489. 



Quadrangle, centre of gravity of, 244. 
Quadruple-expansion engines, cylin- 
der ratios for, 649. 
Quality of steam, 598, 662. 

of superheated steam, 591. 
Quantity of heat, 490. 
Quarter-twist belts, 477. 



Rack teeth, involute, 453. 
Radial valve gears, 680, 681. 
Radiation, 497. 

coefficients of, 497. 

increase with temperature, 498. 

loss of heat by, 497. 
Radii of gear-wheels, 449. 
Radius, hydraulic, 529. 

of gyration, 269, 273, 274, 351, 359. 

of gyration of compound shapes, 
358. 

of oscillation, 269. 
Railway power-plants, circuit-break- 
ers for, 707. 

trains, grade resistance of, 794. 

trains, speed resistance of, 794. 
Railways, curve resistance on, 794. 
Ram, hydraulic, 564. 
Rankine cycle, 667. 
Rate of combustion, 607. 
Rating of electrical apparatus, 746. 
Ratio, expansion, 639. 

of expansion, 665. 

of grate to heating surface, 604. 
Ratios for speed cones and back 

gear, 475. 
Reactive coils, 727. 
Reaumur scale, origin of, 482. 

thermometer, 482. 
Reciprocals, table of, 25-55. 
Recording engine tests, 663. 
Records of boiler trials, 598. 
Rectangles, moment of inertia of, 

355. 
Rectifying machines, 737. 

machines, efficiency of, 740. 
Reducing motion for indicator, 661. 



Regulation of electrical apparatus, 

744. 
Relative value of coal and oil, 580. 

value of heating surface, 603. 
Release, point of, 663. 
Report of boiler trial, 601. 

of engine test, 669. 
Resilience, 348. 
Resistance, 282. 

boxes, 706. 

in pipe angles, 567. 

in pipe bends, 567. 

moment of, 351. 

of elbows to steam flow, 591. 

of globe valve to steam flow, 591. 

of insulation, 729, 743. 

of obstructions in pipes, 567. 

of trains, acceleration, 794. % 

on railway curves, 794. 

testing of insulation, 706. 

to bulging, 614. 

to entrance in steam pipes, 591. 

to flow in steam pipes, 590. 
Results of engine tests, forms for, 

670. 
Retarded motion, 260, 271. 
Return tubular boilers, 603. 

tubular boilers, heating surface of, 
604. 
Return wires, ground, 709. 
Reuleaux valve diagram, 678. 
Revolving bodies, 272. 
Reynolds's pumping engine, effi- 
ciency of, 650. 
Rhenish - Prussian coals, heating 

value of, 574. 
Rheostatic speed control, 756. 
Rigg hydraulic motor, 570. 
Right - angled spherical triangles, 
226. 

triangles, 224. 
Rims of bevel gears, 463. 

of gear-wheels, 462. 
Ring system of hydraulic distribu- 
tion, 571. 
Rise of temperature of electrical 

apparatus, 741. 
Rivers, flow of water in, 535-538. 
Rivet bars, ductility of, 611. 

bars, tensile strength of, 611. 

seams, factor of safety of, 631. 






Index. 



825 



Rivet steel, 403. 


Rotary induction apparatus, 737. 


steel, shearing resistance of, 611. 


motion, formulas for, 270. 


Riveted hydraulic pipe, 327. 


Rules for boiler riveting, 611. 


joints, Lloyd's proportions for, 612. 


for boiler trials, 595. 


joints, proportions of, 419, 612. 


for conducting steam-engine tests, 


Riveting, 416. 


656. 


boiler, 419, 630. 


for safety valves, German, 635. 


butt-joint, 418. 


for safety valves, Lloyd's, 635. 


efficiency of, 417. 


for safety valves, metric, 635. 


factor of safety of, 611. 


for safety valves, Philadelphia, 


group, 418, 419. 


633. 


lap-joint, 418. 


for safety valves, U. S. Treasury 


material for, 611. 


Department, 632. 


pitch of boiler, 611. 


for testing gas and oil engines, 690- 


proportions of, 611. 


698. 


rules for boiler, 611. 




Rivets, boiler, 629. 


S 


weight of bridge, 301. 


Saddles, boiler, 631. 


Rods, connecting, 442. 


Safe load, coefficient for, 359. 


Rolling circle for gear teeth, 451. 


loads for cast-iron columns, 391, 


mill tables, power for driving, 754. 


392. 


Roman cement, coefficient of expan- 


loads on wooden beams, 408, 409. 


sion of, 486. 


Safety valves, 632. 


Roof, double-trussed, 252. 


valves, area of, 633-635. 


multiple-trussed, 253. 


valves, Board of Trade areas, 634. 


simple, 250. 


valves, German rules for, 635. 


single-trussed, 251. 


valves, lever, 632. 


trusses, wind stresses on, 254, 255. 


valves, Lloyd's rules for, 635. 


truss, polygonal, 253. 


valves, metric areas for, 635. 


Roofing slate, 317. 


valves, metric rules for, 635. 


weight of, 321. 


valves, Philadelphia rules for, 633. 


Roots and powers, 24. 


valves, spring, 633. 


and powers, table of, 25-55. 


Salt-well water, 638. 


extraction of, 55. 


Sampling coal, 598. 


Rope brake, 662. 


Sand, radiation from, 497. 


driving, American system, 480. 


Saone coals, heating value of, 575. 


driving, English system, 480. 


Saturated steam, 588. 


driving, Manila, 479. 


Sawdust, radiation from, 497. 


power transmitted by, 479. 


Saw-mill water-wheel, 548. * 


sheaves, 478. 


Saxon coals, heating value of, 574. 


sheaves, grooves for, 480. 


Scale, analyses of boiler, 637. 


standard wire hoisting, 342 


in boilers, 635. 


steel wire, 344. 


making minerals, solubilities of, 


transmission, 478. 


637. 


transmission, cotton, 481. 


Scales, conversion of thermometer, 


transmission, hempen, 469. 


482. 


transmission, sag in, 479. 


Schmid hydraulic motor, 570. 


transmission, wire, 469, 478. 


Scotch boiler, heating surface of, 604. 


weight of Manila, 479. 


boilers, 603. 


wire haulage, 343. 


coal, heating value of, 574. 


wire transmission, 343. 


Screw stay bolts, 614. 



Index. 



Screw threads, U. S. standard, 304. 


Sheet zinc, weight of, 294. 


threads, Whitworth, 305. 


Sheets, boiler tube, 617. 


Screws, lag, 324. 


Shells, materials for boiler, 613. 


machine, 338. 


proportions of boiler, 613. 


set, 338. 


Shingles, number and weight of, 320. 


wood, 324. • 


Short connecting rods, 443. 


Seasoned timber, strength of, 406, 407. 


furnaces, strength of, 615. 


wood, strength of, 405. 


Shrouded pinions, 463. 


Secants, logarithmic, 179-223. 


Sickel roof truss, 253. 


natural, 134-178. 


Signalling systems, electric, 728. 


Seconds, pendulum, length of, 278. 


Silesian coals, heating value of, 575, 


Section modulus, 351, 359. 


577. 


Sections, elements of structural, 353. 


Silver, coefficient of expansion of, 


moments of inertia of standard, 


486, 487. 


356. 


fusing-point of, 489. 


torsion, 393. 


heat transmission through, 497. 


Sector, 128. 


latent heat of, 496. 


centre of gravity of, 245. 


radiation from, 497. 


of sphere, 130. 


specific gravity of, 284. 


Segment, centre of gravity of, 245. 


specific heat of, 495. 


Semi-circular surface, centre of grav- 


Simple condensing engines, perform- 


ity of, 245. 


ance of, 773. 


Series, 77. 


interest, 56. 


arc lighting, 712. 


pendulum, 276. 


Set screws, 338. 


trussed beams, 247. 


Shafting, 428. 


Simpson's rule, 129. 


power required for, 761. 


Sines, logarithmic, 179-223. 


stiffness of, 429. 


natural, 134-178. 


strength of, 429. 


Single-riveted joints, 612. 


table for, 430. 


joints, strength of, 611. 


wrought-iron, 431. 


Sizes of engine parts, 676. 


Shafts, diameter of engine, 676. 


Skylight glass, 320. 


deflection of, 432. 


Slack side of belts, 478. 


graphical analysis of, 433. 


Slate, 317. 


hollow, 431. 


Sleeve couplings, 436. 


Sheared plates, overweight, 404. 


nuts, standard, 339. 


Shearing, 346, 350. 


Slide valve, 681. 


resistance of rivet steel, 611. 


valve, dimensions of, 681, 682. 


Shears, power required for, 760. 


valve engine, performances of, 675. 


Sheaves, chain, 315. 


valve gear, 678. 


for wire rope, 478. 


Slip of pumps, 554. 


grooves for rope, 480. 


Slopes, hydraulic, 534. 


Sheet copper, weight of, 294, 295. 


Slow-burning insulation, 721. 


iron gauge, U. S. standard, 314. 


Slotter, power required for, 752. 


iron, weight of, 294-297. 


Smoke observations, 600. 


lead, weight of, 294, 295. 


Sockets, electric lamp, 725. 


metal, British weight of, 294. 


for incandescent lamps, 716. 


metal, metric weight of, 295. 


Soft steel, 403. 


metal press, power required for, 765. 


Soldering fluid, 730. 


steel gauge, U. S. standard, 314. 


Solids, surfaces of, 129. 


steel, tinned, 318. 


volumes of, 130-132. 



Index. 



827 



Solubilities of scale-making miner- 


Springs, spiral, 400. 


als, 637. 


supporting power of, 399-^02. 


Sparking distances, 749. 


testing indicator, 661. 


Special bolt threads, 421. 


Spring safety valves, 633. 


logarithms, 104. 


water, 638. 


Specifications, American boiler, 628- 


Square-headed bolts, weight of, 302. 


632. 


Square roots, table of, 25-55. 


for structural steel, 403. 


threads, strength of, 421. 


Specific gravity, 282. 


Squares, table of, 25-55. 


gravity hydrometer, 565. 


Standard boiler tubes, 619. 


gravity tables, 284, 285. 


cast-iron pipe, 307-309. 


heat, 495. 


coal unit, 656. 


heat of gases, 496. 


engine tests, 655. 


heat of water, 519. 


flanges, 329. 


heat, table of, 495. 


heat test of steam engine, 658. 


Speed changes by cone pulleys, 474. 


pins and nuts, 341. 


changes for lathes, 476. 


pipe unions, 337. 


cones, 472. 


*screw threads, U. S., 304. 


cones and back gear, 475. 


screw threads, Whitworth, 305. 


control, multi- voltage, 756. 


sleeve nuts, 339. 


control, rheostatic, 756. 


steel eye bars, 340. 


counters, 663. 


temperatures, 482. 


of pumps, 554. 


thermal efficiency for engines, 668. 


of steamships, 795-799. 


Standardization, electrical, 736-749. 


resistance of trains, 794. 


Starting and stopping boiler trials, 


steam-engine, 662. 


597. 


variation of electric motors, 756. 


and stopping steam-engine test, 


Sphere sector, volume of, 131. 


658. 


segment, volume of, 131. 


Statical moment, 242, 267. 


surface of, 129. 


Statics, 234. 


volume of, 130. 


of framed structures, 247. 


Spheres, metric weight of, 298. 


Station conductors, 705. 


volume table, 132. 


Stations and dynamo rooms, 705. 


weight of, 297, 298. 


Stay bolts, boiler, 629. 


Spherical segment, centre of gravity 


bolts, factor of safety of, 631. 


of, 245. 


bolts, proportions of, 620. 


triangles, oblique-angled, 227, 228. 


bolts, screw, 614, 631. 


triangles, right-angled, 226. 


girders, 616. 


Spikes, wire, 323. 


girders, boiler, 616. 


wrought, 323. 


Stayed surfaces, pressures for, 620. 


Spiral-gear formulas, 460. 


surfaces, proportions of, 614. 


Spiral gears, 459. 


Stays, boiler, 614, 616. 


riveted pipe, 326. 


loads on boiler, 616. 


springs, 400. 


tensile strength of, 616. 


Splines, 423. 


Steam, 581-686. 


standard, 424. 


accounted for by indicator, 663. 


Springs, 398. 


boilers, 592. 


deflection of, 399-402. 


boiler details, 611. 


elasticity of, 399-402. 


consumption determined by indi- 


helical, 401. 


cator, 651. 


plate, 399. 


consumption, measurement of, 659. 






S28 



Index. 



Steam consumption, theoretical, 651. 


Steam engine, standard heat test of, 


cylinder proportions, 647. 


658, 


cylinders, bolts for, 420. 


engine test, duration of, 657. 


engine (see Engines). 


engine test, report of, 669. 


engine, 638-686. 


engine tests, 655, 773. 


engine and boiler test, complete, 


engine tests, forms for, 670. 


658. 


engine tests, measurement of coal 


engine capacity, 656. 


in, 660. 


engine, clearance in, 646, 


engine tests, recording, 663. 


engine code, A. S. M. E., 656. 


engine tests, rules for, 656. 


engine, compound, 648. 


engine test, starting and stopping 


engine, cylinder ratios for com- 


of, 658. 


pound, 648. 


engine, thermal efficiency of, 650. 


engine, cylinder ratios for quad- 


engine, water consumption of, 651- 


ruple-expansion, 649. 


655. 


engine, cylinder ratios for triple- 


expansion of, 639. 


expansion, 649. 


flow in pipes, 589, 590. 


engine, diagram for multiple-ex- 


flow of, 588. 


pansion, 649. 


loss of pressure in pipes, 590. 


engine, economical point of cut- 


moisture in, 591. 


off, 646. 


passages, 677. 


engine economy, standard of, 665. 


pipe, 676. 


engine efficiency, standard of, 


pipe diameters, 677. 


665. 


pipes, entrance loss in, 591. 


engine, examination of plant, 656. 


pipe, wrought-iron, 306. 


engine, feed-water test of, 672. 


plant, commercial test of, 657. 


engine, heat analysis of, 666. 


plant installation costs, 774. 


engine, heat-unit standard for, 656. 


port areas, 676. 


engine indicator, 650. 


port, width of, 681. 


engine, leaks in plant, 657. 


power, cost of, 772-775. 


engine, maximum efficiency of, 


properties of, 582. 


650. 


pumps, 558, 559. 


engine, measurement of dimen- 


quality of, 598, 662. 


sions of, 657. 


resistance of elbows to flow, 591. 


engine, multiple-expansion, 648. 


saturated, 588. 


engine performance, 653-655. 


specific heat of, 496. 


engine, performance of automatic, 


superheated, 588, 591. 


675. 


table, British system, 583-585. 


engine, performance of compound, 


table, metric system, 586, 587. 


675. 


tables, introduction to, 582. 


engine, performance of condens- 


used by auxiliaries, measurement 


ing, 675. 


of, 660. 


engine, performance of non-con- 


used by injector, 596. 


densing, 675. 


velocity of efflux, 589. 


engine, performance of slide valve, 


Steamships, powering of, 795-799. 


675. 


speed of, 795. 


engine, pressure on wearing sur- 


Steel, 349. 


faces of, 675. 


chemical properties of, 403. 


engine proportions, 675. 


coefficient of expansion of, 486, 487. 


engine, sizes of parts of, 676. 


eye bar, 404. 


engine speed, 662. 


eye bars, standard, 340. 






Index. 829 


Steel for pins, 404. 


Stresses in gear teeth, 457, 461. 


for rivets, 403. 


Stroke, economical length of, 647. 


furnace charging machine, power 


Structural shapes, elements of, 359. 


for, 765. 


steel, specifications for, 403. 


rt fusing-point of, 489. 


steel, strength of, 349. 


heat transmission through, 497. 


Struts, 368. 


medium, 403. 


hard steel, 374, 375. 


pipe fittings, 332. 


medium steel, 372, 373. 


shearing resistance of rivet, 611. 


wrought- iron, 370, 371. 


soft, 403. 


Stub end for connecting rod, 445. 


specifications for structural, 403. 


Styrian coals, heating value of, 577. 


specific gravity of, 284. 


Sub-production order, 783. 


struts, 372-375. 


Sulphate of lime, solubility of, 637. 


test pieces, 403. 


Sulphur dioxide, latent heat of, 496. 


wire, 345. 


specific gravity of, 284. 


wire rope, 344. 


specific heat of, 495. 


wire, weight of, 313. 


Sulphuric acid, coefficient of expan- 


Stephenson's link motion, 679. 


sion of, 486. 


Stiffened furnaces, 615. 


acid in feed water, 636. 


Stiffness, 348. 


acid, specific heat of, 495. 


of belts, 472. 


Superheated steam, 588, 591. 


of shafting, 429. 


Supplementary thermometer tables, 


Stones, specific gravity of, 284. 


485. 


Stone, strength of, 415. 


Supporting boilers, 632. 


Storage batteries, 708. 


power of springs, 399^02. 


Straight link motion, 680. 


Surface condenser, 685. 


Strain and stress, 345. 


condenser, Wheeler, 686. 


Strands of copper wire, 700. 


Surfaces of solids, 129. 


Strap end for connecting rods, 443. 


Suspension furnace, Morison, 616. 


Straps, eccentric, 446. 


furnaces, strength of, 618. 


Streams, measurement of water in, 


Switchboards, 706. 


539. 


Switches, electric, 724. 


Strength of brick, 415. 


for motors, 707. 


of cements, 415. 


general rules for, 712. 


of corrugated furnaces, 617. 


installation of, 713. 


of double-riveted joint, 611. 


Synchronous commutating electrical 


of gear teeth, 457. 


machines, 737. 


of materials, 345. 


commutating machines, efficiency 


of materials, average, 411-415. 


of, 739. 


of metals, ultimate, 413, 414. 


electrical machines, 737. 


of shafting, 429. 


machines, efficiency of, 739. 


of single-riveted joints, 611. 


System of bodies, centre of gravity 


of square threads, 421. 


of, 246. 


of steel wire, 345. 




of stones, 415. 


T 


of suspension furnaces, 618. 


Tangents, logarithmic, 179-223. 


of timber, 405. 


natural, 134-178. 


of trapezoidal threads, 421. 


Tank measurement, table for, 545. 


of wooden beams, 408, 409. 


Tanks, measuring, 659. 


of wooden pillars, 410. 


Tee sections, moment of inertia of, 


Stress and strain, 345. 


357. 



830 



Index. 



Tees, elements of, 379. 


Thermal value, 572. 


Teeth, epicycloidal, 451. 


Thermometer, centigrade, 482. 


internal, 451. 


conversion tables, 483-485. 


involute, 451. 


Fahrenheit, 482. 


of wheels, 449. 


Reaumur, 482. 


proportions of, 451. 


Thermometers, 482. 


rack, 453. 


Thermometer scales, conversion of, 


Temperature, 481. 


482. 


and pressure of air, 501. 


tables, supplementary, 485. 


and volume of air, 501. 


Thick cylinders, 398. 


barometric corrections for, 516-518. 


Throttling calorimeter, 591. 


entropy diagram, 666. 


Thrust, 368. 


of feed water, 659. 


bearings, 428. 


Temperatures, standard, 482. 


bearings, friction in, 281. 


Tensile strength of boiler plate, 613. 


Thumb-shaped teeth, 457. 


strength of rivet bars, 611. 


Timber, 405. 


strength of stays, 616. 


elastic limit of, 405. 


Tension, 346, 348. 


measurement, 321. 


of belting, 468. 


properties of, 406, 407. 


on wire rope, 479. 


ultimate strength of, 412. 


Test, complete engine and boiler, 658. 


Time, 234, 268. 


duration of engine, 657. 


Tin, coefficient of expansion of, 486, 


of steam plant, commercial, 657. 


487. 


of steel eye bars, 404. 


f using-point of, 489. 


report of engine, 669. 


heat transmission through, 497. 


starting and stopping steam engine, 


plates, 318. 


658. 


radiation from, 497. 


Testing, gas-engine, 690-698. 


specific gravity of, 284. 


indicator springs, 661. 


specific heat of, 495. 


of insulation resistance, 706. 


Tinned sheet steel, 318. 


Tests, electric motor, 760-770. 


Tool grinder, power required for, 752. 


for boiler tubes, 629. 


Torsion, 346, 393. 


for stay bolts, 629. 


sections, 393. 


insulation, 720. 


table, 395, 396. 


measurement of coal in, 660. 


Torus, surface of, 129. 


of steam engines, forms for, 670. 


volume of, 130. 


of wooden posts, 405. 


Toughness, 282. 


recording engine, 663. 


Tower's experiments on friction, 281. 


rules for conducting steam-engine, 


Tractive power of locomotives, 795. 


656. 


Trains, grade resistance of, 794. 


steam-engine, 655, 773. 


speed resistance of, 794. 


Theoretical discharge of water, 527. 


Transfer of heat, 481. 


velocity of water, 528. 


Transformation of arm sections, 440. 


steam consumption, 651. 


Transformers, 727. 


Thermal efficiency curves, 668. 


installation of, 708, 718. 


efficiency of heat motors, 649. 


location of, 709. 


efficiency of steam engines, 650, 


Transmission, belt, 467. 


668. 


gears, 460. 


unit, British, 490. 


long-distance electric, 734. 


units, 490, 572. 


of power, hydraulic, 569. 


units to calories, table, 491. 


rope, 478. 



Index. 831 


Transmission, wire-rope, 343. 


Undershot stream wheel, 546. 


Transylvanian coals, heating value 


wheel, 545. 


of, 577. 


Unions, standard pipe, 337. 


_ 7 Trapezium, 128. 


Unit, Board of Trade, 700. 


Trapezoid, 128. 


equivalents for electric heating, 


Trapezoidal threads, strength of, 421. 


801. 


Treble-riveted joints, 612. 


standard coal, 656. 


Trial, report of boiler, 601. 


United States coal, heating value of, 


Trials of pumping engines, 560. 


574. 


Triangle, centre of gravity of, 244. 


States standard pipe threads, 306. 


Triangles, areas of, 128. 


States standard screw threads, 


formulas for, 224-228. 


304. 


oblique-angled, 225. 


States statutes, boiler specifica- 


right-angled, 224. 


tions, 611-616. 


Trigonometrical formulas, 229. 


States Treasury rules for safety 


Trigonometric tables, 133-223. 


valves, 632. 


Trigonometry, 132-229. 


Units, heat, 490. 


Triple-expansion engines, cylinder 


of power, 269. 


ratios for, 649. 


of work, 269. 


engines, performance of, 773. 


Universal joint, 438. 


Triple-trussed beams, 249. 




Trolley wires, 708. 


V 


Troubles due to water in boilers, 636. 


Vacuum, 684. 


Troy weight, 60. 


Valenciennes coals, heating value of, 


Trussed beams, double, 248. 


576. 


beams, multiple, 249. 


Value, thermal, 572. 


beams, simple, 247. 


Valve action shown by indicator, 


beams, triple, 249. 


651. 


roof, double, 252. 


Allen, 682. 


roof, single, 251. 


diagram, polar, 679. 


Trusses, bridge, 257, 258. 


diagram, Reuleaux, 678. 


Truss, polygonal roof, 253. 


diagram, Zeuner, 679. 


Tube holes, boiler, 630. 


gear, 678. 


plates, boiler, 617. 


gear, Angstrom's, 680. 


setting, boiler, 631. 


gear, Brown's, 680. 


Tubes, boiler, 619, 629. 


gear, Marshall's, 680. 


Turbine, 551-554. 


gear, radial, 680. 


Fourneyron, 551. 


Valves, angle, 335. 


Francis, 552. 


butterfly, 335. 


Jonval, 552. 


check, 335. 


Turpentine, latent heat of, 496. 


dimensions of, 335. 


specific heat of, 495. 


gate, 335. 


Typical indicator diagram, 650. 


globe, 335. 




resistance to flow of water in, 


U 


568. 


Ultimate strength, 348. 


safety, 632. 


strength of metals, 413, 414. 


slide, 681. 


strength of timbers, 412. 


Vaporization, latent heat of, 496. 


strengths of materials, 412-415. 


Variable speed control, electric, 757. 


Underground conductors, rules for, 


Variation in electrical apparatus, 


711. 


745. 



832 



Index. 



Velocities and heads of water, 528. 


Water gas, calorific power of, 581. 


Velocity, 234, 268. 


head, loss by friction in, 533. 


of discharge from orifice, 527. 


heads and air pressures, 609, 610. 


of escape of compressed air, 504. 


heads and pressures, 525. 


Versed cosines, natural, 134-178. 


heads and velocities, 528. 


sines, natural, 134-178. 


heat units in, 520-523. 


Vertical water-wheels, 545. 


horizontal pressure of, 527. 


Virginia coal, heating value of, 574. 


horse-power of, 525. 


Voltages, 743. 


horse-power, table for, 542. 


and frequencies, classified, 746. 


impurities in feed, 635. 


Volume and density of water, 524. 


Kutter's formula for flow of, 535- 


and pressure of air, 502. 


538. 


and temperature of air, 501. 


latent heat of, 496. 


and weight of air, 500. 


measurement by weir, 539. 


measures of, 60, 61. 


measurement of, 538. 


Volumes and velocities of air, 608, 


measurement of feed, 659. 


609. 


measurement of jacket, 659. 


of solids, 130-132. 


molecular weight of, 519. 


of spheres, table, 132. 


Monongahela River, 638. 




plant costs, 771, 772. 


W 


power, cost of, 771. 


Wage systems, 780. 


power, installation costs of, 771. 


Walls, loss of heat through, 499. 


power, operative costs of, 772. 


Walschaert's link motion, 680. 


pressure of, 525. 


Washers, boiler, 614. 


properties of, 520-523. 


Water, 519. 


purification of feed, 635. 


Allegheny River, 638. 


radiation from, 497. 


analyses of boiler feed, 638. 


salt-well, 638. 


Bazin's formula for flow of, 535. 


specific heat of, 519. 


boiler feed, 635. 


spring, 638. 


brake, 662. 


sulphuric acid in feed, 636. 


calcium carbonate in feed, 635. 


table for horse-power of, 542. 


calcium sulphate in feed, 635. 


temperature of feed, 659. 


chemical composition of, 519. 


theoretical discharge of, 527. 


Ch6zy formula for, 529. 


tube boiler, heating surface of, 604. 


coal-mine, 638. 


tube boilers, 603. 


coefficient of expansion of, 486. 


volume of, 520-523. 


columns and air pressures, 610. 


wheel, breast, 547. 


consumption of steam engines, 651- 


wheel, overshot, 548. 


655. 


wheel, Pel ton, 548. 


consumption table, 652. 


wheel, Poncelet, 547. 


density and volume of, 524. 


wheels, 545-554. 


density of, 519. 


wheel, saw-mill, 548. 


discharge from orifice, 527. 


wheels, power of, 545, 546. 


discharge from pipes, 531. 


wheels, vertical, 545. 


flow, Bazin's formula for, 535. 


wheel table, Pelton, 549, 550. 


flow in open channels, 535. 


wheel, undershot, 546. 


flow, Kutter's formula for, 535- 


Wear, coefficient of, 462. 


538. 


on gear teeth, 462. 


flow of, 529-541. 


Wearing surfaces of engines, 675. 


flow through pipes, 529. 


Weather-proof insulation, 721. 



Index. 



833 



Wedge frustum, 132. 

Weight and volume of air, 500. 

measures of, 59. 

of aluminum, 791. 
L of bar iron, metric, 292, 293. 

of bolts, 302, 303. 

of bridge rivets, 301. 

of cast-iron pipe, 299. 

of copper pipe, metric, 328. 

of flagging, 320. 

of flat rolled iron, 286-289. 

of fly-wheels, 676. 

of lead pipe, 319. 

of Manila rope, 479. 

of materials, 286-310. 

of roofing, 321. 

of round bar iron, 290, 291. 

of sheet-copper, 294, 295. 

of sheet-iron, 294-296. 

of sheet-lead, 294-295. 

of sheet-metal, British, 294. 

of sheet-metal, metric, 295. 

of sheet-zinc, 294. 

of shingles, 320. 

of spheres, 297, 298. 

of square bar iron, 290, 291. 

of steel wire, 345. 

of water, molecular, 519. 

of water, tables, 520-523. 

of wire, 313. 

of wood, 578. 

of wrought-iron pipe, metric, 328. 
! Weights and measures, 58-60. 
i Weir, measurement of water by, 539. 

measurement, table for, 540. 
Welsh coals, heating value of, 574. 
j Wetted perimeter, 529. 
i Wheeler surface condenser, 686. 
i W T hitworth standard pipe threads, 
306. 

standard screw threads, 305. 
Width of belts, 468. 

of steam port, 681. 
Window glass, 316. 
Wind stresses, 254, 255. 
Wire, conduit, 722. 

copper, 702. 

gauges, 312. 

haulage rope, 343. 

nails, 323. 

rope, extra strong, 344. 



Wire rope, standard hoisting, 342. 

rope, tension on, 479. 

rope transmission, 469, 478. 

spikes, 323. 

strength of steel, 345. 

table of copper, 702. 

transmission rope, 343. 

weight of, 313. 

weight of steel, 345. 
Wires, carrying capacity of, 711. 

general rules for electrical, 710. 

return ground, 709. 

trolley, 708. 
Wiring, car, 717. 

for constant-current systems, 712. 

for extra high-potential systems, 
719. 

for high-potential systems, 718. 

for low-potential systems, 714. 

formulas, electric, 733-736. 

marine electric, 730. 

outside, 708. 
Wood, elastic limit of, 405. 

heating value of, 578. 

moisture in, 405. 

physical properties of, 406, 407. 

screws, 324. 

strength of, 405-407. 

weight of, 578. 
Wooden beams, 405. 

beams, strength of, 408, 409. 

gear teeth, 462. 

pillars, strength of, 410. 
Woods, specific gravity of, 284. 
Wood-stave pipe, water flow in, 529. 
Wood-working machinery, electric 
group driving of, 770. 

tools, power required for, 763, 770. 
Work, 234, 268, 269. 

required to compress air, 505. 

units of, 269. 
Working pressures for boilers, 622- 
626. 

pressures on corrugated furnaces, 
617. 

stresses on gear teeth, 457. 
Workmanship on boilers, 629. 
Works management, 780-787. 
Worthington jet condenser, 685. 

steam pumps, 559. 
Wrenches, 422. 



53 



834 



Index. 



Wrought-iron pipe, metric weight of, 


Zero, absolute, 489. 


328. 




Zeuner diagram for Allen valve, 


pipe, water flow in, 529. 




682. 


steam pipe, 306. 




valve diagram, 679. 


struts, 370, 371. 




Zinc, coefficient of expansion of, 486. 


Wrought spikes, 323. 




487. 
f using-point of, 489. 


Y 




heat transmission through, 497. 


Yield point, 347. 




latent heat of, 496. 


Yorkshire coal, heating value of, 574. 


specific gravity of, 284. 






specific heat of, 495. 


Z 




weight of sheet, 294. 


Z-bar columns, elements of, 


378. 


Zurich water-power plant costs, 


bars, elements of, 376, 377. 




771. 



THE END. 



A SYSTEM OF ELECTRIC DRIVE FOR MACHINE 

TOOLS, WITH METHODS OF VARIABLE 

SPEED CONTROL. 

(As furnished by the Crocker- Wheeler Company.) 

Besides eliminating the disadvantages of line shafting, belting, and the 
inflexibility of location, the individual drive of machine tools by electric 
motors increases the efficiency and output of a machine shop. The ordi- 
nary belt-driven tool usually has a speed range obtained by mechanical 
means of from 20 : 1 to 50 : 1, with increasing speed steps of about 30 to 50 
per cent. The Crocker- Wheeler system for the multiple- voltage operation 
of machine shops not only extends the speed range, but also reduces the 
speed increment per step to about 10 per cent., which has been found by 
experience to be as small an amount as would be desirable to use. This 
system is a method of electric-power distribution at different voltages, 
which enables standard motors to be operated at variable speed by chang- 
ing the potential of the current at their terminals. The generating plant 
supplies the highest voltage of the system. This voltage may be termed 
the primary, and is divided by a 3-unit balancing transformer into three 
unvarying voltages of unequal value, which are maintained between the 
wires of a 4- wire circuit, various connections of which afford six different 
and distinct voltages. 

The principle on which this system of speed control is based is that in 
a separately-excited shunt motor the speed of the armature is proportional 
to the voltage supplied to its terminals. If this voltage remains constant, 
the speed will remain constant even with varying load. 

It is the function of the balancer to maintain these voltages constant 
and to accommodate the unbalance of currents between the four wires of 
the distribution circuit. 

As the conditions of machine-tool operation will result in the various 
motors of the system being nearly equally distributed on the various cir- 
cuits, the unbalanced current will be but a small percentage of the total 
current taken by all the motors. 

The intermediate wires of the system are extended to the variable speed 
motors only, the constant-speed and crane motors and the lighting being 
supplied in the usual manner from the outside wires at the generator 
voltage. 

Those motors requiring variable speed are connected to the 4-wire cir- 
cuit by means of a controller of the drum type adapted for mounting on 
the tool in a place convenient for the operator. The action of this con- 
troller is such that, as the drum revolves, the armature terminals of the 
motor are connected to the six circuits — afforded by this system — in the 
proper sequence, and the travel of the drum from one position to the next 
is so quickened by the action of a spring that contacts are made and 
broken at a high rate of speed, preventing the formation of arcs and elimi- 
nating the possibility of the drum stopping between contacts. This gives 
six fundamental motor speeds, which are subj ect to a further refinement 
by varying the motor's field strength sufficiently to cover the gaps between 
them. 

The speed range obtained on the voltage points alone is 6 : 1, being pro- 
portional to the ratio of maximum to minimum voltages. The addition of 
field-resistance points above the highest voltage points extends the total 
range in the controller to a value of 10 : 1, For exceptional cases the range 

1 




I 



may be increased to a maximum of 12 : 1, the proper range in any case 
being determined by the character of the machine tool and the work 
which it performs. 

The Crocker-Wheeler system, as outlined, has certain positive advan- 
tages, of which the most important are the following : 

_— , 1. Variable speed, under instant control, over any range. 

2. Every speed constant, regardless of the load. 

3. Controllers simple and convenient of attachment. 

4. The horse-power of the motor but slightly in excess of that re- 

quired by the tool. 

5. Output of machine tools much greater than when they are belt- 

driven. 

6. Easy of adaptation to existing shops with 2-wire system of electric- 

power distribution. 

7. Employment of standard motors. 

8. Ability to maintain high cutting speeds due to superior facilities 

for manipulation. 

Motors used in an ordinary shop equipment may be divided into classes 
A, B, C, or D, according to the nature of their duty. 

Class A being constant-speed motors, such as drive groups of small tools 
by shafting. 

Class B, controllable-speed motors, generally of the series-wound type, 
as used on cranes. 

The duty which the motors in both these classes have to perform is such 
that their demand for current is intermittent and often excessive, conse- 
quently they are best suited for connection to the outside mains, and such 
speed regulation as they may require can be obtained by rheostatic con- 
trol. 

The other two classes, C and D, are controllable-speed motors for the 
drive of individual tools, where the speed should be maintained constant 
at any one of a number of fixed values. 

Class C is formed of motors driving pressure blowers, punch presses, 
planers, etc., which demand approximately constant torque at all speeds, 
the horse-power diminishing with the speed. This characteristic of the 
tool being identical with the power characteristic of the motor on this 
system, the normal horse-power of the motor need not be greater than the 
maximum demanded by the tool. 

Class D covers those motors operating lathes, boring mills, etc., where 
the torque increases as the speed diminishes. If the range required by 
these tools is to be obtained by using a motor through its maximum range, 
the motor would be very large and unnecessarily expensive. For this class 
a speed range of approximately 3 : 1 has been selected as a basis for the 
determination of the most suitable sizes of motors, with respect to the 
duty which they have to perform. A motor, therefore, to give a constant 
horse-power throughout this range, must -have a normal rating of about 
twice the horse-power required by the tool. This range, however, may be 
extended to cover the entire range required by the tool by using one or 
more additional gear runs. The method is an advantageous compromise 
between the use of an excessively large motor with no gears and the 
constant-speed motor with many gears. 

The extreme facility of manipulation which this system affords enables 
the machinist to push his tool to the highest limit of cutting speed and 
gives large increases in output. Results show that as much as 20 per cent, 
increase in output over a belt-driven tool may be obtained by this system 
of motor drive. As by actual test in commercial plants it has been shown 
that 2% per cent, increase is sufficient to warrant the outlay necessary for 
individual drive, the possibility of large saving in operating expense is at 
once apparent. 



IF YOUR ENGINEER 

TELLS YOU THAT A 

GREEN'S 

ECONOMIZER 

In your boiler-room will produce the greatest 
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kept from the boiler, beside saving from 10 
to 20 per cent, on coal bill. Ask for our booklet. 

THE GREEN FUEL ECONOMIZER CO. 

Sole Manufacturers in the U.S.A. MATTEAWAN, N. Y, 

BALDWIN LOCOMOTIVE WORKS 

BROAD AND NARROW GAUGE 
SINGLE EXPANSION AND COMPOUND 

LOCOMOTIVES 



, 0> * 




Mine, Furnace, and Industrial Locomotives 

Electric Locomotives with Westinghouse Motors 

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BURNHAM, WILLIAMS & CO., ^ l n a a del uTa a 



AMERICAN VACUUM 
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NEW YORK OFFICE, J20 LIBERTY ST, 



AIR COMPRESSORS 



STEAM, BELT, OR ELECTRIC DRIVEN 



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™ INGERS0LL-SER6EANT 

26 Cortlandt Street, New York 



X 

DRILL 
CO. 



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FOR HEAVY LIFTING 




a compact powerful Tri- 
plex Block enables a man 
to handle a load that five 
men cannot handle with 
the old slow and danger- 
ous hoisting methods. 

Compare the Triplex 

with any other block and 

it gives double speed at the 

same pull. 



TO COMPARE SPEEDS: 

start both hooks together 
at the floor and pull 
equally on both hand 
chains ; note the rise of 
the hooks. 

TO DETERMINE EASE : 

hoist the same load with 
one block then with the 
other; note ease of pulling 
with the Triplex. 

We make three types 
of Blocks, each the best 
for its purpose. Write 
(Try This) us an( j we w [\\ send you 

a book containing some valuable hoisting hints. 

We send Blocks for trial use, and pay the freight 
if not kept. )4 to 2 ° tons > an y height of hoist. 

THE YALE & TOWNE MFG. CO., 

GENERAL OFFICES: 
9-11-13 MURRAY ST., NEW YORK CITY. 



Wt)t 

(Engineering 

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and construction of large 
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THE CONSTRUCTOR 

A HANDBOOK OF MACHINE DESIGN 

By F. REULEAUX 

Authorized translation, complete and unabridged, from the 
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By HENRY HARRISON SUPLEE, B.Sc. 

Quarto, 312 pages, 1200 illustrations 
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Compressed Air for Machine Shop and 

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Factory Depreciation 

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Limit Gauges in the Work Shop . 

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The Question of Apprentices . 

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Training Apprentices 

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The Trend of Machine Tool Design 

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Machine Shop Floors 

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19 





RITHMETIC 
& ALGEBRA 

By H. B. LUBSEN 

Adapted from the German 

By HENRY HARRISON SUPLEE, B.Sc. 

J2mo* Cloth* $2*00 by mail, postpaid 



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Sixth Edition. 8vo. 1042 Pages. 900 Figures 
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A TEXT=BOOK 

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By WILFRID J. LINEHAM 

M.INST.C.E., M.I.MECH.E., M.I.E.E. 

Part I WORKSHOP PRACTICE 

Part II.— THEORY AND EXAMPLES 



Sixth Edition, Revised and Enlarged 



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